Average Error: 0.3 → 0.0
Time: 20.8s
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 r173089 = 1.0;
        double r173090 = 4.0;
        double r173091 = x;
        double r173092 = y;
        double r173093 = 0.75;
        double r173094 = r173092 * r173093;
        double r173095 = r173091 + r173094;
        double r173096 = z;
        double r173097 = r173095 - r173096;
        double r173098 = r173090 * r173097;
        double r173099 = r173098 / r173092;
        double r173100 = r173089 + r173099;
        return r173100;
}
↓
double f(double x, double y, double z) {
        double r173101 = 4.0;
        double r173102 = x;
        double r173103 = z;
        double r173104 = r173102 - r173103;
        double r173105 = y;
        double r173106 = r173104 / r173105;
        double r173107 = fma(r173101, r173106, r173101);
        return r173107;
}

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. Simplified0.0

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

  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 2019306 +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)))