Average Error: 0.1 → 0.0
Time: 9.9s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x - y}{z}, -2\right)
double f(double x, double y, double z) {
        double r797141 = 4.0;
        double r797142 = x;
        double r797143 = y;
        double r797144 = r797142 - r797143;
        double r797145 = z;
        double r797146 = 0.5;
        double r797147 = r797145 * r797146;
        double r797148 = r797144 - r797147;
        double r797149 = r797141 * r797148;
        double r797150 = r797149 / r797145;
        return r797150;
}
↓
double f(double x, double y, double z) {
        double r797151 = 4.0;
        double r797152 = x;
        double r797153 = y;
        double r797154 = r797152 - r797153;
        double r797155 = z;
        double r797156 = r797154 / r797155;
        double r797157 = 2.0;
        double r797158 = -r797157;
        double r797159 = fma(r797151, r797156, r797158);
        return r797159;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.0
Herbie0.0
\[4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right)\]

Derivation

  1. Initial program 0.1

    \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"

  :herbie-target
  (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))