Average Error: 0.1 → 0.0
Time: 13.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[1 + \mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 1\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
1 + \mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 1\right)
double f(double x, double y, double z) {
        double r241904 = 1.0;
        double r241905 = 4.0;
        double r241906 = x;
        double r241907 = y;
        double r241908 = 0.25;
        double r241909 = r241907 * r241908;
        double r241910 = r241906 + r241909;
        double r241911 = z;
        double r241912 = r241910 - r241911;
        double r241913 = r241905 * r241912;
        double r241914 = r241913 / r241907;
        double r241915 = r241904 + r241914;
        return r241915;
}
↓
double f(double x, double y, double z) {
        double r241916 = 1.0;
        double r241917 = 4.0;
        double r241918 = x;
        double r241919 = y;
        double r241920 = r241918 / r241919;
        double r241921 = z;
        double r241922 = r241921 / r241919;
        double r241923 = r241920 - r241922;
        double r241924 = fma(r241917, r241923, r241916);
        double r241925 = r241916 + r241924;
        return r241925;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto 1 + \color{blue}{\left(\left(4 \cdot \frac{x}{y} + 1\right) - 4 \cdot \frac{z}{y}\right)}\]

  3. Simplified0.0

    \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 1\right)}\]

  4. Final simplification0.0

    \[\leadsto 1 + \mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 1\right)\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))