Average Error: 0.1 → 0.0
Time: 13.2s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)
double f(double x, double y, double z) {
        double r132791 = 1.0;
        double r132792 = 4.0;
        double r132793 = x;
        double r132794 = y;
        double r132795 = 0.25;
        double r132796 = r132794 * r132795;
        double r132797 = r132793 + r132796;
        double r132798 = z;
        double r132799 = r132797 - r132798;
        double r132800 = r132792 * r132799;
        double r132801 = r132800 / r132794;
        double r132802 = r132791 + r132801;
        return r132802;
}
↓
double f(double x, double y, double z) {
        double r132803 = 4.0;
        double r132804 = x;
        double r132805 = z;
        double r132806 = r132804 - r132805;
        double r132807 = y;
        double r132808 = r132806 / r132807;
        double r132809 = 2.0;
        double r132810 = fma(r132803, r132808, r132809);
        return r132810;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)\]

Reproduce

herbie shell --seed 2019303 +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)))