Computing Spherical Harmonics

math_constexpr.h

#pragma once

#include <limits>
#define _USE_MATH_DEFINES
#include <math.h>

namespace framework {
  namespace math_constexpr {

    int constexpr abs(int x) {
      return x < 0 ? -x : x;
    }

    float constexpr abs(float x) {
      return x < 0 ? -x : x;
    }

    template <typename T> T constexpr cube(T x) {
      return x*x*x;
    }

    namespace detail {
      static constexpr float epsilon = 0.00001f;

      // newton raphson
      float constexpr sqrt_step(float x, float c, float p) {
        return abs(c - p) < epsilon ? c : sqrt_step(x, 0.5f * (c + x / c), c);
      }

      // triple angle formula
      float constexpr sin_step(float x) {
        return x < epsilon ? x : 3 * sin_step(x / 3.f) - 4 * cube(sin_step(x / 3.f));
      }
    }


    float constexpr pow(float b, int e) {
      return e <  0 ? 1.f / pow(b, -e) 
           : e == 0 ? 1.f 
           : b * pow(b, e - 1);
    }

    float constexpr sqrt(float x) {
      return x >= 0 && x < std::numeric_limits<float>::infinity() ? detail::sqrt_step(x, x, 0) : std::numeric_limits<float>::quiet_NaN();
    }

    float constexpr sin(float x) {
      return detail::sin_step(x < 0 ? float(M_PI) - x : x);
    }

    float constexpr cos(float x) {
      return sin(float(M_PI_2) - x);
    }

    int constexpr factorial_power(int x, int n, int h = 1) {
      return (n > 0) ? x * factorial_power(x - h, n - 1, h) : 1;
    }

    int constexpr factorial(int x, int h = 1) {
      return (x > 1) ? x * factorial(x - h, h) : 1;
    }

    float constexpr K(int l, int m) {
      return sqrt((2 * l + 1) / (4 * float(M_PI) * factorial_power(l + abs(m), abs(m) + abs(m))));
    }

    float constexpr legendre(int l, int m, float x) {
      return l == m + 1 ? x * (2 * m + 1) * legendre(m, m, x)
           : l == m     ? pow(-1, m) * factorial(2 * m - 1, 2) * pow(1 - x*x, m / 2)
                        : (x * (2 * l - 1) * legendre(l - 1, m, x) - (l + m - 1) * legendre(l - 2, m, x)) / (l - m);
    }
    
    float constexpr spherical_harmonic(int l, int m, float theta, float phi) {
      return m > 0 ? sqrt(2) * K(l, m) * cos(m*phi) * legendre(l, m, cos(theta))
           : m < 0 ? sqrt(2) * K(l, m) * sin(-m*phi) * legendre(l, -m, cos(theta))
                   : K(l, m) * legendre(l, 0, cos(theta));
    }
  }
}

/* 

Copyright (c) 2016 Edward Kmett
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions
are met:

1. Redistributions of source code must retain the above copyright
   notice, this list of conditions and the following disclaimer.

2. Redistributions in binary form must reproduce the above copyright
   notice, this list of conditions and the following disclaimer in the
   documentation and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE AUTHORS ``AS IS'' AND ANY EXPRESS OR
IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED.  IN NO EVENT SHALL THE AUTHORS OR CONTRIBUTORS BE LIABLE FOR
ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
POSSIBILITY OF SUCH DAMAGE.

*/