From 08c2574864a184e66e9a1d9157dfe03dda2353a8 Mon Sep 17 00:00:00 2001
From: Zachary Ferguson <zy.fergus@gmail.com>
Date: Sun, 20 Jul 2025 22:09:20 -0400
Subject: [PATCH] Replace smooth mu with a piecewise quadratic function

---
 docs/source/_static/img/int_mu_f1_dx.png    |  Bin 35498 -> 35063 bytes
 docs/source/_static/img/int_mu_f1a_dx.png   |  Bin 30365 -> 30084 bytes
 docs/source/_static/img/mu.png              |  Bin 29614 -> 29745 bytes
 docs/source/_static/img/mu_f1.png           |  Bin 29445 -> 29507 bytes
 docs/source/_static/img/mu_f1a.png          |  Bin 30506 -> 30526 bytes
 docs/source/tutorials/adhesion.rst          |    2 +-
 docs/source/tutorials/advanced_friction.rst |   23 +-
 notebooks/separate_mu.ipynb                 | 1570 +++++++++++++++----
 src/ipc/adhesion/adhesion.cpp               |   32 +-
 src/ipc/friction/smooth_mu.cpp              |   46 +-
 10 files changed, 1314 insertions(+), 359 deletions(-)

diff --git a/docs/source/_static/img/int_mu_f1_dx.png b/docs/source/_static/img/int_mu_f1_dx.png
index 578d55e73cf5809f3c869ca802e8e0d182b856f4..f4c86ab0c6adae08c18cdd712fb2ee44244bc7a1 100644
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diff --git a/docs/source/tutorials/adhesion.rst b/docs/source/tutorials/adhesion.rst
index 5772cf4da..29e4901bf 100644
--- a/docs/source/tutorials/adhesion.rst
+++ b/docs/source/tutorials/adhesion.rst
@@ -201,4 +201,4 @@ We integrate their product to obtain a smooth tangential adhesion mollifier:
 .. figure:: /_static/img/int_mu_f1a_dx.png
    :align: center
 
-   Integrated mollifier :math:`\int \mu(y) f_1^a(y) \mathrm{d} y` where :math:`\mu_s = 1`, :math:`\mu_k = 0.1`, and :math:`\epsilon_v = 0.001`.
\ No newline at end of file
+   Integrated mollifier :math:`\int \mu(y) f_1^a(y) \mathrm{d} y` where :math:`y=\|\mathbf{u}\|`, :math:`\mu_s = 1`, :math:`\mu_k = 0.1`, and :math:`\epsilon_v = 0.001`.
\ No newline at end of file
diff --git a/docs/source/tutorials/advanced_friction.rst b/docs/source/tutorials/advanced_friction.rst
index 554be8dfc..01c1a9968 100644
--- a/docs/source/tutorials/advanced_friction.rst
+++ b/docs/source/tutorials/advanced_friction.rst
@@ -108,8 +108,10 @@ Smooth :math:`\mu`
 When adding separate coefficients for static and kinetic friction, we need to maintain the :math:`C^1` continuity of the friction force. This lead us to define a smooth coefficient of friction :math:`\mu(y)` that transitions between the static and kinetic coefficients based on the magnitude of the relative velocity :math:`y = \|\mathbf{u}\|`. The smooth coefficient of friction is defined as
 
 .. math::
-    \mu(y) = \begin{cases} \mu_{s} + \left(\mu_{k} - \mu_{s}\right) \left(\frac{3 y^{2}}{\epsilon_{v}^{2}} - \frac{2 y^{3}}{\epsilon_{v}^{3}}\right) & \text{for}\: 0 \leq y \leq \epsilon_{v}\\
-        \mu_{k} & \text{otherwise}.
+    \mu(y) = \begin{cases}
+        2(\mu_{k} - \mu_{s})\frac{y^{2}}{\epsilon_{v}^{2}} + \mu_{s} & \text{for}\: y \leq \frac{\epsilon_{v}}{2} \\
+        -2(\mu_{k} - \mu_{s})\frac{\left(\epsilon_{v} - y\right)^{2}}{\epsilon_{v}^{2}} + \mu_k & \text{for}\: y \leq \epsilon_{v} \\
+        \mu_{k} & \text{for}\: y > \epsilon_{v}
     \end{cases}
 
 We plot the smooth coefficient of friction :math:`\mu(y)` below:
@@ -132,13 +134,9 @@ Replacing the constant coefficient of friction :math:`\mu` with a smooth functio
 However, :math:`\frac{\mathrm{d}}{\mathrm{d}y} \mu(y) f_0(y) \neq \mu(y) f_1(y)`, so we need to adjust the integrated mollifier by integrating the product of the smooth coefficient of friction and the mollifier:
 
 .. math::
-    \int \mu(y) f_1(y) \mathrm{d} y = \begin{cases}
-        \frac{\epsilon_{v}^{6} \left(17 \mu_{k} - 7 \mu_{s}\right) + 30 \epsilon_{v}^{4} \mu_{s} y^{2} - 10 \epsilon_{v}^{3} \mu_{s} y^{3} + 45 \epsilon_{v}^{2} \left(\mu_{k} - \mu_{s}\right) y^{4} + 42 \epsilon_{v} \left(- \mu_{k} + \mu_{s}\right) y^{5} + 10 \left(\mu_{k} - \mu_{s}\right) y^{6}}{30 \epsilon_{v}^{5}} & \text{for}\: \epsilon_{v} > y \\
-        \mu_{k} y & \text{otherwise}
-    \end{cases}.
-
-
-
+    \int \mu(y) f_1(y) \mathrm{d} y = \begin{cases} \frac{\frac{\epsilon_{v}^{5} \left(27 \mu_{k} - 11 \mu_{s}\right)}{48} + \epsilon_{v}^{3} \mu_{s} y^{2} - \frac{\epsilon_{v}^{2} \mu_{s} y^{3}}{3} + \epsilon_{v} y^{4} \left(\mu_{k} - \mu_{s}\right) + \frac{2 y^{5} \left(- \mu_{k} + \mu_{s}\right)}{5}}{\epsilon_{v}^{4}} & \text{for}\: y \leq \frac{\epsilon_{v}}{2} \\
+    \frac{\epsilon_{v}^{5} \left(9 \mu_{k} - 4 \mu_{s}\right) + 15 \epsilon_{v}^{3} y^{2} \left(- \mu_{k} + 2 \mu_{s}\right) + 5 \epsilon_{v}^{2} y^{3} \left(9 \mu_{k} - 10 \mu_{s}\right) - 30 \epsilon_{v} y^{4} \left(\mu_{k} - \mu_{s}\right) + 6 y^{5} \left(\mu_{k} - \mu_{s}\right)}{15 \epsilon_{v}^{4}} & \text{for}\: y \leq \epsilon_{v} \\
+    \mu_{k} y & \text{for}\: y > \epsilon_v \end{cases}
 
 The following plot shows the behavior of the integrated mollifier multiplied by the smooth coefficient of friction:
 
@@ -155,11 +153,10 @@ While this approach provides a smooth transition between static and kinetic fric
 1. The product :math:`\mu(y) f_1(y)` underestimates the friction force in the static regime, which may lead to less accurate simulations of static friction.
     - This is, when :math:`\mu_k < \mu_s` and :math:`|y| \leq \epsilon_v`, :math:`\max(\mu(y) f_1(y)) < \mu_s`.
     - We could address this by scaling by :math:`\frac{\max(\mu(y) f_1(y))}{\mu_s}`, but computing the maximum is non-trivial.
-2. The combined function :math:`\mu(y) f_1(y)` is a degree 5 polynomial, which is more complex than the original mollifier :math:`f_1(y)` (degree 2). This may lead to more difficult to optimize problems. There are two options to address this:
-    a. Replacing :math:`\mu(y)` with a piecewise quadratic function would reduce the degree to 4, but it would still be more complex than the original mollifier.
-    b. Replacing :math:`\mu(y) f_1(y)` with a piecewise quadratic function that has the desired behavior. This help with 1. as well. However making it backwards compatible with the original mollifier is challenging.
+2. The combined function :math:`\mu(y) f_1(y)` is a degree 4 polynomial, which is more complex than the original mollifier :math:`f_1(y)` (degree 2). This may lead to more difficult to optimize problems.
+    - We could replace :math:`\mu(y) f_1(y)` with a piecewise quadratic function that has the desired behavior. This help with (1) as well. However making it backwards compatible with the original mollifier is challenging.
 
-If you have suggestions for improving this approach or alternative methods, please reach out on our `GitHub Discussions <https://github.com/ipc-sim/ipc-toolkit/discussions>`.
+If you have suggestions for improving this approach or alternative methods, please reach out on our `GitHub Discussions <https://github.com/ipc-sim/ipc-toolkit/discussions>`_.
 
 Future Directions
 -----------------
diff --git a/notebooks/separate_mu.ipynb b/notebooks/separate_mu.ipynb
index 20b79b68c..d4be9c39d 100644
--- a/notebooks/separate_mu.ipynb
+++ b/notebooks/separate_mu.ipynb
@@ -9,7 +9,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 562,
+   "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
@@ -20,7 +20,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 563,
+   "execution_count": 2,
    "metadata": {},
    "outputs": [
     {
@@ -3941,31 +3941,32 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 564,
+   "execution_count": 3,
    "metadata": {},
    "outputs": [],
    "source": [
-    "x = Symbol('x')\n",
-    "eps_v = Symbol(r'\\epsilon_v')\n",
-    "mu_s, mu_k = symbols(r'\\mu_s \\mu_k')"
+    "x = Symbol('x', real=True)\n",
+    "eps_v = Symbol(r'\\epsilon_v', real=True, positive=True)\n",
+    "mu_s, mu_k = symbols(r'\\mu_s \\mu_k', real=True, positive=True)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 565,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [],
    "source": [
     "import re\n",
     "\n",
     "\n",
-    "def plot(*fs, title=None, min_x=-2, max_x=2, **subs):\n",
+    "def plot(*fs, title=None, min_x=-2, max_x=2, even=False, odd=False, **subs):\n",
     "    subs = {eps_v: subs.get(\n",
     "        \"eps_v\", 1e-3), mu_s: subs.get(\"mu_s\", 1), mu_k: subs.get(\"mu_k\", 0.1)}\n",
     "    xs = np.linspace(min_x*subs[eps_v], max_x*subs[eps_v], 201)\n",
+    "    s = np.sign(xs) if odd else np.ones_like(xs)\n",
     "\n",
     "    def ys(f):\n",
-    "        return np.array([f.subs({x: xi} | subs) for xi in xs], dtype=float)\n",
+    "        return s * np.array([f.subs({x: abs(xi) if (even or odd) else xi} | subs) for xi in xs], dtype=float)\n",
     "\n",
     "    fig = go.Figure(\n",
     "        [go.Scatter(x=xs, y=ys(f)) for f in fs],\n",
@@ -3975,9 +3976,17 @@
     "        ))\n",
     "    file_name = re.sub(\n",
     "        r'[\\$\\_\\{\\}\\^\\\\]|\\(x\\)', '', title).replace(\"mathrm\", \"\").replace(' ', '_').strip()\n",
-    "    fig.write_image(\n",
-    "        f\"../docs/source/_static/img/{file_name}.png\")\n",
-    "    fig.show()"
+    "    # fig.write_image(f\"../docs/source/_static/img/{file_name}.png\")\n",
+    "    fig.show()\n",
+    "\n",
+    "\n",
+    "def print_latex(expr):\n",
+    "    print(latex(expr).replace(r\"\\\\\", r\"\\\\\" + \"\\n\"))\n",
+    "\n",
+    "\n",
+    "def print_code(expr):\n",
+    "    print(cxxcode(expr).replace(r\"\\epsilon\", \"eps\").replace(\n",
+    "        r\"\\mu\", \"mu\").replace(\"x\", \"y\"))"
    ]
   },
   {
@@ -3989,16 +3998,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 566,
+   "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle f_{0}(x) = \\begin{cases} \\left|{x}\\right| & \\text{for}\\: \\epsilon_{v} \\leq \\left|{x}\\right| \\\\\\frac{\\epsilon_{v}}{3} + \\frac{\\left|{x}\\right|^{2}}{\\epsilon_{v}} - \\frac{\\left|{x}\\right|^{3}}{3 \\epsilon_{v}^{2}} & \\text{otherwise} \\end{cases}$"
+       "$\\displaystyle f_{0}(x) = \\begin{cases} x & \\text{for}\\: \\epsilon_{v} \\leq x \\\\\\frac{\\epsilon_{v}}{3} + \\frac{x^{2}}{\\epsilon_{v}} - \\frac{x^{3}}{3 \\epsilon_{v}^{2}} & \\text{otherwise} \\end{cases}$"
       ],
       "text/plain": [
-       "Eq(f_{0}(x), Piecewise((Abs(x), \\epsilon_v <= Abs(x)), (\\epsilon_v/3 + Abs(x)**2/\\epsilon_v - Abs(x)**3/(3*\\epsilon_v**2), True)))"
+       "Eq(f_{0}(x), Piecewise((x, \\epsilon_v <= x), (\\epsilon_v/3 + x**2/\\epsilon_v - x**3/(3*\\epsilon_v**2), True)))"
       ]
      },
      "metadata": {},
@@ -4008,7 +4017,8 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "\\begin{cases} \\left|{x}\\right| & \\text{for}\\: \\epsilon_{v} \\leq \\left|{x}\\right| \\\\\\frac{\\epsilon_{v}}{3} + \\frac{\\left|{x}\\right|^{2}}{\\epsilon_{v}} - \\frac{\\left|{x}\\right|^{3}}{3 \\epsilon_{v}^{2}} & \\text{otherwise} \\end{cases}\n"
+      "\\begin{cases} x & \\text{for}\\: \\epsilon_{v} \\leq x \\\\\n",
+      "\\frac{\\epsilon_{v}}{3} + \\frac{x^{2}}{\\epsilon_{v}} - \\frac{x^{3}}{3 \\epsilon_{v}^{2}} & \\text{otherwise} \\end{cases}\n"
      ]
     },
     {
@@ -4829,9 +4839,892 @@
        }
       },
       "text/html": [
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+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "def f0(x):\n",
+    "    \"\"\"Smooth friction mollifier\"\"\"\n",
+    "    return x * x * (1 - x / (3 * eps_v)) / eps_v + eps_v / 3\n",
+    "\n",
+    "\n",
+    "sym_f0 = Piecewise(\n",
+    "    (x, x >= eps_v),\n",
+    "    (f0(x), x < eps_v)\n",
+    ")\n",
+    "\n",
+    "display(Eq(Symbol(\"f_{0}(x)\"), sym_f0.expand()))\n",
+    "print_latex(sym_f0.expand())\n",
+    "\n",
+    "plot(sym_f0, title=r'$f_0(x)$', even=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle f_{1}(x) = \\begin{cases} \\frac{2 x}{\\epsilon_{v}} - \\frac{x^{2}}{\\epsilon_{v}^{2}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\1 & \\text{otherwise} \\end{cases}$"
+      ],
+      "text/plain": [
+       "Eq(f_{1}(x), Piecewise((2*x/\\epsilon_v - x**2/\\epsilon_v**2, \\epsilon_v >= x), (1, True)))"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
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+            "zerolinecolor": "#C8D4E3"
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+            "ticks": ""
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+           "baxis": {
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+           },
+           "bgcolor": "rgb(17,17,17)",
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+          },
+          "title": {
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+          },
+          "updatemenudefaults": {
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+           },
+           "zerolinecolor": "#283442",
+           "zerolinewidth": 2
+          }
+         }
+        },
+        "title": {
+         "text": "$f_1(x)$"
+        },
+        "width": 800,
+        "xaxis": {
+         "title": {
+          "text": "speed"
+         }
+        }
+       }
+      },
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                       {\"responsive\": true}                    ).then(function(){\n",
        "                            \n",
-       "var gd = document.getElementById('3bd6f704-47a7-42af-b06f-4261d9712cb2');\n",
+       "var gd = document.getElementById('cbe48c32-389e-4ae9-960d-a90129d25bce');\n",
        "var x = new MutationObserver(function (mutations, observer) {{\n",
        "        var display = window.getComputedStyle(gd).display;\n",
        "        if (!display || display === 'none') {{\n",
@@ -4861,39 +5754,60 @@
     }
    ],
    "source": [
-    "def f0(x):\n",
-    "    \"\"\"Smooth friction mollifier\"\"\"\n",
-    "    return x * x * (1 - x / (3 * eps_v)) / eps_v + eps_v / 3\n",
+    "def f1(x):\n",
+    "    \"\"\"Derivative of f0\"\"\"\n",
+    "    return x * (2 - x / eps_v) / eps_v\n",
     "\n",
     "\n",
-    "sym_f0 = Piecewise(\n",
-    "    (abs(x), abs(x) >= eps_v),\n",
-    "    (f0(abs(x)), abs(x) < eps_v)\n",
+    "# def sign(x):\n",
+    "#     return Piecewise((1, x > 0), (-1, x < 0), (0, True))\n",
+    "\n",
+    "\n",
+    "sym_f1 = Piecewise(\n",
+    "    (f1(x), x <= eps_v),\n",
+    "    (1, x > eps_v)\n",
     ")\n",
     "\n",
-    "display(Eq(Symbol(\"f_{0}(x)\"), sym_f0.expand()))\n",
-    "print(latex(sym_f0.expand()))\n",
+    "display(Eq(Symbol(\"f_{1}(x)\"), sym_f1.expand()))\n",
+    "\n",
+    "plot(sym_f1, title=r\"$f_1(x)$\", odd=True)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Smooth Coefficient of Friction\n",
     "\n",
-    "plot(sym_f0, title=r'$f_0(x)$')"
+    "We can use a polynomial to model a smooth transition from static to kinematic coefficient of friction."
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 567,
+   "execution_count": 7,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle f_{1}(x) = \\begin{cases} -1 & \\text{for}\\: \\epsilon_{v} < - x \\\\\\frac{2 x}{\\epsilon_{v}} + \\frac{x^{2}}{\\epsilon_{v}^{2}} & \\text{for}\\: x \\leq 0 \\\\\\frac{2 x}{\\epsilon_{v}} - \\frac{x^{2}}{\\epsilon_{v}^{2}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\1 & \\text{otherwise} \\end{cases}$"
+       "$\\displaystyle \\mu(x) = \\begin{cases} \\frac{\\epsilon_{v}^{2} \\mu_{s} + 2 x^{2} \\left(\\mu_{k} - \\mu_{s}\\right)}{\\epsilon_{v}^{2}} & \\text{for}\\: \\epsilon_{v} \\geq 2 x \\\\\\frac{\\epsilon_{v}^{2} \\mu_{k} + 2 \\left(\\epsilon_{v} - x\\right)^{2} \\left(- \\mu_{k} + \\mu_{s}\\right)}{\\epsilon_{v}^{2}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\\mu_{k} & \\text{otherwise} \\end{cases}$"
       ],
       "text/plain": [
-       "Eq(f_{1}(x), Piecewise((-1, \\epsilon_v < -x), (2*x/\\epsilon_v + x**2/\\epsilon_v**2, x <= 0), (2*x/\\epsilon_v - x**2/\\epsilon_v**2, \\epsilon_v >= x), (1, True)))"
+       "Eq(\\mu(x), Piecewise(((\\epsilon_v**2*\\mu_s + 2*x**2*(\\mu_k - \\mu_s))/\\epsilon_v**2, \\epsilon_v >= 2*x), ((\\epsilon_v**2*\\mu_k + 2*(\\epsilon_v - x)**2*(-\\mu_k + \\mu_s))/\\epsilon_v**2, \\epsilon_v >= x), (\\mu_k, True)))"
       ]
      },
      "metadata": {},
      "output_type": "display_data"
     },
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\\begin{cases} \\frac{\\epsilon_{v}^{2} \\mu_{s} + 2 x^{2} \\left(\\mu_{k} - \\mu_{s}\\right)}{\\epsilon_{v}^{2}} & \\text{for}\\: \\epsilon_{v} \\geq 2 x \\\\\n",
+      "\\frac{\\epsilon_{v}^{2} \\mu_{k} + 2 \\left(\\epsilon_{v} - x\\right)^{2} \\left(- \\mu_{k} + \\mu_{s}\\right)}{\\epsilon_{v}^{2}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\n",
+      "\\mu_{k} & \\text{otherwise} \\end{cases}\n"
+     ]
+    },
     {
      "data": {
       "application/vnd.plotly.v1+json": {
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         }
@@ -5701,7 +6615,7 @@
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         },
         "title": {
-         "text": "$f_1(x)$"
+         "text": "$\\mu(x)$"
         },
         "width": 800,
         "xaxis": {
@@ -5712,9 +6626,9 @@
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       },
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                       {\"responsive\": true}                    ).then(function(){\n",
        "                            \n",
-       "var gd = document.getElementById('2aee6964-3fc6-4fd2-97bb-7ada061c1b63');\n",
+       "var gd = document.getElementById('f20ebe0b-f7c1-49cd-837e-e8a86da9dace');\n",
        "var x = new MutationObserver(function (mutations, observer) {{\n",
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@@ -5741,62 +6655,6 @@
      },
      "metadata": {},
      "output_type": "display_data"
-    }
-   ],
-   "source": [
-    "def f1(x):\n",
-    "    \"\"\"Derivative of f0\"\"\"\n",
-    "    return x * (2 - x / eps_v) / eps_v\n",
-    "\n",
-    "\n",
-    "# def sign(x):\n",
-    "#     return Piecewise((1, x > 0), (-1, x < 0), (0, True))\n",
-    "\n",
-    "\n",
-    "sym_f1 = Piecewise(\n",
-    "    (-1, x < -eps_v),\n",
-    "    (-f1(-x), x <= 0),\n",
-    "    (f1(x), x <= eps_v),\n",
-    "    (1, x > eps_v)\n",
-    ")\n",
-    "\n",
-    "display(Eq(Symbol(\"f_{1}(x)\"), sym_f1.expand()))\n",
-    "\n",
-    "plot(sym_f1, title=r\"$f_1(x)$\")"
-   ]
-  },
-  {
-   "cell_type": "markdown",
-   "metadata": {},
-   "source": [
-    "## Smooth Coefficient of Friction\n",
-    "\n",
-    "We can use a polynomial to model a smooth transition from static to kinematic coefficient of friction."
-   ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 568,
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\mu(x) = \\begin{cases} \\mu_{k} & \\text{for}\\: \\epsilon_{v} \\leq \\left|{x}\\right| \\\\\\mu_{s} + \\left(\\mu_{k} - \\mu_{s}\\right) \\left(\\frac{3 \\left|{x}\\right|^{2}}{\\epsilon_{v}^{2}} - \\frac{2 \\left|{x}\\right|^{3}}{\\epsilon_{v}^{3}}\\right) & \\text{otherwise} \\end{cases}$"
-      ],
-      "text/plain": [
-       "Eq(\\mu(x), Piecewise((\\mu_k, \\epsilon_v <= Abs(x)), (\\mu_s + (\\mu_k - \\mu_s)*(3*Abs(x)**2/\\epsilon_v**2 - 2*Abs(x)**3/\\epsilon_v**3), True)))"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "\\begin{cases} \\mu_{k} & \\text{for}\\: \\epsilon_{v} \\leq \\left|{x}\\right| \\\\\\mu_{s} + \\left(\\mu_{k} - \\mu_{s}\\right) \\left(\\frac{3 \\left|{x}\\right|^{2}}{\\epsilon_{v}^{2}} - \\frac{2 \\left|{x}\\right|^{3}}{\\epsilon_{v}^{3}}\\right) & \\text{otherwise} \\end{cases}\n"
-     ]
     },
     {
      "data": {
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@@ -6605,7 +7463,7 @@
          }
         },
         "title": {
-         "text": "$\\mu(x)$"
+         "text": "$\\mu'(x)$"
         },
         "width": 800,
         "xaxis": {
@@ -6616,9 +7474,9 @@
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                       {\"responsive\": true}                    ).then(function(){\n",
        "                            \n",
-       "var gd = document.getElementById('f0f8c3d2-f0f7-42cc-801b-b4bf9429274d');\n",
+       "var gd = document.getElementById('d7c9744f-5c4d-423c-af58-b7eaed1d51a1');\n",
        "var x = new MutationObserver(function (mutations, observer) {{\n",
        "        var display = window.getComputedStyle(gd).display;\n",
        "        if (!display || display === 'none') {{\n",
@@ -6648,21 +7506,34 @@
     }
    ],
    "source": [
-    "def mu(x):\n",
-    "    # return (mu_k - mu_s) * (0.5 * (-cos(pi * x / eps_v) + 1)) + mu_s\n",
+    "def mu_cos(x):\n",
+    "    return (mu_k - mu_s) * (0.5 * (-cos(pi * x / eps_v) + 1)) + mu_s\n",
+    "\n",
+    "\n",
+    "def mu_cubic(x):\n",
     "    return (mu_k - mu_s) * (3 * (x / eps_v)**2 - 2 * (x / eps_v)**3) + mu_s\n",
     "\n",
     "\n",
+    "def mu_quadratic_part1(x):\n",
+    "    return (2 / eps_v**2) * (mu_k - mu_s) * x**2 + mu_s\n",
+    "\n",
+    "\n",
+    "def mu_quadratic_part2(x):\n",
+    "    return -2 * (mu_k - mu_s) / eps_v**2 * (x - eps_v)**2 + mu_k\n",
+    "\n",
+    "\n",
     "sym_mu = Piecewise(\n",
-    "    (mu_k, abs(x) >= eps_v),\n",
-    "    (mu(abs(x)), abs(x) < eps_v)\n",
-    ")\n",
+    "    (mu_quadratic_part1(x), x <= eps_v / 2),\n",
+    "    (mu_quadratic_part2(x), x <= eps_v),\n",
+    "    (mu_k, True)\n",
+    ").simplify()\n",
     "\n",
     "display(Eq(Symbol(r\"\\mu(x)\"), sym_mu))\n",
     "\n",
-    "print(latex(sym_mu))\n",
+    "print_latex(sym_mu)\n",
     "\n",
-    "plot(sym_mu, title=r\"$\\mu(x)$\")"
+    "plot(sym_mu, title=r\"$\\mu(x)$\", even=True)\n",
+    "plot(sym_mu.diff(x), title=r\"$\\mu'(x)$\", odd=True)"
    ]
   },
   {
@@ -6675,16 +7546,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 569,
+   "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\mu(x) f_{1(x)} = \\begin{cases} - \\mu_{k} & \\text{for}\\: \\epsilon_{v} < - x \\\\\\frac{x \\left(2 + \\frac{x}{\\epsilon_{v}}\\right) \\left(\\mu_{s} + \\left(\\mu_{k} - \\mu_{s}\\right) \\left(\\frac{3 x^{2}}{\\epsilon_{v}^{2}} + \\frac{2 x^{3}}{\\epsilon_{v}^{3}}\\right)\\right)}{\\epsilon_{v}} & \\text{for}\\: x \\leq 0 \\\\\\frac{x \\left(2 - \\frac{x}{\\epsilon_{v}}\\right) \\left(\\mu_{s} + \\left(\\mu_{k} - \\mu_{s}\\right) \\left(\\frac{3 x^{2}}{\\epsilon_{v}^{2}} - \\frac{2 x^{3}}{\\epsilon_{v}^{3}}\\right)\\right)}{\\epsilon_{v}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\\mu_{k} & \\text{otherwise} \\end{cases}$"
+       "$\\displaystyle \\mu(x) f_{1}(x) = \\begin{cases} \\frac{x \\left(2 \\epsilon_{v} - x\\right) \\left(\\epsilon_{v}^{2} \\mu_{s} + 2 x^{2} \\left(\\mu_{k} - \\mu_{s}\\right)\\right)}{\\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq 2 x \\\\\\frac{x \\left(2 \\epsilon_{v} - x\\right) \\left(\\epsilon_{v}^{2} \\mu_{k} + 2 \\left(\\epsilon_{v} - x\\right)^{2} \\left(- \\mu_{k} + \\mu_{s}\\right)\\right)}{\\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\\mu_{k} & \\text{otherwise} \\end{cases}$"
       ],
       "text/plain": [
-       "Eq(\\mu(x) f_1(x), Piecewise((-\\mu_k, \\epsilon_v < -x), (x*(2 + x/\\epsilon_v)*(\\mu_s + (\\mu_k - \\mu_s)*(3*x**2/\\epsilon_v**2 + 2*x**3/\\epsilon_v**3))/\\epsilon_v, x <= 0), (x*(2 - x/\\epsilon_v)*(\\mu_s + (\\mu_k - \\mu_s)*(3*x**2/\\epsilon_v**2 - 2*x**3/\\epsilon_v**3))/\\epsilon_v, \\epsilon_v >= x), (\\mu_k, True)))"
+       "Eq(\\mu(x) f_{1}(x), Piecewise((x*(2*\\epsilon_v - x)*(\\epsilon_v**2*\\mu_s + 2*x**2*(\\mu_k - \\mu_s))/\\epsilon_v**4, \\epsilon_v >= 2*x), (x*(2*\\epsilon_v - x)*(\\epsilon_v**2*\\mu_k + 2*(\\epsilon_v - x)**2*(-\\mu_k + \\mu_s))/\\epsilon_v**4, \\epsilon_v >= x), (\\mu_k, True)))"
       ]
      },
      "metadata": {},
@@ -6694,7 +7565,9 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "\\begin{cases} - \\mu_{k} & \\text{for}\\: \\epsilon_{v} < - x \\\\\\frac{x \\left(2 + \\frac{x}{\\epsilon_{v}}\\right) \\left(\\mu_{s} + \\left(\\mu_{k} - \\mu_{s}\\right) \\left(\\frac{3 x^{2}}{\\epsilon_{v}^{2}} + \\frac{2 x^{3}}{\\epsilon_{v}^{3}}\\right)\\right)}{\\epsilon_{v}} & \\text{for}\\: x \\leq 0 \\\\\\frac{x \\left(2 - \\frac{x}{\\epsilon_{v}}\\right) \\left(\\mu_{s} + \\left(\\mu_{k} - \\mu_{s}\\right) \\left(\\frac{3 x^{2}}{\\epsilon_{v}^{2}} - \\frac{2 x^{3}}{\\epsilon_{v}^{3}}\\right)\\right)}{\\epsilon_{v}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\\mu_{k} & \\text{otherwise} \\end{cases}\n"
+      "\\begin{cases} \\frac{x \\left(2 \\epsilon_{v} - x\\right) \\left(\\epsilon_{v}^{2} \\mu_{s} + 2 x^{2} \\left(\\mu_{k} - \\mu_{s}\\right)\\right)}{\\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq 2 x \\\\\n",
+      "\\frac{x \\left(2 \\epsilon_{v} - x\\right) \\left(\\epsilon_{v}^{2} \\mu_{k} + 2 \\left(\\epsilon_{v} - x\\right)^{2} \\left(- \\mu_{k} + \\mu_{s}\\right)\\right)}{\\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\n",
+      "\\mu_{k} & \\text{otherwise} \\end{cases}\n"
      ]
     },
     {
@@ -6711,7 +7584,7 @@
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@@ -7515,9 +8388,9 @@
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        "                            \n",
-       "var gd = document.getElementById('8c33989f-a443-42c1-845e-32941e1f1471');\n",
+       "var gd = document.getElementById('feb0de4c-b70f-4588-8883-5d9f06d1cdf1');\n",
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@@ -7547,22 +8420,25 @@
     }
    ],
    "source": [
-    "def mu_f1(x):\n",
-    "    return mu(x) * f1(x)\n",
+    "def mu_f1_part1(x):\n",
+    "    return mu_quadratic_part1(x) * f1(x)\n",
+    "\n",
+    "\n",
+    "def mu_f1_part2(x):\n",
+    "    return mu_quadratic_part2(x) * f1(x)\n",
     "\n",
     "\n",
     "sym_mu_f1 = Piecewise(\n",
-    "    (-mu_k, x < -eps_v),\n",
-    "    (-mu_f1(-x), x <= 0),\n",
-    "    (mu_f1(x), x <= eps_v),\n",
-    "    (mu_k, x > eps_v)\n",
-    ")\n",
+    "    (mu_f1_part1(x), x <= eps_v/2),\n",
+    "    (mu_f1_part2(x), x <= eps_v),\n",
+    "    (mu_k, True)\n",
+    ").simplify()\n",
     "\n",
-    "display(Eq(Symbol(r\"\\mu(x) f_1(x)\"), sym_mu_f1))\n",
+    "display(Eq(Symbol(r\"\\mu(x) f_{1}(x)\"), sym_mu_f1))\n",
     "\n",
-    "print(latex(sym_mu_f1))\n",
+    "print_latex(sym_mu_f1)\n",
     "\n",
-    "plot(sym_mu_f1, title=r'$\\mu(x) f_1(x)$')"
+    "plot(sym_mu_f1, title=r'$\\mu(x) f_1(x)$', odd=True)"
    ]
   },
   {
@@ -7574,28 +8450,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 570,
+   "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\frac{\\epsilon_{v} \\left(17 \\mu_{k} - 7 \\mu_{s}\\right)}{30}$"
-      ],
-      "text/plain": [
-       "\\epsilon_v*(17*\\mu_k - 7*\\mu_s)/30"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\int \\mu(x) f_1(x) \\mathrm{d}x = \\begin{cases} \\frac{17 \\epsilon_{v} \\mu_{k}}{30} - \\frac{7 \\epsilon_{v} \\mu_{s}}{30} + \\frac{\\mu_{s} \\left|{x}\\right|^{2}}{\\epsilon_{v}} - \\frac{\\mu_{s} \\left|{x}\\right|^{3}}{3 \\epsilon_{v}^{2}} + \\frac{3 \\mu_{k} \\left|{x}\\right|^{4}}{2 \\epsilon_{v}^{3}} - \\frac{3 \\mu_{s} \\left|{x}\\right|^{4}}{2 \\epsilon_{v}^{3}} - \\frac{7 \\mu_{k} \\left|{x}\\right|^{5}}{5 \\epsilon_{v}^{4}} + \\frac{7 \\mu_{s} \\left|{x}\\right|^{5}}{5 \\epsilon_{v}^{4}} + \\frac{\\mu_{k} \\left|{x}\\right|^{6}}{3 \\epsilon_{v}^{5}} - \\frac{\\mu_{s} \\left|{x}\\right|^{6}}{3 \\epsilon_{v}^{5}} & \\text{for}\\: \\epsilon_{v} > \\left|{x}\\right| \\\\\\mu_{k} \\left|{x}\\right| & \\text{otherwise} \\end{cases}$"
+       "$\\displaystyle \\int \\mu(x) f_1(x) \\mathrm{d}x = \\begin{cases} \\frac{\\frac{\\epsilon_{v}^{5} \\left(27 \\mu_{k} - 11 \\mu_{s}\\right)}{48} + \\epsilon_{v}^{3} \\mu_{s} x^{2} - \\frac{\\epsilon_{v}^{2} \\mu_{s} x^{3}}{3} + \\epsilon_{v} x^{4} \\left(\\mu_{k} - \\mu_{s}\\right) + \\frac{2 x^{5} \\left(- \\mu_{k} + \\mu_{s}\\right)}{5}}{\\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq 2 x \\\\\\frac{\\epsilon_{v}^{5} \\left(9 \\mu_{k} - 4 \\mu_{s}\\right) + 15 \\epsilon_{v}^{3} x^{2} \\left(- \\mu_{k} + 2 \\mu_{s}\\right) + 5 \\epsilon_{v}^{2} x^{3} \\left(9 \\mu_{k} - 10 \\mu_{s}\\right) - 30 \\epsilon_{v} x^{4} \\left(\\mu_{k} - \\mu_{s}\\right) + 6 x^{5} \\left(\\mu_{k} - \\mu_{s}\\right)}{15 \\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\\mu_{k} x & \\text{otherwise} \\end{cases}$"
       ],
       "text/plain": [
-       "Eq(\\int \\mu(x) f_1(x) \\mathrm{d}x, Piecewise((17*\\epsilon_v*\\mu_k/30 - 7*\\epsilon_v*\\mu_s/30 + \\mu_s*Abs(x)**2/\\epsilon_v - \\mu_s*Abs(x)**3/(3*\\epsilon_v**2) + 3*\\mu_k*Abs(x)**4/(2*\\epsilon_v**3) - 3*\\mu_s*Abs(x)**4/(2*\\epsilon_v**3) - 7*\\mu_k*Abs(x)**5/(5*\\epsilon_v**4) + 7*\\mu_s*Abs(x)**5/(5*\\epsilon_v**4) + \\mu_k*Abs(x)**6/(3*\\epsilon_v**5) - \\mu_s*Abs(x)**6/(3*\\epsilon_v**5), \\epsilon_v > Abs(x)), (\\mu_k*Abs(x), True)))"
+       "Eq(\\int \\mu(x) f_1(x) \\mathrm{d}x, Piecewise(((\\epsilon_v**5*(27*\\mu_k - 11*\\mu_s)/48 + \\epsilon_v**3*\\mu_s*x**2 - \\epsilon_v**2*\\mu_s*x**3/3 + \\epsilon_v*x**4*(\\mu_k - \\mu_s) + 2*x**5*(-\\mu_k + \\mu_s)/5)/\\epsilon_v**4, \\epsilon_v >= 2*x), ((\\epsilon_v**5*(9*\\mu_k - 4*\\mu_s) + 15*\\epsilon_v**3*x**2*(-\\mu_k + 2*\\mu_s) + 5*\\epsilon_v**2*x**3*(9*\\mu_k - 10*\\mu_s) - 30*\\epsilon_v*x**4*(\\mu_k - \\mu_s) + 6*x**5*(\\mu_k - \\mu_s))/(15*\\epsilon_v**4), \\epsilon_v >= x), (\\mu_k*x, True)))"
       ]
      },
      "metadata": {},
@@ -7605,7 +8469,9 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "\\begin{cases} \\frac{\\epsilon_{v}^{6} \\left(17 \\mu_{k} - 7 \\mu_{s}\\right) + 30 \\epsilon_{v}^{4} \\mu_{s} \\left|{x}\\right|^{2} - 10 \\epsilon_{v}^{3} \\mu_{s} \\left|{x}\\right|^{3} + 45 \\epsilon_{v}^{2} \\left(\\mu_{k} - \\mu_{s}\\right) \\left|{x}\\right|^{4} + 42 \\epsilon_{v} \\left(- \\mu_{k} + \\mu_{s}\\right) \\left|{x}\\right|^{5} + 10 \\left(\\mu_{k} - \\mu_{s}\\right) \\left|{x}\\right|^{6}}{30 \\epsilon_{v}^{5}} & \\text{for}\\: \\epsilon_{v} > \\left|{x}\\right| \\\\\\mu_{k} \\left|{x}\\right| & \\text{otherwise} \\end{cases}\n"
+      "\\begin{cases} \\frac{\\frac{\\epsilon_{v}^{5} \\left(27 \\mu_{k} - 11 \\mu_{s}\\right)}{48} + \\epsilon_{v}^{3} \\mu_{s} x^{2} - \\frac{\\epsilon_{v}^{2} \\mu_{s} x^{3}}{3} + \\epsilon_{v} x^{4} \\left(\\mu_{k} - \\mu_{s}\\right) + \\frac{2 x^{5} \\left(- \\mu_{k} + \\mu_{s}\\right)}{5}}{\\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq 2 x \\\\\n",
+      "\\frac{\\epsilon_{v}^{5} \\left(9 \\mu_{k} - 4 \\mu_{s}\\right) + 15 \\epsilon_{v}^{3} x^{2} \\left(- \\mu_{k} + 2 \\mu_{s}\\right) + 5 \\epsilon_{v}^{2} x^{3} \\left(9 \\mu_{k} - 10 \\mu_{s}\\right) - 30 \\epsilon_{v} x^{4} \\left(\\mu_{k} - \\mu_{s}\\right) + 6 x^{5} \\left(\\mu_{k} - \\mu_{s}\\right)}{15 \\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\n",
+      "\\mu_{k} x & \\text{otherwise} \\end{cases}\n"
      ]
     },
     {
@@ -7622,7 +8488,7 @@
           "dtype": "f8"
          },
          "y": {
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           "dtype": "f8"
          }
         }
@@ -8426,9 +9292,9 @@
        }
       },
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        "                            \n",
-       "var gd = document.getElementById('ebca4712-6a0f-41b0-914d-34b272cc9ebc');\n",
+       "var gd = document.getElementById('9f0349c6-7013-464a-a16d-77a331a7a946');\n",
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@@ -8458,85 +9324,84 @@
     }
    ],
    "source": [
-    "mu_f0 = integrate(mu_f1(x), x)\n",
+    "mu_f0_part1 = integrate(mu_f1_part1(x), x)\n",
+    "mu_f0_part2 = integrate(mu_f1_part2(x), x)\n",
     "\n",
-    "# Constant of integration s.t. mu_f0(eps_v) = mu_k * eps_v\n",
-    "c = mu_k * eps_v - mu_f0.subs(x, eps_v)\n",
-    "display(c.simplify())\n",
-    "mu_f0 = mu_f0 + c\n",
+    "# Constant of integration s.t. mu_f0_part2(eps_v) = mu_k * eps_v\n",
+    "c = mu_k * eps_v - mu_f0_part2.subs(x, eps_v)\n",
+    "mu_f0_part2 = mu_f0_part2 + c\n",
     "\n",
-    "mu_f0.simplify()\n",
+    "# Constant of integration s.t. mu_f0_part1(eps_v/2) = mu_f0_part2(eps_v/2)\n",
+    "c = mu_f0_part2.subs(x, eps_v / 2) - mu_f0_part1.subs(x, eps_v / 2)\n",
+    "mu_f0_part1 = mu_f0_part1 + c\n",
     "\n",
     "sym_mu_f0 = Piecewise(\n",
-    "    (mu_f0.expand().subs(x, abs(x)), abs(x) < eps_v),\n",
-    "    (mu_k * abs(x), abs(x) >= eps_v)\n",
-    ")\n",
+    "    (mu_f0_part1.simplify().subs(x, x), x <= eps_v/2),\n",
+    "    (mu_f0_part2.simplify().subs(x, x), x <= eps_v),\n",
+    "    (mu_k * x, True)\n",
+    ").simplify()\n",
     "\n",
     "display(Eq(Symbol(r\"\\int \\mu(x) f_1(x) \\mathrm{d}x\"), sym_mu_f0))\n",
     "\n",
-    "print(latex(sym_mu_f0.simplify()))\n",
+    "print_latex(sym_mu_f0.simplify())\n",
     "\n",
-    "plot(sym_mu_f0, title=r'$\\int \\mu(x) f_1(x) \\mathrm{d}x$')"
+    "plot(sym_mu_f0, title=r'$\\int \\mu(x) f_1(x) \\mathrm{d}x$', even=True)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 571,
+   "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\operatorname{Poly}{\\left( \\frac{\\mu_{k} - \\mu_{s}}{3 \\epsilon_{v}^{5}} x^{6} + \\frac{- 7 \\mu_{k} + 7 \\mu_{s}}{5 \\epsilon_{v}^{4}} x^{5} + \\frac{3 \\mu_{k} - 3 \\mu_{s}}{2 \\epsilon_{v}^{3}} x^{4} -  \\frac{\\mu_{s}}{3 \\epsilon_{v}^{2}} x^{3} + \\frac{\\mu_{s}}{\\epsilon_{v}} x^{2} + \\frac{17 \\epsilon_{v} \\mu_{k}}{30} - \\frac{7 \\epsilon_{v} \\mu_{s}}{30}, x, domain=\\mathbb{Z}\\left(\\mu_{k}, \\mu_{s}, \\epsilon_{v}\\right) \\right)}$"
+       "$\\displaystyle \\operatorname{Poly}{\\left( \\frac{- 2 \\mu_{k} + 2 \\mu_{s}}{5 \\epsilon_{v}^{4}} x^{5} + \\frac{\\mu_{k} - \\mu_{s}}{\\epsilon_{v}^{3}} x^{4} -  \\frac{\\mu_{s}}{3 \\epsilon_{v}^{2}} x^{3} + \\frac{\\mu_{s}}{\\epsilon_{v}} x^{2} + \\frac{9 \\epsilon_{v} \\mu_{k}}{16} - \\frac{11 \\epsilon_{v} \\mu_{s}}{48}, x, domain=\\mathbb{Z}\\left(\\mu_{k}, \\mu_{s}, \\epsilon_{v}\\right) \\right)}$"
       ],
       "text/plain": [
-       "Poly((\\mu_k - \\mu_s)/(3*\\epsilon_v**5)*x**6 + (-7*\\mu_k + 7*\\mu_s)/(5*\\epsilon_v**4)*x**5 + (3*\\mu_k - 3*\\mu_s)/(2*\\epsilon_v**3)*x**4 - \\mu_s/(3*\\epsilon_v**2)*x**3 + \\mu_s/\\epsilon_v*x**2 + 17*\\epsilon_v*\\mu_k/30 - 7*\\epsilon_v*\\mu_s/30, x, domain='ZZ(\\mu_k,\\mu_s,\\epsilon_v)')"
+       "Poly((-2*\\mu_k + 2*\\mu_s)/(5*\\epsilon_v**4)*x**5 + (\\mu_k - \\mu_s)/\\epsilon_v**3*x**4 - \\mu_s/(3*\\epsilon_v**2)*x**3 + \\mu_s/\\epsilon_v*x**2 + 9*\\epsilon_v*\\mu_k/16 - 11*\\epsilon_v*\\mu_s/48, x, domain='ZZ(\\mu_k,\\mu_s,\\epsilon_v)')"
       ]
      },
-     "execution_count": 571,
      "metadata": {},
-     "output_type": "execute_result"
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\operatorname{Poly}{\\left( \\frac{2 \\mu_{k} - 2 \\mu_{s}}{5 \\epsilon_{v}^{4}} x^{5} + \\frac{- 2 \\mu_{k} + 2 \\mu_{s}}{\\epsilon_{v}^{3}} x^{4} + \\frac{9 \\mu_{k} - 10 \\mu_{s}}{3 \\epsilon_{v}^{2}} x^{3} + \\frac{- \\mu_{k} + 2 \\mu_{s}}{\\epsilon_{v}} x^{2} + \\frac{3 \\epsilon_{v} \\mu_{k}}{5} - \\frac{4 \\epsilon_{v} \\mu_{s}}{15}, x, domain=\\mathbb{Z}\\left(\\mu_{k}, \\mu_{s}, \\epsilon_{v}\\right) \\right)}$"
+      ],
+      "text/plain": [
+       "Poly((2*\\mu_k - 2*\\mu_s)/(5*\\epsilon_v**4)*x**5 + (-2*\\mu_k + 2*\\mu_s)/\\epsilon_v**3*x**4 + (9*\\mu_k - 10*\\mu_s)/(3*\\epsilon_v**2)*x**3 + (-\\mu_k + 2*\\mu_s)/\\epsilon_v*x**2 + 3*\\epsilon_v*\\mu_k/5 - 4*\\epsilon_v*\\mu_s/15, x, domain='ZZ(\\mu_k,\\mu_s,\\epsilon_v)')"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     }
    ],
    "source": [
-    "poly(mu_f0, x)"
+    "display(poly(mu_f0_part1, x))\n",
+    "display(poly(mu_f0_part2, x))"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 572,
+   "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\operatorname{Poly}{\\left( \\frac{\\mu_{k} - \\mu_{s}}{3 \\epsilon_{v}^{5}} x^{6} + \\frac{- 7 \\mu_{k} + 7 \\mu_{s}}{5 \\epsilon_{v}^{4}} x^{5} + \\frac{3 \\mu_{k} - 3 \\mu_{s}}{2 \\epsilon_{v}^{3}} x^{4} -  \\frac{\\mu_{s}}{3 \\epsilon_{v}^{2}} x^{3} + \\frac{\\mu_{s}}{\\epsilon_{v}} x^{2} + \\frac{17 \\epsilon_{v} \\mu_{k}}{30} - \\frac{7 \\epsilon_{v} \\mu_{s}}{30}, x, domain=\\mathbb{Z}\\left(\\mu_{k}, \\mu_{s}, \\epsilon_{v}\\right) \\right)}$"
-      ],
-      "text/plain": [
-       "Poly((\\mu_k - \\mu_s)/(3*\\epsilon_v**5)*x**6 + (-7*\\mu_k + 7*\\mu_s)/(5*\\epsilon_v**4)*x**5 + (3*\\mu_k - 3*\\mu_s)/(2*\\epsilon_v**3)*x**4 - \\mu_s/(3*\\epsilon_v**2)*x**3 + \\mu_s/\\epsilon_v*x**2 + 17*\\epsilon_v*\\mu_k/30 - 7*\\epsilon_v*\\mu_s/30, x, domain='ZZ(\\mu_k,\\mu_s,\\epsilon_v)')"
-      ]
-     },
-     "execution_count": 572,
-     "metadata": {},
-     "output_type": "execute_result"
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(9.0/16.0)*eps_v*mu_k - 11.0/48.0*eps_v*mu_s + z*(mu_s*x + z*(-1.0/3.0*mu_s*x + z*(mu_k*x - mu_s*x + z*(-2.0/5.0*mu_k*x + (2.0/5.0)*mu_s*x))))\n",
+      "(3.0/5.0)*eps_v*mu_k - 4.0/15.0*eps_v*mu_s + z*(-mu_k*x + 2*mu_s*x + z*(3*mu_k*x - 10.0/3.0*mu_s*x + z*(-2*mu_k*x + 2*mu_s*x + z*((2.0/5.0)*mu_k*x - 2.0/5.0*mu_s*x))))\n"
+     ]
     }
    ],
    "source": [
-    "def mu_f0(x, eps_v, mu_s, mu_k):\n",
-    "    \"\"\"Integral of mu(x) f1(x).\"\"\"\n",
-    "    d_mu = mu_k - mu_s\n",
-    "    x_over_eps_v = x / eps_v\n",
-    "    return x * x_over_eps_v * (\n",
-    "        x_over_eps_v * (\n",
-    "            x_over_eps_v * (\n",
-    "                x_over_eps_v * (\n",
-    "                    x_over_eps_v * d_mu / 3 - 1.4 * d_mu\n",
-    "                ) + 1.5 * d_mu\n",
-    "            ) - mu_s/3\n",
-    "        ) + mu_s) + eps_v * (17 * mu_k - 7 * mu_s) / 30\n",
-    "\n",
-    "\n",
-    "poly(nsimplify(mu_f0(x, eps_v, mu_s, mu_k)), x)"
+    "print(cxxcode(horner(poly(mu_f0_part1.subs({x/eps_v: Symbol('z')}), Symbol(\"z\")))).replace(\n",
+    "    r\"\\epsilon\", \"eps\").replace(r\"\\mu\", \"mu\"))\n",
+    "print(cxxcode(horner(poly(mu_f0_part2.subs({x/eps_v: Symbol('z')}), Symbol(\"z\")))).replace(\n",
+    "    r\"\\epsilon\", \"eps\").replace(r\"\\mu\", \"mu\"))"
    ]
   },
   {
@@ -8548,16 +9413,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 573,
+   "execution_count": 12,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\frac{\\epsilon_{v}^{6} \\left(17 \\mu_{k} - 7 \\mu_{s}\\right) + 10 x^{2} \\left(3 \\epsilon_{v}^{4} \\mu_{s} - x \\left(\\epsilon_{v}^{3} \\mu_{s} - x \\left(4.5 \\epsilon_{v}^{2} \\left(\\mu_{k} - \\mu_{s}\\right) + x \\left(4.2 \\epsilon_{v} \\left(- \\mu_{k} + \\mu_{s}\\right) + x \\left(\\mu_{k} - \\mu_{s}\\right)\\right)\\right)\\right)\\right)}{30 \\epsilon_{v}^{5}}$"
+       "$\\displaystyle \\frac{\\frac{\\epsilon_{v}^{5} \\left(27 \\mu_{k} - 11 \\mu_{s}\\right)}{48} + \\epsilon_{v}^{3} \\mu_{s} x^{2} - \\frac{\\epsilon_{v}^{2} \\mu_{s} x^{3}}{3} + \\epsilon_{v} x^{4} \\left(\\mu_{k} - \\mu_{s}\\right) + \\frac{2 x^{5} \\left(- \\mu_{k} + \\mu_{s}\\right)}{5}}{\\epsilon_{v}^{4}}$"
       ],
       "text/plain": [
-       "(\\epsilon_v**6*(17*\\mu_k - 7*\\mu_s) + 10*x**2*(3*\\epsilon_v**4*\\mu_s - x*(\\epsilon_v**3*\\mu_s - x*(4.5*\\epsilon_v**2*(\\mu_k - \\mu_s) + x*(4.2*\\epsilon_v*(-\\mu_k + \\mu_s) + x*(\\mu_k - \\mu_s))))))/(30*\\epsilon_v**5)"
+       "(\\epsilon_v**5*(27*\\mu_k - 11*\\mu_s)/48 + \\epsilon_v**3*\\mu_s*x**2 - \\epsilon_v**2*\\mu_s*x**3/3 + \\epsilon_v*x**4*(\\mu_k - \\mu_s) + 2*x**5*(-\\mu_k + \\mu_s)/5)/\\epsilon_v**4"
       ]
      },
      "metadata": {},
@@ -8577,23 +9442,23 @@
     }
    ],
    "source": [
-    "f = mu_f0(x, eps_v, mu_s, mu_k).simplify()\n",
+    "f = mu_f0_part1.simplify()\n",
     "display(f)\n",
     "display(f.subs(mu_s, mu_k).simplify().expand())"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 574,
+   "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\frac{6 \\epsilon_{v}^{4} \\mu_{s} - 3 \\epsilon_{v}^{3} \\mu_{s} x + 18 \\epsilon_{v}^{2} \\mu_{k} x^{2} - 18 \\epsilon_{v}^{2} \\mu_{s} x^{2} - 21 \\epsilon_{v} \\mu_{k} x^{3} + 21 \\epsilon_{v} \\mu_{s} x^{3} + 6 \\mu_{k} x^{4} - 6 \\mu_{s} x^{4}}{3 \\epsilon_{v}^{5}}$"
+       "$\\displaystyle \\frac{2 \\epsilon_{v}^{3} \\mu_{s} - \\epsilon_{v}^{2} \\mu_{s} x + 4 \\epsilon_{v} x^{2} \\left(\\mu_{k} - \\mu_{s}\\right) + 2 x^{3} \\left(- \\mu_{k} + \\mu_{s}\\right)}{\\epsilon_{v}^{4}}$"
       ],
       "text/plain": [
-       "(6*\\epsilon_v**4*\\mu_s - 3*\\epsilon_v**3*\\mu_s*x + 18*\\epsilon_v**2*\\mu_k*x**2 - 18*\\epsilon_v**2*\\mu_s*x**2 - 21*\\epsilon_v*\\mu_k*x**3 + 21*\\epsilon_v*\\mu_s*x**3 + 6*\\mu_k*x**4 - 6*\\mu_s*x**4)/(3*\\epsilon_v**5)"
+       "(2*\\epsilon_v**3*\\mu_s - \\epsilon_v**2*\\mu_s*x + 4*\\epsilon_v*x**2*(\\mu_k - \\mu_s) + 2*x**3*(-\\mu_k + \\mu_s))/\\epsilon_v**4"
       ]
      },
      "metadata": {},
@@ -8620,16 +9485,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 575,
+   "execution_count": 14,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\frac{- 30 \\epsilon_{v}^{3} \\mu_{s} + 360 \\epsilon_{v}^{2} \\mu_{k} x - 360 \\epsilon_{v}^{2} \\mu_{s} x - 630 \\epsilon_{v} \\mu_{k} x^{2} + 630 \\epsilon_{v} \\mu_{s} x^{2} + 240 \\mu_{k} x^{3} - 240 \\mu_{s} x^{3}}{30 \\epsilon_{v}^{5} x}$"
+       "$\\displaystyle \\frac{- \\epsilon_{v}^{2} \\mu_{s} + 8 \\epsilon_{v} \\mu_{k} x - 8 \\epsilon_{v} \\mu_{s} x - 6 \\mu_{k} x^{2} + 6 \\mu_{s} x^{2}}{\\epsilon_{v}^{4} x}$"
       ],
       "text/plain": [
-       "(-30*\\epsilon_v**3*\\mu_s + 360*\\epsilon_v**2*\\mu_k*x - 360*\\epsilon_v**2*\\mu_s*x - 630*\\epsilon_v*\\mu_k*x**2 + 630*\\epsilon_v*\\mu_s*x**2 + 240*\\mu_k*x**3 - 240*\\mu_s*x**3)/(30*\\epsilon_v**5*x)"
+       "(-\\epsilon_v**2*\\mu_s + 8*\\epsilon_v*\\mu_k*x - 8*\\epsilon_v*\\mu_s*x - 6*\\mu_k*x**2 + 6*\\mu_s*x**2)/(\\epsilon_v**4*x)"
       ]
      },
      "metadata": {},
@@ -8665,73 +9530,134 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 576,
+   "execution_count": 15,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle \\frac{- \\epsilon_{v}^{2} \\mu_{s} + 2 x^{2} \\left(- \\mu_{k} + \\mu_{s}\\right) + 4 x \\left(2 \\epsilon_{v} - x\\right) \\left(\\mu_{k} - \\mu_{s}\\right)}{\\epsilon_{v}^{4} x}$"
+      ],
+      "text/plain": [
+       "(-\\epsilon_v**2*\\mu_s + 2*x**2*(-\\mu_k + \\mu_s) + 4*x*(2*\\epsilon_v - x)*(\\mu_k - \\mu_s))/(\\epsilon_v**4*x)"
+      ]
+     },
+     "execution_count": 15,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "mu_f2_x_minus_mu_f1_over_x3 = \\\n",
+    "    ((mu_f1_part1(x).diff(x) * x - mu_f1_part1(x)) / x**3).simplify()\n",
+    "mu_f2_x_minus_mu_f1_over_x3.simplify()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\frac{- \\epsilon_{v}^{3} \\mu_{s} - x^{2} \\left(3 \\epsilon_{v} - 2 x\\right) \\left(\\mu_{k} - \\mu_{s}\\right) + 6 x \\left(\\epsilon_{v} - x\\right) \\left(2 \\epsilon_{v} - x\\right) \\left(\\mu_{k} - \\mu_{s}\\right)}{\\epsilon_{v}^{5} x}$"
+       "$\\displaystyle - \\frac{\\mu_{s}}{\\epsilon_{v}^{2} x} + \\frac{8 \\mu_{k}}{\\epsilon_{v}^{3}} - \\frac{8 \\mu_{s}}{\\epsilon_{v}^{3}} - \\frac{6 \\mu_{k} x}{\\epsilon_{v}^{4}} + \\frac{6 \\mu_{s} x}{\\epsilon_{v}^{4}}$"
       ],
       "text/plain": [
-       "(-\\epsilon_v**3*\\mu_s - x**2*(3*\\epsilon_v - 2*x)*(\\mu_k - \\mu_s) + 6*x*(\\epsilon_v - x)*(2*\\epsilon_v - x)*(\\mu_k - \\mu_s))/(\\epsilon_v**5*x)"
+       "-\\mu_s/(\\epsilon_v**2*x) + 8*\\mu_k/\\epsilon_v**3 - 8*\\mu_s/\\epsilon_v**3 - 6*\\mu_k*x/\\epsilon_v**4 + 6*\\mu_s*x/\\epsilon_v**4"
       ]
      },
-     "execution_count": 576,
+     "execution_count": 16,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "smooth_mu_f2_x_minus_f1_over_x3 = \\\n",
-    "    (((mu(x) * f1(x)).diff(x) * x - mu(x) * f1(x)) / x**3).simplify()\n",
-    "smooth_mu_f2_x_minus_f1_over_x3"
+    "mu_f2_x_minus_mu_f1_over_x3.expand()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "std::pow(z, 2)*(-\\mu_s/x + z*(8*\\mu_k - 8*\\mu_s + z*(-6*\\mu_k*x + 6*\\mu_s*x)))\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(cxxcode(horner(Poly(mu_f2_x_minus_mu_f1_over_x3.expand().subs(\n",
+    "    {1/eps_v: Symbol('z')}), Symbol(\"z\")))))"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 577,
+   "execution_count": 18,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle - \\frac{\\mu_{s}}{\\epsilon_{v}^{2} x} + \\frac{12 \\mu_{k}}{\\epsilon_{v}^{3}} - \\frac{12 \\mu_{s}}{\\epsilon_{v}^{3}} - \\frac{21 \\mu_{k} x}{\\epsilon_{v}^{4}} + \\frac{21 \\mu_{s} x}{\\epsilon_{v}^{4}} + \\frac{8 \\mu_{k} x^{2}}{\\epsilon_{v}^{5}} - \\frac{8 \\mu_{s} x^{2}}{\\epsilon_{v}^{5}}$"
+       "$\\displaystyle \\frac{- \\epsilon_{v}^{2} \\mu_{k} + 2 \\left(\\epsilon_{v} - x\\right)^{2} \\left(\\mu_{k} - \\mu_{s}\\right) + 4 \\left(\\epsilon_{v} - x\\right) \\left(2 \\epsilon_{v} - x\\right) \\left(\\mu_{k} - \\mu_{s}\\right)}{\\epsilon_{v}^{4} x}$"
       ],
       "text/plain": [
-       "-\\mu_s/(\\epsilon_v**2*x) + 12*\\mu_k/\\epsilon_v**3 - 12*\\mu_s/\\epsilon_v**3 - 21*\\mu_k*x/\\epsilon_v**4 + 21*\\mu_s*x/\\epsilon_v**4 + 8*\\mu_k*x**2/\\epsilon_v**5 - 8*\\mu_s*x**2/\\epsilon_v**5"
+       "(-\\epsilon_v**2*\\mu_k + 2*(\\epsilon_v - x)**2*(\\mu_k - \\mu_s) + 4*(\\epsilon_v - x)*(2*\\epsilon_v - x)*(\\mu_k - \\mu_s))/(\\epsilon_v**4*x)"
       ]
      },
-     "execution_count": 577,
+     "execution_count": 18,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "smooth_mu_f2_x_minus_f1_over_x3.expand()"
+    "mu_f2_x_minus_mu_f1_over_x3 = \\\n",
+    "    ((mu_f1_part2(x).diff(x) * x - mu_f1_part2(x)) / x**3).simplify()\n",
+    "mu_f2_x_minus_mu_f1_over_x3.simplify()"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 578,
+   "execution_count": 19,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle 8 \\mu_{k} x^{2} z^{5} - 21 \\mu_{k} x z^{4} + 12 \\mu_{k} z^{3} - 8 \\mu_{s} x^{2} z^{5} + 21 \\mu_{s} x z^{4} - 12 \\mu_{s} z^{3} - \\frac{\\mu_{s} z^{2}}{x}$"
+       "$\\displaystyle \\frac{9 \\mu_{k}}{\\epsilon_{v}^{2} x} - \\frac{10 \\mu_{s}}{\\epsilon_{v}^{2} x} - \\frac{16 \\mu_{k}}{\\epsilon_{v}^{3}} + \\frac{16 \\mu_{s}}{\\epsilon_{v}^{3}} + \\frac{6 \\mu_{k} x}{\\epsilon_{v}^{4}} - \\frac{6 \\mu_{s} x}{\\epsilon_{v}^{4}}$"
       ],
       "text/plain": [
-       "8*\\mu_k*x**2*z**5 - 21*\\mu_k*x*z**4 + 12*\\mu_k*z**3 - 8*\\mu_s*x**2*z**5 + 21*\\mu_s*x*z**4 - 12*\\mu_s*z**3 - \\mu_s*z**2/x"
+       "9*\\mu_k/(\\epsilon_v**2*x) - 10*\\mu_s/(\\epsilon_v**2*x) - 16*\\mu_k/\\epsilon_v**3 + 16*\\mu_s/\\epsilon_v**3 + 6*\\mu_k*x/\\epsilon_v**4 - 6*\\mu_s*x/\\epsilon_v**4"
       ]
      },
-     "execution_count": 578,
+     "execution_count": 19,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "smooth_mu_f2_x_minus_f1_over_x3.expand().subs({1/eps_v: Symbol('z')})"
+    "mu_f2_x_minus_mu_f1_over_x3.expand()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "std::pow(z, 2)*(z*(-16*\\mu_k + 16*\\mu_s + z*(6*\\mu_k*x - 6*\\mu_s*x)) + (9*\\mu_k - 10*\\mu_s)/x)\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(cxxcode(horner(Poly(mu_f2_x_minus_mu_f1_over_x3.expand().subs(\n",
+    "    {1/eps_v: Symbol('z')}), Symbol(\"z\")))))"
    ]
   },
   {
@@ -8743,7 +9669,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 579,
+   "execution_count": 21,
    "metadata": {},
    "outputs": [
     {
@@ -9576,9 +10502,9 @@
        }
       },
       "text/html": [
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        "                            \n",
-       "var gd = document.getElementById('9dd7e5ac-e2bd-402d-bec6-7f46938e1be2');\n",
+       "var gd = document.getElementById('489526be-0728-453e-9a47-131967f48515');\n",
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@@ -9631,16 +10557,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 580,
+   "execution_count": 22,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle f_{1}^{a}(x) = \\begin{cases} 0 & \\text{for}\\: x \\leq 0 \\\\\\frac{2 x}{\\epsilon_{v}} - \\frac{x^{2}}{\\epsilon_{v}^{2}} & \\text{for}\\: x < 2 \\epsilon_{v} \\\\0 & \\text{otherwise} \\end{cases}$"
+       "$\\displaystyle f_{1}^{a}(x) = \\begin{cases} 0 & \\text{for}\\: x < 0 \\\\\\frac{2 x}{\\epsilon_{v}} - \\frac{x^{2}}{\\epsilon_{v}^{2}} & \\text{for}\\: x \\leq 2 \\epsilon_{v} \\\\0 & \\text{otherwise} \\end{cases}$"
       ],
       "text/plain": [
-       "Eq(f_{1}^{a}(x), Piecewise((0, x <= 0), (2*x/\\epsilon_v - x**2/\\epsilon_v**2, x < 2*\\epsilon_v), (0, True)))"
+       "Eq(f_{1}^{a}(x), Piecewise((0, x < 0), (2*x/\\epsilon_v - x**2/\\epsilon_v**2, x <= 2*\\epsilon_v), (0, True)))"
       ]
      },
      "metadata": {},
@@ -10464,9 +11390,9 @@
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       },
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                       {\"responsive\": true}                    ).then(function(){\n",
        "                            \n",
-       "var gd = document.getElementById('1f1015c2-02a6-427a-8abb-51d463539d6a');\n",
+       "var gd = document.getElementById('d7dcb2d7-f612-4f0e-a23a-da6e3678c0ff');\n",
        "var x = new MutationObserver(function (mutations, observer) {{\n",
        "        var display = window.getComputedStyle(gd).display;\n",
        "        if (!display || display === 'none') {{\n",
@@ -10506,9 +11432,9 @@
     "\n",
     "\n",
     "sym_a1 = Piecewise(\n",
-    "    (0, x <= 0),\n",
-    "    (a1(x), x < 2 * eps_v),\n",
-    "    (0, x >= 2 * eps_v),\n",
+    "    (0, x < 0),\n",
+    "    (a1(x), x <= 2 * eps_v),\n",
+    "    (0, True),\n",
     ")\n",
     "\n",
     "display(Eq(Symbol(\"f_{1}^{a}(x)\"), sym_a1.expand()))\n",
@@ -10525,16 +11451,16 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 590,
+   "execution_count": 23,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\mu(x) f_{1}^{a}(x) = \\begin{cases} 0 & \\text{for}\\: x \\leq 0 \\\\\\frac{x \\left(2 - \\frac{x}{\\epsilon_{v}}\\right) \\left(\\mu_{s} + \\left(\\mu_{k} - \\mu_{s}\\right) \\left(\\frac{3 x^{2}}{\\epsilon_{v}^{2}} - \\frac{2 x^{3}}{\\epsilon_{v}^{3}}\\right)\\right)}{\\epsilon_{v}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\\frac{\\mu_{k} x \\left(2 - \\frac{x}{\\epsilon_{v}}\\right)}{\\epsilon_{v}} & \\text{for}\\: x \\leq 2 \\epsilon_{v} \\\\0 & \\text{otherwise} \\end{cases}$"
+       "$\\displaystyle \\mu(x) f_{1}^{a}(x) = \\begin{cases} 0 & \\text{for}\\: x \\leq 0 \\\\\\frac{x \\left(2 \\epsilon_{v} - x\\right) \\left(\\epsilon_{v}^{2} \\mu_{s} + 2 x^{2} \\left(\\mu_{k} - \\mu_{s}\\right)\\right)}{\\epsilon_{v}^{4}} & \\text{for}\\: x \\leq \\frac{\\epsilon_{v}}{2} \\\\\\frac{x \\left(2 \\epsilon_{v} - x\\right) \\left(\\epsilon_{v}^{2} \\mu_{k} + 2 \\left(\\epsilon_{v} - x\\right)^{2} \\left(- \\mu_{k} + \\mu_{s}\\right)\\right)}{\\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\\frac{\\mu_{k} x \\left(2 \\epsilon_{v} - x\\right)}{\\epsilon_{v}^{2}} & \\text{for}\\: x \\leq 2 \\epsilon_{v} \\\\0 & \\text{otherwise} \\end{cases}$"
       ],
       "text/plain": [
-       "Eq(\\mu(x) f_{1}^{a}(x), Piecewise((0, x <= 0), (x*(2 - x/\\epsilon_v)*(\\mu_s + (\\mu_k - \\mu_s)*(3*x**2/\\epsilon_v**2 - 2*x**3/\\epsilon_v**3))/\\epsilon_v, \\epsilon_v >= x), (\\mu_k*x*(2 - x/\\epsilon_v)/\\epsilon_v, x <= 2*\\epsilon_v), (0, True)))"
+       "Eq(\\mu(x) f_{1}^{a}(x), Piecewise((0, x <= 0), (x*(2*\\epsilon_v - x)*(\\epsilon_v**2*\\mu_s + 2*x**2*(\\mu_k - \\mu_s))/\\epsilon_v**4, x <= \\epsilon_v/2), (x*(2*\\epsilon_v - x)*(\\epsilon_v**2*\\mu_k + 2*(\\epsilon_v - x)**2*(-\\mu_k + \\mu_s))/\\epsilon_v**4, \\epsilon_v >= x), (\\mu_k*x*(2*\\epsilon_v - x)/\\epsilon_v**2, x <= 2*\\epsilon_v), (0, True)))"
       ]
      },
      "metadata": {},
@@ -10554,7 +11480,7 @@
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f_{1}^{a}(x)$\"},\"width\":800,\"xaxis\":{\"title\":{\"text\":\"speed\"}}},                        {\"responsive\": true}                    ).then(function(){\n",
        "                            \n",
-       "var gd = document.getElementById('a1fe613f-4e2d-4ff8-963e-4090e8314e15');\n",
+       "var gd = document.getElementById('d978ca8d-96e3-4482-b592-541b89c4a4c8');\n",
        "var x = new MutationObserver(function (mutations, observer) {{\n",
        "        var display = window.getComputedStyle(gd).display;\n",
        "        if (!display || display === 'none') {{\n",
@@ -11390,16 +12316,23 @@
     }
    ],
    "source": [
-    "def mu_a1(x):\n",
-    "    return mu(x) * a1(x)\n",
+    "def mu_a1_part1(x):\n",
+    "    return mu_quadratic_part1(x) * a1(x)\n",
+    "\n",
+    "\n",
+    "def mu_a1_part2(x):\n",
+    "    return mu_quadratic_part2(x) * a1(x)\n",
     "\n",
     "\n",
     "sym_mu_a1 = Piecewise(\n",
     "    (0, x <= 0),\n",
-    "    (mu_a1(x), x <= eps_v),\n",
+    "    (mu_a1_part1(x), x <= eps_v / 2),\n",
+    "    (mu_a1_part2(x), x <= eps_v),\n",
     "    (mu_k * a1(x), x <= 2 * eps_v),\n",
-    "    (0, x > 2 * eps_v)\n",
-    ")\n",
+    "    (0, True)\n",
+    ").simplify()\n",
+    "\n",
+    "# sym_mu_a1 = (sym_mu * sym_a1).simplify()\n",
     "\n",
     "display(Eq(Symbol(r\"\\mu(x) f_{1}^{a}(x)\"), sym_mu_a1))\n",
     "\n",
@@ -11408,16 +12341,24 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 593,
+   "execution_count": 24,
    "metadata": {},
    "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(3.0/80.0)*eps_a*(-mu_k + mu_s)\n",
+      "(11.0/48.0)*eps_a*(mu_k - mu_s)\n"
+     ]
+    },
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\int \\mu(x) f_{1}^{a}(x) \\mathrm{d}x = \\begin{cases} 0 & \\text{for}\\: x \\leq 0 \\\\\\frac{\\mu_{s} x^{2}}{\\epsilon_{v}} - \\frac{\\mu_{s} x^{3}}{3 \\epsilon_{v}^{2}} + \\frac{x^{4} \\left(3 \\mu_{k} - 3 \\mu_{s}\\right)}{2 \\epsilon_{v}^{3}} + \\frac{x^{5} \\left(- 7 \\mu_{k} + 7 \\mu_{s}\\right)}{5 \\epsilon_{v}^{4}} + \\frac{x^{6} \\left(\\mu_{k} - \\mu_{s}\\right)}{3 \\epsilon_{v}^{5}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\\\frac{7 \\epsilon_{v}^{3} \\left(- \\mu_{k} + \\mu_{s}\\right) + 10 \\mu_{k} x^{2} \\left(3 \\epsilon_{v} - x\\right)}{30 \\epsilon_{v}^{2}} & \\text{for}\\: x \\leq 2 \\epsilon_{v} \\\\\\frac{\\epsilon_{v} \\left(33 \\mu_{k} + 7 \\mu_{s}\\right)}{30} & \\text{otherwise} \\end{cases}$"
+       "$\\displaystyle \\int \\mu(x) f_{1}^{a}(x) \\mathrm{d}x = \\begin{cases} 0 & \\text{for}\\: x \\leq 0 \\\\\\frac{\\mu_{s} x^{2}}{\\epsilon_{v}} - \\frac{\\mu_{s} x^{3}}{3 \\epsilon_{v}^{2}} + \\frac{x^{4} \\left(\\mu_{k} - \\mu_{s}\\right)}{\\epsilon_{v}^{3}} + \\frac{x^{5} \\left(- 2 \\mu_{k} + 2 \\mu_{s}\\right)}{5 \\epsilon_{v}^{4}} & \\text{for}\\: x \\leq \\frac{\\epsilon_{v}}{2} \\\\- \\frac{3 \\epsilon_{v} \\left(- \\mu_{k} + \\mu_{s}\\right)}{80} + \\frac{x^{2} \\left(- \\mu_{k} + 2 \\mu_{s}\\right)}{\\epsilon_{v}} + \\frac{x^{3} \\left(9 \\mu_{k} - 10 \\mu_{s}\\right)}{3 \\epsilon_{v}^{2}} + \\frac{x^{4} \\left(- 2 \\mu_{k} + 2 \\mu_{s}\\right)}{\\epsilon_{v}^{3}} + \\frac{x^{5} \\left(2 \\mu_{k} - 2 \\mu_{s}\\right)}{5 \\epsilon_{v}^{4}} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\- \\frac{3 \\epsilon_{v} \\left(- \\mu_{k} + \\mu_{s}\\right)}{80} - \\frac{4 \\epsilon_{v} \\left(\\mu_{k} - \\mu_{s}\\right)}{15} + \\frac{\\mu_{k} x^{2} \\left(3 \\epsilon_{v} - x\\right)}{3 \\epsilon_{v}^{2}} & \\text{for}\\: x \\leq 2 \\epsilon_{v} \\\\\\frac{4 \\epsilon_{v} \\mu_{k}}{3} - \\frac{3 \\epsilon_{v} \\left(- \\mu_{k} + \\mu_{s}\\right)}{80} - \\frac{4 \\epsilon_{v} \\left(\\mu_{k} - \\mu_{s}\\right)}{15} & \\text{otherwise} \\end{cases}$"
       ],
       "text/plain": [
-       "Eq(\\int \\mu(x) f_{1}^{a}(x) \\mathrm{d}x, Piecewise((0, x <= 0), (\\mu_s*x**2/\\epsilon_v - \\mu_s*x**3/(3*\\epsilon_v**2) + x**4*(3*\\mu_k - 3*\\mu_s)/(2*\\epsilon_v**3) + x**5*(-7*\\mu_k + 7*\\mu_s)/(5*\\epsilon_v**4) + x**6*(\\mu_k - \\mu_s)/(3*\\epsilon_v**5), \\epsilon_v >= x), ((7*\\epsilon_v**3*(-\\mu_k + \\mu_s) + 10*\\mu_k*x**2*(3*\\epsilon_v - x))/(30*\\epsilon_v**2), x <= 2*\\epsilon_v), (\\epsilon_v*(33*\\mu_k + 7*\\mu_s)/30, True)))"
+       "Eq(\\int \\mu(x) f_{1}^{a}(x) \\mathrm{d}x, Piecewise((0, x <= 0), (\\mu_s*x**2/\\epsilon_v - \\mu_s*x**3/(3*\\epsilon_v**2) + x**4*(\\mu_k - \\mu_s)/\\epsilon_v**3 + x**5*(-2*\\mu_k + 2*\\mu_s)/(5*\\epsilon_v**4), x <= \\epsilon_v/2), (-3*\\epsilon_v*(-\\mu_k + \\mu_s)/80 + x**2*(-\\mu_k + 2*\\mu_s)/\\epsilon_v + x**3*(9*\\mu_k - 10*\\mu_s)/(3*\\epsilon_v**2) + x**4*(-2*\\mu_k + 2*\\mu_s)/\\epsilon_v**3 + x**5*(2*\\mu_k - 2*\\mu_s)/(5*\\epsilon_v**4), \\epsilon_v >= x), (-3*\\epsilon_v*(-\\mu_k + \\mu_s)/80 - 4*\\epsilon_v*(\\mu_k - \\mu_s)/15 + \\mu_k*x**2*(3*\\epsilon_v - x)/(3*\\epsilon_v**2), x <= 2*\\epsilon_v), (4*\\epsilon_v*\\mu_k/3 - 3*\\epsilon_v*(-\\mu_k + \\mu_s)/80 - 4*\\epsilon_v*(\\mu_k - \\mu_s)/15, True)))"
       ]
      },
      "metadata": {},
@@ -11437,7 +12378,7 @@
           "dtype": "f8"
          },
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{\"height\":600,\"template\":{\"data\":{\"barpolar\":[{\"marker\":{\"line\":{\"color\":\"rgb(17,17,17)\",\"width\":0.5},\"pattern\":{\"fillmode\":\"overlay\",\"size\":10,\"solidity\":0.2}},\"type\":\"barpolar\"}],\"bar\":[{\"error_x\":{\"color\":\"#f2f5fa\"},\"error_y\":{\"color\":\"#f2f5fa\"},\"marker\":{\"line\":{\"color\":\"rgb(17,17,17)\",\"width\":0.5},\"pattern\":{\"fillmode\":\"overlay\",\"size\":10,\"solidity\":0.2}},\"type\":\"bar\"}],\"carpet\":[{\"aaxis\":{\"endlinecolor\":\"#A2B1C6\",\"gridcolor\":\"#506784\",\"linecolor\":\"#506784\",\"minorgridcolor\":\"#506784\",\"startlinecolor\":\"#A2B1C6\"},\"baxis\":{\"endlinecolor\":\"#A2B1C6\",\"gridcolor\":\"#506784\",\"linecolor\":\"#506784\",\"minorgridcolor\":\"#506784\",\"startlinecolor\":\"#A2B1C6\"},\"type\":\"carpet\"}],\"choropleth\":[{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"},\"type\":\"choropleth\"}],\"contourcarpet\":[{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"},\"type\":\"contourcarpet\"}],\"contour\":[{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"},\"colorscale\":[[0.0,\"#0d0887\"],[0.1111111111111111,\"#46039f\"],[0.2222222222222222,\"#7201a8\"],[0.3333333333333333,\"#9c179e\"],[0.4444444444444444,\"#bd3786\"],[0.5555555555555556,\"#d8576b\"],[0.6666666666666666,\"#ed7953\"],[0.7777777777777778,\"#fb9f3a\"],[0.8888888888888888,\"#fdca26\"],[1.0,\"#f0f921\"]],\"type\":\"contour\"}],\"heatmap\":[{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"},\"colorscale\":[[0.0,\"#0d0887\"],[0.1111111111111111,\"#46039f\"],[0.2222222222222222,\"#7201a8\"],[0.3333333333333333,\"#9c179e\"],[0.4444444444444444,\"#bd3786\"],[0.5555555555555556,\"#d8576b\"],[0.6666666666666666,\"#ed7953\"],[0.7777777777777778,\"#fb9f3a\"],[0.8888888888888888,\"#fdca26\"],[1.0,\"#f0f921\"]],\"type\":\"heatmap\"}],\"histogram2dcontour\":[{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"},\"colorscale\":[[0.0,\"#0d0887\"],[0.1111111111111111,\"#46039f\"],[0.2222222222222222,\"#7201a8\"],[0.3333333333333333,\"#9c179e\"],[0.4444444444444444,\"#bd3786\"],[0.5555555555555556,\"#d8576b\"],[0.6666666666666666,\"#ed7953\"],[0.7777777777777778,\"#fb9f3a\"],[0.8888888888888888,\"#fdca26\"],[1.0,\"#f0f921\"]],\"type\":\"histogram2dcontour\"}],\"histogram2d\":[{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"},\"colorscale\":[[0.0,\"#0d0887\"],[0.1111111111111111,\"#46039f\"],[0.2222222222222222,\"#7201a8\"],[0.3333333333333333,\"#9c179e\"],[0.4444444444444444,\"#bd3786\"],[0.5555555555555556,\"#d8576b\"],[0.6666666666666666,\"#ed7953\"],[0.7777777777777778,\"#fb9f3a\"],[0.8888888888888888,\"#fdca26\"],[1.0,\"#f0f921\"]],\"type\":\"histogram2d\"}],\"histogram\":[{\"marker\":{\"pattern\":{\"fillmode\":\"overlay\",\"size\":10,\"solidity\":0.2}},\"type\":\"histogram\"}],\"mesh3d\":[{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"},\"type\":\"mesh3d\"}],\"parcoords\":[{\"line\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"parcoords\"}],\"pie\":[{\"automargin\":true,\"type\":\"pie\"}],\"scatter3d\":[{\"line\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"marker\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"scatter3d\"}],\"scattercarpet\":[{\"marker\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"scattercarpet\"}],\"scattergeo\":[{\"marker\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"scattergeo\"}],\"scattergl\":[{\"marker\":{\"line\":{\"color\":\"#283442\"}},\"type\":\"scattergl\"}],\"scattermapbox\":[{\"marker\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"scattermapbox\"}],\"scattermap\":[{\"marker\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"scattermap\"}],\"scatterpolargl\":[{\"marker\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"scatterpolargl\"}],\"scatterpolar\":[{\"marker\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"scatterpolar\"}],\"scatter\":[{\"marker\":{\"line\":{\"color\":\"#283442\"}},\"type\":\"scatter\"}],\"scatterternary\":[{\"marker\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"type\":\"scatterternary\"}],\"surface\":[{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"},\"colorscale\":[[0.0,\"#0d0887\"],[0.1111111111111111,\"#46039f\"],[0.2222222222222222,\"#7201a8\"],[0.3333333333333333,\"#9c179e\"],[0.4444444444444444,\"#bd3786\"],[0.5555555555555556,\"#d8576b\"],[0.6666666666666666,\"#ed7953\"],[0.7777777777777778,\"#fb9f3a\"],[0.8888888888888888,\"#fdca26\"],[1.0,\"#f0f921\"]],\"type\":\"surface\"}],\"table\":[{\"cells\":{\"fill\":{\"color\":\"#506784\"},\"line\":{\"color\":\"rgb(17,17,17)\"}},\"header\":{\"fill\":{\"color\":\"#2a3f5f\"},\"line\":{\"color\":\"rgb(17,17,17)\"}},\"type\":\"table\"}]},\"layout\":{\"annotationdefaults\":{\"arrowcolor\":\"#f2f5fa\",\"arrowhead\":0,\"arrowwidth\":1},\"autotypenumbers\":\"strict\",\"coloraxis\":{\"colorbar\":{\"outlinewidth\":0,\"ticks\":\"\"}},\"colorscale\":{\"diverging\":[[0,\"#8e0152\"],[0.1,\"#c51b7d\"],[0.2,\"#de77ae\"],[0.3,\"#f1b6da\"],[0.4,\"#fde0ef\"],[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\\\\mu(x) f_{1}^{a}(x) \\\\mathrm{d}x$\"},\"width\":800,\"xaxis\":{\"title\":{\"text\":\"speed\"}}},                        {\"responsive\": true}                    ).then(function(){\n",
        "                            \n",
-       "var gd = document.getElementById('ba42a656-895b-49e3-92fa-09000ca065aa');\n",
+       "var gd = document.getElementById('744a18d6-7cb9-4b57-bb17-0974e5c4ae36');\n",
        "var x = new MutationObserver(function (mutations, observer) {{\n",
        "        var display = window.getComputedStyle(gd).display;\n",
        "        if (!display || display === 'none') {{\n",
@@ -12273,19 +13214,26 @@
     }
    ],
    "source": [
-    "mu_a0 = integrate(mu_a1(x), x)\n",
+    "mu_a0_part1 = integrate(mu_a1_part1(x), x)\n",
+    "mu_a0_part2 = integrate(mu_a1_part2(x), x)\n",
     "\n",
     "# Constant of integration s.t. mu_a0(eps_v) = mu_k * a0_part1(eps_v)\n",
-    "c = (mu_k * a0_part1(eps_v) - mu_a0.subs(x, eps_v)).simplify()\n",
-    "mu_a0 = mu_a0\n",
+    "c2 = (mu_k * a0_part1(eps_v) - mu_a0_part2.subs(x, eps_v)).simplify()\n",
+    "c1 = (mu_a0_part2.subs(x, eps_v/2) - mu_a0_part1.subs(x, eps_v/2)).simplify()\n",
+    "\n",
+    "mu_a0_part2 -= c1\n",
     "\n",
-    "mu_a0.simplify()\n",
+    "print(cxxcode(c1.simplify()).replace(\n",
+    "    r\"\\epsilon_v\", \"eps_a\").replace(r\"\\mu\", \"mu\"))\n",
+    "print(cxxcode((c2+c1).simplify()).replace(r\"\\epsilon_v\",\n",
+    "      \"eps_a\").replace(r\"\\mu\", \"mu\"))\n",
     "\n",
     "sym_mu_a0 = Piecewise(\n",
     "    (0, x <= 0),\n",
-    "    (mu_a0, x <= eps_v),\n",
-    "    ((mu_k * a0_part1(x) - c).simplify(), x <= 2 * eps_v),\n",
-    "    ((mu_k * a0_part2(x) - c).simplify(), x > 2 * eps_v),\n",
+    "    (mu_a0_part1, x <= eps_v/2),\n",
+    "    (mu_a0_part2, x <= eps_v),\n",
+    "    ((mu_k * a0_part1(x)).simplify()-c2-c1, x <= 2 * eps_v),\n",
+    "    ((mu_k * a0_part2(x)).simplify()-c2-c1, x > 2 * eps_v),\n",
     ")\n",
     "\n",
     "display(Eq(Symbol(r\"\\int \\mu(x) f_{1}^{a}(x) \\mathrm{d}x\"), sym_mu_a0))\n",
@@ -12296,16 +13244,28 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 583,
+   "execution_count": 25,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\frac{\\mu_{s} x^{2}}{\\epsilon_{v}} - \\frac{\\mu_{s} x^{3}}{3 \\epsilon_{v}^{2}} + \\frac{x^{4} \\left(3 \\mu_{k} - 3 \\mu_{s}\\right)}{2 \\epsilon_{v}^{3}} + \\frac{x^{5} \\left(- 7 \\mu_{k} + 7 \\mu_{s}\\right)}{5 \\epsilon_{v}^{4}} + \\frac{x^{6} \\left(\\mu_{k} - \\mu_{s}\\right)}{3 \\epsilon_{v}^{5}}$"
+       "$\\displaystyle \\frac{\\mu_{s} x^{2}}{\\epsilon_{v}} - \\frac{\\mu_{s} x^{3}}{3 \\epsilon_{v}^{2}} + \\frac{x^{4} \\left(\\mu_{k} - \\mu_{s}\\right)}{\\epsilon_{v}^{3}} + \\frac{x^{5} \\left(- 2 \\mu_{k} + 2 \\mu_{s}\\right)}{5 \\epsilon_{v}^{4}}$"
+      ],
+      "text/plain": [
+       "\\mu_s*x**2/\\epsilon_v - \\mu_s*x**3/(3*\\epsilon_v**2) + x**4*(\\mu_k - \\mu_s)/\\epsilon_v**3 + x**5*(-2*\\mu_k + 2*\\mu_s)/(5*\\epsilon_v**4)"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\displaystyle - \\frac{3 \\epsilon_{v} \\left(- \\mu_{k} + \\mu_{s}\\right)}{40} + \\frac{x^{2} \\left(- \\mu_{k} + 2 \\mu_{s}\\right)}{\\epsilon_{v}} + \\frac{x^{3} \\left(9 \\mu_{k} - 10 \\mu_{s}\\right)}{3 \\epsilon_{v}^{2}} + \\frac{x^{4} \\left(- 2 \\mu_{k} + 2 \\mu_{s}\\right)}{\\epsilon_{v}^{3}} + \\frac{x^{5} \\left(2 \\mu_{k} - 2 \\mu_{s}\\right)}{5 \\epsilon_{v}^{4}}$"
       ],
       "text/plain": [
-       "\\mu_s*x**2/\\epsilon_v - \\mu_s*x**3/(3*\\epsilon_v**2) + x**4*(3*\\mu_k - 3*\\mu_s)/(2*\\epsilon_v**3) + x**5*(-7*\\mu_k + 7*\\mu_s)/(5*\\epsilon_v**4) + x**6*(\\mu_k - \\mu_s)/(3*\\epsilon_v**5)"
+       "-3*\\epsilon_v*(-\\mu_k + \\mu_s)/40 + x**2*(-\\mu_k + 2*\\mu_s)/\\epsilon_v + x**3*(9*\\mu_k - 10*\\mu_s)/(3*\\epsilon_v**2) + x**4*(-2*\\mu_k + 2*\\mu_s)/\\epsilon_v**3 + x**5*(2*\\mu_k - 2*\\mu_s)/(5*\\epsilon_v**4)"
       ]
      },
      "metadata": {},
@@ -12313,134 +13273,96 @@
     }
    ],
    "source": [
-    "# print(pycode(mu_a0.subs({eps_v: \"eps_v\", mu_s: \"mu_s\", mu_k: \"mu_k\"})))\n",
-    "display(mu_a0)"
+    "display(mu_a0_part1, mu_a0_part2 - c1)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 584,
+   "execution_count": 26,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "z*(mu_s*y + z*(-1.0/3.0*mu_s*y + z*(mu_k*y - mu_s*y + z*(-2.0/5.0*mu_k*y + (2.0/5.0)*mu_s*y))))\n",
+      "(3.0/80.0)*eps_v*mu_k - 3.0/80.0*eps_v*mu_s + z*(-mu_k*y + 2*mu_s*y + z*(3*mu_k*y - 10.0/3.0*mu_s*y + z*(-2*mu_k*y + 2*mu_s*y + z*((2.0/5.0)*mu_k*y - 2.0/5.0*mu_s*y))))\n"
+     ]
+    }
+   ],
    "source": [
-    "def mu_a0(x, eps_v, mu_s, mu_k):\n",
-    "    \"\"\"Integral of mu(x) a1(x).\"\"\"\n",
-    "    d_mu = mu_k - mu_s\n",
-    "    x_over_eps_v = x / eps_v\n",
-    "    c = 7 * eps_v * d_mu / 30\n",
-    "    return x * x_over_eps_v * (\n",
-    "        x_over_eps_v * (\n",
-    "            x_over_eps_v * (\n",
-    "                x_over_eps_v * (\n",
-    "                    x_over_eps_v * d_mu / 3 - 7 * d_mu / 5\n",
-    "                ) + 3 * d_mu / 2\n",
-    "            ) - mu_s/3\n",
-    "        ) + mu_s) + c"
+    "print_code(horner(poly(mu_a0_part1.subs({x/eps_v: Symbol('z')}), Symbol(\"z\"))))\n",
+    "print_code(horner(poly(mu_a0_part2.subs({x/eps_v: Symbol('z')}), Symbol(\"z\"))))"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 585,
+   "execution_count": 27,
    "metadata": {},
    "outputs": [
     {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\frac{7 \\epsilon_{v} \\mu_{k}}{30} - \\frac{7 \\epsilon_{v} \\mu_{s}}{30} + \\frac{\\mu_{s} x^{2}}{\\epsilon_{v}} - \\frac{\\mu_{s} x^{3}}{3 \\epsilon_{v}^{2}} + \\frac{3 \\mu_{k} x^{4}}{2 \\epsilon_{v}^{3}} - \\frac{3 \\mu_{s} x^{4}}{2 \\epsilon_{v}^{3}} - \\frac{7 \\mu_{k} x^{5}}{5 \\epsilon_{v}^{4}} + \\frac{7 \\mu_{s} x^{5}}{5 \\epsilon_{v}^{4}} + \\frac{\\mu_{k} x^{6}}{3 \\epsilon_{v}^{5}} - \\frac{\\mu_{s} x^{6}}{3 \\epsilon_{v}^{5}}$"
-      ],
-      "text/plain": [
-       "7*\\epsilon_v*\\mu_k/30 - 7*\\epsilon_v*\\mu_s/30 + \\mu_s*x**2/\\epsilon_v - \\mu_s*x**3/(3*\\epsilon_v**2) + 3*\\mu_k*x**4/(2*\\epsilon_v**3) - 3*\\mu_s*x**4/(2*\\epsilon_v**3) - 7*\\mu_k*x**5/(5*\\epsilon_v**4) + 7*\\mu_s*x**5/(5*\\epsilon_v**4) + \\mu_k*x**6/(3*\\epsilon_v**5) - \\mu_s*x**6/(3*\\epsilon_v**5)"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "std::pow(z, 2)*(-mu_s/y + z*(8*mu_k - 8*mu_s + z*(-6*mu_k*y + 6*mu_s*y)))\n"
+     ]
     }
    ],
    "source": [
-    "display(a := mu_a0(x, eps_v, mu_s, mu_k).expand())"
+    "a2_part1 = ((mu_a1_part1(x).diff(x) * x - mu_a1_part1(x)) / x**3).simplify()\n",
+    "print_code(horner(Poly(a2_part1.expand().subs(\n",
+    "    {1/eps_v: Symbol('z')}), Symbol('z'))))"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 586,
+   "execution_count": 28,
    "metadata": {},
    "outputs": [
     {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\frac{2 \\mu_{s} x}{\\epsilon_{v}} - \\frac{\\mu_{s} x^{2}}{\\epsilon_{v}^{2}} + \\frac{6 \\mu_{k} x^{3}}{\\epsilon_{v}^{3}} - \\frac{6 \\mu_{s} x^{3}}{\\epsilon_{v}^{3}} - \\frac{7 \\mu_{k} x^{4}}{\\epsilon_{v}^{4}} + \\frac{7 \\mu_{s} x^{4}}{\\epsilon_{v}^{4}} + \\frac{2 \\mu_{k} x^{5}}{\\epsilon_{v}^{5}} - \\frac{2 \\mu_{s} x^{5}}{\\epsilon_{v}^{5}}$"
-      ],
-      "text/plain": [
-       "2*\\mu_s*x/\\epsilon_v - \\mu_s*x**2/\\epsilon_v**2 + 6*\\mu_k*x**3/\\epsilon_v**3 - 6*\\mu_s*x**3/\\epsilon_v**3 - 7*\\mu_k*x**4/\\epsilon_v**4 + 7*\\mu_s*x**4/\\epsilon_v**4 + 2*\\mu_k*x**5/\\epsilon_v**5 - 2*\\mu_s*x**5/\\epsilon_v**5"
-      ]
-     },
-     "execution_count": 586,
-     "metadata": {},
-     "output_type": "execute_result"
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "std::pow(z, 2)*(z*(-16*mu_k + 16*mu_s + z*(6*mu_k*y - 6*mu_s*y)) + (9*mu_k - 10*mu_s)/y)\n"
+     ]
     }
    ],
    "source": [
-    "a.diff(x)"
+    "a2_part2 = ((mu_a1_part2(x).diff(x) * x - mu_a1_part2(x)) / x**3).simplify()\n",
+    "print_code(horner(Poly(a2_part2.expand().subs(\n",
+    "    {1/eps_v: Symbol('z')}), Symbol('z'))))"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 587,
+   "execution_count": 29,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/latex": [
-       "$\\displaystyle \\frac{2 \\mu_{s} x}{\\epsilon_{v}} - \\frac{\\mu_{s} x^{2}}{\\epsilon_{v}^{2}} + \\frac{6 \\mu_{k} x^{3}}{\\epsilon_{v}^{3}} - \\frac{6 \\mu_{s} x^{3}}{\\epsilon_{v}^{3}} - \\frac{7 \\mu_{k} x^{4}}{\\epsilon_{v}^{4}} + \\frac{7 \\mu_{s} x^{4}}{\\epsilon_{v}^{4}} + \\frac{2 \\mu_{k} x^{5}}{\\epsilon_{v}^{5}} - \\frac{2 \\mu_{s} x^{5}}{\\epsilon_{v}^{5}}$"
+       "$\\displaystyle \\begin{cases} 0 & \\text{for}\\: x \\leq 0 \\\\\\frac{- \\epsilon_{v}^{2} \\mu_{s} + 8 \\epsilon_{v} \\mu_{k} x - 8 \\epsilon_{v} \\mu_{s} x - 6 \\mu_{k} x^{2} + 6 \\mu_{s} x^{2}}{\\epsilon_{v}^{4} x} & \\text{for}\\: x \\leq \\frac{\\epsilon_{v}}{2} \\\\\\frac{- \\epsilon_{v}^{2} \\mu_{k} - 2 \\left(\\epsilon_{v} - x\\right)^{2} \\left(- \\mu_{k} + \\mu_{s}\\right) + 4 \\left(\\epsilon_{v} - x\\right) \\left(2 \\epsilon_{v} - x\\right) \\left(\\mu_{k} - \\mu_{s}\\right)}{\\epsilon_{v}^{4} x} & \\text{for}\\: \\epsilon_{v} \\geq x \\\\- \\frac{\\mu_{k}}{\\epsilon_{v}^{2} x} & \\text{for}\\: x \\leq 2 \\epsilon_{v} \\\\0 & \\text{otherwise} \\end{cases}$"
       ],
       "text/plain": [
-       "2*\\mu_s*x/\\epsilon_v - \\mu_s*x**2/\\epsilon_v**2 + 6*\\mu_k*x**3/\\epsilon_v**3 - 6*\\mu_s*x**3/\\epsilon_v**3 - 7*\\mu_k*x**4/\\epsilon_v**4 + 7*\\mu_s*x**4/\\epsilon_v**4 + 2*\\mu_k*x**5/\\epsilon_v**5 - 2*\\mu_s*x**5/\\epsilon_v**5"
+       "Piecewise((0, x <= 0), ((-\\epsilon_v**2*\\mu_s + 8*\\epsilon_v*\\mu_k*x - 8*\\epsilon_v*\\mu_s*x - 6*\\mu_k*x**2 + 6*\\mu_s*x**2)/(\\epsilon_v**4*x), x <= \\epsilon_v/2), ((-\\epsilon_v**2*\\mu_k - 2*(\\epsilon_v - x)**2*(-\\mu_k + \\mu_s) + 4*(\\epsilon_v - x)*(2*\\epsilon_v - x)*(\\mu_k - \\mu_s))/(\\epsilon_v**4*x), \\epsilon_v >= x), (-\\mu_k/(\\epsilon_v**2*x), x <= 2*\\epsilon_v), (0, True))"
       ]
      },
-     "execution_count": 587,
+     "execution_count": 29,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "(mu(x) * a1(x)).expand()"
+    "((sym_mu_a1.diff(x) * x - sym_mu_a1) / x**3).simplify()"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 588,
+   "execution_count": 30,
    "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle 8 \\mu_{k} x^{2} z^{5} - 21 \\mu_{k} x z^{4} + 12 \\mu_{k} z^{3} - 8 \\mu_{s} x^{2} z^{5} + 21 \\mu_{s} x z^{4} - 12 \\mu_{s} z^{3} - \\frac{\\mu_{s} z^{2}}{x}$"
-      ],
-      "text/plain": [
-       "8*\\mu_k*x**2*z**5 - 21*\\mu_k*x*z**4 + 12*\\mu_k*z**3 - 8*\\mu_s*x**2*z**5 + 21*\\mu_s*x*z**4 - 12*\\mu_s*z**3 - \\mu_s*z**2/x"
-      ]
-     },
-     "metadata": {},
-     "output_type": "display_data"
-    },
-    {
-     "data": {
-      "text/latex": [
-       "$\\displaystyle \\operatorname{Poly}{\\left( \\left(8 \\mu_{k} x^{2} - 8 \\mu_{s} x^{2}\\right) z^{5} + \\left(- 21 \\mu_{k} x + 21 \\mu_{s} x\\right) z^{4} + \\left(12 \\mu_{k} - 12 \\mu_{s}\\right) z^{3} -  \\frac{\\mu_{s}}{x} z^{2}, z, domain=\\mathbb{Z}\\left(x, \\mu_{k}, \\mu_{s}\\right) \\right)}$"
-      ],
-      "text/plain": [
-       "Poly((8*\\mu_k*x**2 - 8*\\mu_s*x**2)*z**5 + (-21*\\mu_k*x + 21*\\mu_s*x)*z**4 + (12*\\mu_k - 12*\\mu_s)*z**3 - \\mu_s/x*z**2, z, domain='ZZ(x,\\mu_k,\\mu_s)')"
-      ]
-     },
-     "execution_count": 588,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
+   "outputs": [],
    "source": [
-    "a2 = (((mu(x) * a1(x)).diff(x) * x - mu(x) * a1(x)) / x**3).simplify()\n",
-    "display(a2.expand().subs({1/eps_v: Symbol('z')}))\n",
-    "Poly(a2.expand().subs({1/eps_v: Symbol('z')}), Symbol('z'))"
+    "# plot(((sym_mu_a1.diff(x) * x - sym_mu_a1) / x**3).simplify(), title=\"a2\")"
    ]
   },
   {
diff --git a/src/ipc/adhesion/adhesion.cpp b/src/ipc/adhesion/adhesion.cpp
index 336d7236a..17dcf161b 100644
--- a/src/ipc/adhesion/adhesion.cpp
+++ b/src/ipc/adhesion/adhesion.cpp
@@ -135,15 +135,25 @@ double smooth_mu_a0(
 {
     assert(eps_a > 0);
     const double delta_mu = mu_k - mu_s;
-    if (mu_s == mu_k || y <= 0 || y >= eps_a) {
+    if (y <= 0) {
+        return 0;
+    } else if (mu_s == mu_k || y >= eps_a) {
         // If the static and kinetic friction coefficients are equal, simplify.
-        const double c = (7 / 30.) * eps_a * delta_mu;
+        const double c = (11 / 48.) * eps_a * delta_mu;
         return mu_k * tangential_adhesion_f0(y, eps_a) - c;
     } else {
         const double z = y / eps_a;
-        return y * z
-            * (z * (z * (z * (z / 3 - 1.4) + 1.5) * delta_mu - mu_s / 3)
-               + mu_s);
+        if (y < 0.5 * eps_a) {
+            return y * z
+                * (z * (z * (1 - 0.4 * z) * delta_mu - mu_s / 3.0) + mu_s);
+        } else {
+            return y * z
+                * (z
+                       * (z * (0.4 * z - 2) * delta_mu + 3 * mu_k
+                          - (10.0 / 3.0) * mu_s)
+                   - mu_k + 2 * mu_s)
+                + (3.0 / 80.0) * eps_a * delta_mu;
+        }
     }
 }
 
@@ -174,14 +184,20 @@ double smooth_mu_a2_x_minus_mu_a1_over_x3(
     const double y, const double mu_s, const double mu_k, const double eps_a)
 {
     assert(eps_a > 0);
-    if (mu_s == mu_k || y <= 0 || y >= eps_a) {
+    assert(y >= 0);
+    if (mu_s == mu_k || y >= eps_a) {
         // If the static and kinetic friction coefficients are equal, simplify.
         return mu_k * tangential_adhesion_f2_x_minus_f1_over_x3(y, eps_a);
     } else {
         const double delta_mu = mu_k - mu_s;
         const double z = 1 / eps_a;
-        return z * z
-            * (z * (z * y * (z * y * 8 - 21) + 12) * delta_mu - mu_s / y);
+        if (y < 0.5 * eps_a) {
+            return z * z * (z * (8 - 6 * z * y) * delta_mu - mu_s / y);
+        } else {
+            return z * z
+                * (z * (6 * z * y - 16) * delta_mu
+                   + (9 * mu_k - 10 * mu_s) / y);
+        }
     }
 }
 
diff --git a/src/ipc/friction/smooth_mu.cpp b/src/ipc/friction/smooth_mu.cpp
index 8416aa9c2..4c4a88625 100644
--- a/src/ipc/friction/smooth_mu.cpp
+++ b/src/ipc/friction/smooth_mu.cpp
@@ -15,10 +15,12 @@ double smooth_mu(
         // If the static and kinetic friction coefficients are equal, simplify.
         return mu_k;
     } else {
-        const double y_over_eps_v = std::abs(y) / eps_v;
-        return (mu_k - mu_s) * (3 - 2 * y_over_eps_v) * y_over_eps_v
-            * y_over_eps_v
-            + mu_s;
+        const double z = std::abs(y) / eps_v;
+        if (std::abs(y) < 0.5 * eps_v) {
+            return 2 * (mu_k - mu_s) * z * z + mu_s;
+        } else {
+            return -2 * (mu_k - mu_s) * (z * (z - 2) + 1) + mu_k;
+        }
     }
 }
 
@@ -30,8 +32,12 @@ double smooth_mu_derivative(
         // If the static and kinetic friction coefficients are equal, simplify.
         return 0;
     } else {
-        const double y_over_eps_v = std::abs(y) / eps_v;
-        return (mu_k - mu_s) * (6 - 6 * y_over_eps_v) * y_over_eps_v / eps_v;
+        const double z = std::abs(y) / eps_v;
+        if (std::abs(y) < 0.5 * eps_v) {
+            return 4 * (mu_k - mu_s) * z / eps_v;
+        } else {
+            return -4 * (mu_k - mu_s) * (z - 1) / eps_v;
+        }
     }
 }
 
@@ -44,11 +50,19 @@ double smooth_mu_f0(
         return mu_k * smooth_friction_f0(y, eps_v);
     } else {
         const double delta_mu = mu_k - mu_s;
-        const double c = eps_v * (17 * mu_k - 7 * mu_s) / 30;
         const double z = std::abs(y) / eps_v;
-        return y * z
-            * (z * (z * (z * (z / 3 - 1.4) + 1.5) * delta_mu - mu_s / 3) + mu_s)
-            + c;
+        if (std::abs(y) < 0.5 * eps_v) {
+            return y * z
+                * (z * (z * (1 - 0.4 * z) * delta_mu - mu_s / 3.0) + mu_s)
+                + (9.0 / 16.0) * eps_v * mu_k - (11.0 / 48.0) * eps_v * mu_s;
+        } else {
+            return y * z
+                * (z
+                       * (z * (0.4 * z - 2) * delta_mu
+                          + (3 * mu_k - (10.0 / 3.0) * mu_s))
+                   + (2 * mu_s - mu_k))
+                + 0.6 * eps_v * mu_k - (4.0 / 15.0) * eps_v * mu_s;
+        }
     }
 }
 
@@ -82,13 +96,19 @@ double smooth_mu_f2_x_minus_mu_f1_over_x3(
 {
     assert(eps_v > 0);
     if (mu_s == mu_k || std::abs(y) >= eps_v) {
-        // If the static and kinetic friction coefficients are equal, simplify.
+        // If the static and kinetic friction coefficients are equal,
+        // simplify.
         return mu_k * smooth_friction_f2_x_minus_f1_over_x3(y, eps_v);
     } else {
         const double delta_mu = mu_k - mu_s;
         const double z = 1 / eps_v;
-        return z * z
-            * (z * (z * y * (z * y * 8 - 21) + 12) * delta_mu - mu_s / y);
+        if (std::abs(y) < 0.5 * eps_v) {
+            return z * z * (z * (8 - 6 * y * z) * delta_mu - mu_s / y);
+        } else {
+            return z * z
+                * (z * (6 * y * z - 16) * delta_mu
+                   + (9 * mu_k - 10 * mu_s) / y);
+        }
     }
 }