Average Error: 0.1 → 0.0
Time: 1.9s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[4 \cdot \frac{x - \mathsf{fma}\left(0.5, z, y\right)}{z}\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
4 \cdot \frac{x - \mathsf{fma}\left(0.5, z, y\right)}{z}
double f(double x, double y, double z) {
        double r885074 = 4.0;
        double r885075 = x;
        double r885076 = y;
        double r885077 = r885075 - r885076;
        double r885078 = z;
        double r885079 = 0.5;
        double r885080 = r885078 * r885079;
        double r885081 = r885077 - r885080;
        double r885082 = r885074 * r885081;
        double r885083 = r885082 / r885078;
        return r885083;
}
↓
double f(double x, double y, double z) {
        double r885084 = 4.0;
        double r885085 = x;
        double r885086 = 0.5;
        double r885087 = z;
        double r885088 = y;
        double r885089 = fma(r885086, r885087, r885088);
        double r885090 = r885085 - r885089;
        double r885091 = r885090 / r885087;
        double r885092 = r885084 * r885091;
        return r885092;
}

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. Simplified0.2

    \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(x - \mathsf{fma}\left(0.5, z, y\right)\right)}\]
  3. Using strategy rm

  4. Applied div-inv0.2

    \[\leadsto \color{blue}{\left(4 \cdot \frac{1}{z}\right)} \cdot \left(x - \mathsf{fma}\left(0.5, z, y\right)\right)\]

  5. Applied associate-*l*0.2

    \[\leadsto \color{blue}{4 \cdot \left(\frac{1}{z} \cdot \left(x - \mathsf{fma}\left(0.5, z, y\right)\right)\right)}\]

  6. Simplified0.0

    \[\leadsto 4 \cdot \color{blue}{\frac{x - \mathsf{fma}\left(0.5, z, y\right)}{z}}\]

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z))))

  (/ (* 4 (- (- x y) (* z 0.5))) z))