Average Error: 0.3 → 0.0
Time: 4.8s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r357906 = 1.0;
        double r357907 = 4.0;
        double r357908 = x;
        double r357909 = y;
        double r357910 = 0.75;
        double r357911 = r357909 * r357910;
        double r357912 = r357908 + r357911;
        double r357913 = z;
        double r357914 = r357912 - r357913;
        double r357915 = r357907 * r357914;
        double r357916 = r357915 / r357909;
        double r357917 = r357906 + r357916;
        return r357917;
}
↓
double f(double x, double y, double z) {
        double r357918 = 4.0;
        double r357919 = x;
        double r357920 = y;
        double r357921 = r357919 / r357920;
        double r357922 = z;
        double r357923 = r357922 / r357920;
        double r357924 = r357918 * r357923;
        double r357925 = r357918 - r357924;
        double r357926 = fma(r357918, r357921, r357925);
        return r357926;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.3

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))