Average Error: 0.1 → 0.0
Time: 9.6s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)
double f(double x, double y, double z) {
        double r287720 = 1.0;
        double r287721 = 4.0;
        double r287722 = x;
        double r287723 = y;
        double r287724 = 0.75;
        double r287725 = r287723 * r287724;
        double r287726 = r287722 + r287725;
        double r287727 = z;
        double r287728 = r287726 - r287727;
        double r287729 = r287721 * r287728;
        double r287730 = r287729 / r287723;
        double r287731 = r287720 + r287730;
        return r287731;
}
↓
double f(double x, double y, double z) {
        double r287732 = 4.0;
        double r287733 = x;
        double r287734 = z;
        double r287735 = r287733 - r287734;
        double r287736 = y;
        double r287737 = r287735 / r287736;
        double r287738 = fma(r287732, r287737, r287732);
        return r287738;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75 - \frac{z - x}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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)))