@luma.gl/shadertools/src/modules/fp64/fp64-arithmetic.glsl.js

Shader module system for luma.gl

// Copyright (c) 2015 - 2017 Uber Technologies, Inc.
//
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//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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// THE SOFTWARE.

export default `\
uniform float ONE;

/*
About LUMA_FP64_CODE_ELIMINATION_WORKAROUND

The purpose of this workaround is to prevent shader compilers from
optimizing away necessary arithmetic operations by swapping their sequences
or transform the equation to some 'equivalent' from.

The method is to multiply an artifical variable, ONE, which will be known to
the compiler to be 1 only at runtime. The whole expression is then represented
as a polynomial with respective to ONE. In the coefficients of all terms, only one a
and one b should appear

err = (a + b) * ONE^6 - a * ONE^5 - (a + b) * ONE^4 + a * ONE^3 - b - (a + b) * ONE^2 + a * ONE
*/

// Divide float number to high and low floats to extend fraction bits
vec2 split(float a) {
  const float SPLIT = 4097.0;
  float t = a * SPLIT;
#if defined(LUMA_FP64_CODE_ELIMINATION_WORKAROUND)
  float a_hi = t * ONE - (t - a);
  float a_lo = a * ONE - a_hi;
#else
  float a_hi = t - (t - a);
  float a_lo = a - a_hi;
#endif
  return vec2(a_hi, a_lo);
}

// Divide float number again when high float uses too many fraction bits
vec2 split2(vec2 a) {
  vec2 b = split(a.x);
  b.y += a.y;
  return b;
}

// Special sum operation when a > b
vec2 quickTwoSum(float a, float b) {
#if defined(LUMA_FP64_CODE_ELIMINATION_WORKAROUND)
  float sum = (a + b) * ONE;
  float err = b - (sum - a) * ONE;
#else
  float sum = a + b;
  float err = b - (sum - a);
#endif
  return vec2(sum, err);
}

// General sum operation
vec2 twoSum(float a, float b) {
  float s = (a + b);
#if defined(LUMA_FP64_CODE_ELIMINATION_WORKAROUND)
  float v = (s * ONE - a) * ONE;
  float err = (a - (s - v) * ONE) * ONE * ONE * ONE + (b - v);
#else
  float v = s - a;
  float err = (a - (s - v)) + (b - v);
#endif
  return vec2(s, err);
}

vec2 twoSub(float a, float b) {
  float s = (a - b);
#if defined(LUMA_FP64_CODE_ELIMINATION_WORKAROUND)
  float v = (s * ONE - a) * ONE;
  float err = (a - (s - v) * ONE) * ONE * ONE * ONE - (b + v);
#else
  float v = s - a;
  float err = (a - (s - v)) - (b + v);
#endif
  return vec2(s, err);
}

vec2 twoSqr(float a) {
  float prod = a * a;
  vec2 a_fp64 = split(a);
#if defined(LUMA_FP64_CODE_ELIMINATION_WORKAROUND)
  float err = ((a_fp64.x * a_fp64.x - prod) * ONE + 2.0 * a_fp64.x *
    a_fp64.y * ONE * ONE) + a_fp64.y * a_fp64.y * ONE * ONE * ONE;
#else
  float err = ((a_fp64.x * a_fp64.x - prod) + 2.0 * a_fp64.x * a_fp64.y) + a_fp64.y * a_fp64.y;
#endif
  return vec2(prod, err);
}

vec2 twoProd(float a, float b) {
  float prod = a * b;
  vec2 a_fp64 = split(a);
  vec2 b_fp64 = split(b);
  float err = ((a_fp64.x * b_fp64.x - prod) + a_fp64.x * b_fp64.y +
    a_fp64.y * b_fp64.x) + a_fp64.y * b_fp64.y;
  return vec2(prod, err);
}

vec2 sum_fp64(vec2 a, vec2 b) {
  vec2 s, t;
  s = twoSum(a.x, b.x);
  t = twoSum(a.y, b.y);
  s.y += t.x;
  s = quickTwoSum(s.x, s.y);
  s.y += t.y;
  s = quickTwoSum(s.x, s.y);
  return s;
}

vec2 sub_fp64(vec2 a, vec2 b) {
  vec2 s, t;
  s = twoSub(a.x, b.x);
  t = twoSub(a.y, b.y);
  s.y += t.x;
  s = quickTwoSum(s.x, s.y);
  s.y += t.y;
  s = quickTwoSum(s.x, s.y);
  return s;
}

vec2 mul_fp64(vec2 a, vec2 b) {
  vec2 prod = twoProd(a.x, b.x);
  // y component is for the error
  prod.y += a.x * b.y;
#if defined(LUMA_FP64_HIGH_BITS_OVERFLOW_WORKAROUND)
  prod = split2(prod);
#endif
  prod = quickTwoSum(prod.x, prod.y);
  prod.y += a.y * b.x;
#if defined(LUMA_FP64_HIGH_BITS_OVERFLOW_WORKAROUND)
  prod = split2(prod);
#endif
  prod = quickTwoSum(prod.x, prod.y);
  return prod;
}

vec2 div_fp64(vec2 a, vec2 b) {
  float xn = 1.0 / b.x;
#if defined(LUMA_FP64_HIGH_BITS_OVERFLOW_WORKAROUND)
  vec2 yn = mul_fp64(a, vec2(xn, 0));
#else
  vec2 yn = a * xn;
#endif
  float diff = (sub_fp64(a, mul_fp64(b, yn))).x;
  vec2 prod = twoProd(xn, diff);
  return sum_fp64(yn, prod);
}

vec2 sqrt_fp64(vec2 a) {
  if (a.x == 0.0 && a.y == 0.0) return vec2(0.0, 0.0);
  if (a.x < 0.0) return vec2(0.0 / 0.0, 0.0 / 0.0);

  float x = 1.0 / sqrt(a.x);
  float yn = a.x * x;
#if defined(LUMA_FP64_CODE_ELIMINATION_WORKAROUND)
  vec2 yn_sqr = twoSqr(yn) * ONE;
#else
  vec2 yn_sqr = twoSqr(yn);
#endif
  float diff = sub_fp64(a, yn_sqr).x;
  vec2 prod = twoProd(x * 0.5, diff);
#if defined(LUMA_FP64_HIGH_BITS_OVERFLOW_WORKAROUND)
  return sum_fp64(split(yn), prod);
#else
  return sum_fp64(vec2(yn, 0.0), prod);
#endif
}
`;