{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# `mtflib` Tutorial: Advanced Functionality\n",
    "\n",
    "This notebook explores advanced topics in `mtflib`, including substitution, composition, accessing coefficients, and the use of other elementary functions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Initialization\n",
    "\n",
    "As always, we begin by initializing the global settings for `mtflib`."
   ]
  },
  {
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initializing MTF globals with: _MAX_ORDER=8, _MAX_DIMENSION=3\n",
      "Loading/Precomputing Taylor coefficients up to order 8\n",
      "Global precomputed coefficients loading/generation complete.\n",
      "Size of precomputed_coefficients dictionary in memory: 464 bytes, 0.45 KB, 0.00 MB\n",
      "MTF globals initialized: _MAX_ORDER=8, _MAX_DIMENSION=3, _INITIALIZED=True\n",
      "Max coefficient count (order=8, nvars=3): 165\n",
      "Precomputed coefficients loaded and ready for use.\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import sympy as sp\n",
    "from IPython.display import display\n",
    "\n",
    "from mtflib import ComplexMultivariateTaylorFunction, mtf\n",
    "\n",
    "if not mtf.get_mtf_initialized_status():\n",
    "    mtf.initialize_mtf(max_order=8, max_dimension=3)\n",
    "else:\n",
    "    print(\"MTF globals are already initialized.\")\n",
    "\n",
    "# Define some variables for our examples\n",
    "x = mtf.var(1)\n",
    "y = mtf.var(2)\n",
    "z = mtf.var(3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Substitution and Composition\n",
    "\n",
    "A powerful feature of `mtflib` is the ability to substitute variables with constants or even other Taylor functions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Substitution with a Constant\n",
    "\n",
    "Replace a variable in a Taylor function with a constant value."
   ]
  },
  {
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Original function f(x, y, z):\n",
      "         Coefficient  Order  Exponents\n",
      "0 1.000000000000e+00      1  (1, 0, 0)\n",
      "1 1.000000000000e+00      1  (0, 1, 0)\n",
      "2 1.000000000000e+00      4  (0, 0, 4)\n",
      "\n",
      "\n",
      "After substituting x=0.1, we get g(y, z):\n",
      "                             Coefficient  Order  Exponents\n",
      "0 1.000000000000e-01+0.000000000000e+00j      0  (0, 0, 0)\n",
      "1 1.000000000000e+00+0.000000000000e+00j      1  (0, 1, 0)\n",
      "2 1.000000000000e+00+0.000000000000e+00j      4  (0, 0, 4)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "f_xyz = x + y + z**4\n",
    "print(f\"Original function f(x, y, z):\\n{f_xyz}\\n\")\n",
    "\n",
    "# Substitute x with 0.1\n",
    "g_yz = f_xyz.substitute_variable(1, 0.1)\n",
    "print(f\"After substituting x=0.1, we get g(y, z):\\n{g_yz}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Composition with other Taylor Functions\n",
    "\n",
    "Substitute variables with other Taylor functions to compose them."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Composed function f(g(y), h(y,z)):\n",
      "         Coefficient  Order  Exponents\n",
      "0 2.000000000000e+00      0  (0, 0, 0)\n",
      "1 2.000000000000e+00      1  (0, 1, 0)\n",
      "2 1.000000000000e+00      2  (0, 2, 0)\n",
      "3 3.000000000000e+00      2  (0, 1, 1)\n",
      "4 3.000000000000e+00      4  (0, 2, 2)\n",
      "5 1.000000000000e+00      6  (0, 3, 3)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Outer function: f(x, y) = x^2 + y^3\n",
    "f_xy = x**2 + y**3\n",
    "\n",
    "# Substituting functions:\n",
    "# g(y) = 1 + y\n",
    "# h(y, z) = 1 + y*z\n",
    "g_y = 1 + y\n",
    "h_yz = 1 + y * z\n",
    "\n",
    "# Perform composition: f(g(y), h(y,z))\n",
    "composed_f = f_xy.compose({1: g_y, 2: h_yz})\n",
    "print(f\"Composed function f(g(y), h(y,z)):\\n{composed_f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Accessing Coefficients\n",
    "\n",
    "You can directly access the coefficients and exponents of a Taylor function."
   ]
  },
  {
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   "execution_count": 4,
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Coefficients of sin(x):\n",
      "  Exponent (np.int32(1), np.int32(0), np.int32(0)): 1.0\n",
      "  Exponent (np.int32(3), np.int32(0), np.int32(0)): -0.16666666666666666\n",
      "  Exponent (np.int32(5), np.int32(0), np.int32(0)): 0.008333333333333333\n",
      "  Exponent (np.int32(7), np.int32(0), np.int32(0)): -0.0001984126984126984\n"
     ]
    }
   ],
   "source": [
    "sin_x = mtf.sin(x)\n",
    "coefficients = {tuple(exp): coeff for exp, coeff in zip(sin_x.exponents, sin_x.coeffs)}\n",
    "\n",
    "print(\"Coefficients of sin(x):\")\n",
    "for exp, coeff in coefficients.items():\n",
    "    print(f\"  Exponent {exp}: {coeff}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Other Elementary Functions\n",
    "\n",
    "`mtflib` supports a variety of other elementary functions."
   ]
  },
  {
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   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- Example: mtf.exp(x+2*y) ---\n",
      "The following is a SymPy object, which will render beautifully in a notebook.\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 0.000198413 x^{7} - 0.00277778 x^{6} y - 0.0166667 x^{5} y^{2} + 0.00833333 x^{5} - 0.0555556 x^{4} y^{3} + 0.0833333 x^{4} y - 0.111111 x^{3} y^{4} + 0.333333 x^{3} y^{2} - 0.166667 x^{3} - 0.133333 x^{2} y^{5} + 0.666667 x^{2} y^{3} - 1.0 x^{2} y - 0.0888889 x y^{6} + 0.666667 x y^{4} - 2.0 x y^{2} + 1.0 x - 0.0253968 y^{7} + 0.266667 y^{5} - 1.33333 y^{3} + 2.0 y$"
      ],
      "text/plain": [
       "-0.000198413*x**7 - 0.00277778*x**6*y - 0.0166667*x**5*y**2 + 0.00833333*x**5 - 0.0555556*x**4*y**3 + 0.0833333*x**4*y - 0.111111*x**3*y**4 + 0.333333*x**3*y**2 - 0.166667*x**3 - 0.133333*x**2*y**5 + 0.666667*x**2*y**3 - 1.0*x**2*y - 0.0888889*x*y**6 + 0.666667*x*y**4 - 2.0*x*y**2 + 1.0*x - 0.0253968*y**7 + 0.266667*y**5 - 1.33333*y**3 + 2.0*y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- Example: mtf.exp(i*x) ---\n",
      "The following is a SymPy object for a complex function.\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.48016 \\cdot 10^{-5} x^{8} - 0.000198413 i x^{7} - 0.00138889 x^{6} + 0.00833333 i x^{5} + 0.0416667 x^{4} - 0.166667 i x^{3} - 0.5 x^{2} + 1.0 i x + 1.0$"
      ],
      "text/plain": [
       "2.48016e-5*x**8 - 0.000198413*I*x**7 - 0.00138889*x**6 + 0.00833333*I*x**5 + 0.0416667*x**4 - 0.166667*I*x**3 - 0.5*x**2 + 1.0*I*x + 1.0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "--- Example: Custom coefficient formatting ---\n",
      "The following is a SymPy object with custom rational formatting.\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{x^{6} y}{360} - \\frac{x^{5} y^{2}}{60} + \\frac{x^{5}}{120} - \\frac{x^{4} y^{3}}{18} + \\frac{x^{4} y}{12} - \\frac{x^{3} y^{4}}{9} + \\frac{x^{3} y^{2}}{3} - \\frac{x^{3}}{6} - \\frac{2 x^{2} y^{5}}{15} + \\frac{2 x^{2} y^{3}}{3} - x^{2} y - \\frac{4 x y^{6}}{45} + \\frac{2 x y^{4}}{3} - 2 x y^{2} + x - \\frac{8 y^{7}}{315} + \\frac{4 y^{5}}{15} - \\frac{4 y^{3}}{3} + 2 y$"
      ],
      "text/plain": [
       "-x**6*y/360 - x**5*y**2/60 + x**5/120 - x**4*y**3/18 + x**4*y/12 - x**3*y**4/9 + x**3*y**2/3 - x**3/6 - 2*x**2*y**5/15 + 2*x**2*y**3/3 - x**2*y - 4*x*y**6/45 + 2*x*y**4/3 - 2*x*y**2 + x - 8*y**7/315 + 4*y**5/15 - 4*y**3/3 + 2*y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"--- Example: mtf.exp(x+2*y) ---\")\n",
    "x = mtf.var(1)\n",
    "y = mtf.var(2)\n",
    "mtf_sin = mtf.sin(x + 2 * y)\n",
    "sympy_sin_expr = mtf_sin.symprint()\n",
    "print(\"The following is a SymPy object, which will render beautifully in a notebook.\")\n",
    "display(sympy_sin_expr)\n",
    "\n",
    "# --- Example with ComplexMultivariateTaylorFunction ---\n",
    "print(\"--- Example: mtf.exp(i*x) ---\")\n",
    "i = 1j\n",
    "\n",
    "x_complex = ComplexMultivariateTaylorFunction.from_variable(var_index=1, dimension=1)\n",
    "mtf_complex_exp = mtf.exp(i * x_complex)\n",
    "sympy_complex_expr = mtf_complex_exp.symprint(symbols=[\"x\"])\n",
    "print(\"The following is a SymPy object for a complex function.\")\n",
    "display(sympy_complex_expr)\n",
    "\n",
    "# --- Example with custom coefficient formatting ---\n",
    "print(\"--- Example: Custom coefficient formatting ---\")\n",
    "\n",
    "\n",
    "def custom_formatter(c, p):\n",
    "    # A simple rational formatter\n",
    "    if np.iscomplexobj(c):\n",
    "        return sp.Rational(c.real).limit_denominator(10**p) + sp.I * sp.Rational(\n",
    "            c.imag\n",
    "        ).limit_denominator(10**p)\n",
    "    else:\n",
    "        return sp.Rational(c).limit_denominator(10**p)\n",
    "\n",
    "\n",
    "# Re-initialize for the 2D mtf_sin\n",
    "x = mtf.var(1)\n",
    "y = mtf.var(2)\n",
    "mtf_sin = mtf.sin(x + 2 * y)\n",
    "\n",
    "sympy_custom_format_expr = mtf_sin.symprint(\n",
    "    precision=3, coeff_formatter=custom_formatter\n",
    ")\n",
    "print(\"The following is a SymPy object with custom rational formatting.\")\n",
    "display(sympy_custom_format_expr)"
   ]
  }
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