Average Error: 0.2 → 0.0
Time: 1.7s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x}{z} - \frac{y}{z}, -2\right)
double f(double x, double y, double z) {
        double r1438920 = 4.0;
        double r1438921 = x;
        double r1438922 = y;
        double r1438923 = r1438921 - r1438922;
        double r1438924 = z;
        double r1438925 = 0.5;
        double r1438926 = r1438924 * r1438925;
        double r1438927 = r1438923 - r1438926;
        double r1438928 = r1438920 * r1438927;
        double r1438929 = r1438928 / r1438924;
        return r1438929;
}
↓
double f(double x, double y, double z) {
        double r1438930 = 4.0;
        double r1438931 = x;
        double r1438932 = z;
        double r1438933 = r1438931 / r1438932;
        double r1438934 = y;
        double r1438935 = r1438934 / r1438932;
        double r1438936 = r1438933 - r1438935;
        double r1438937 = 2.0;
        double r1438938 = -r1438937;
        double r1438939 = fma(r1438930, r1438936, r1438938);
        return r1438939;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

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

Derivation

  1. Initial program 0.2

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

  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. Using strategy rm

  5. Applied div-sub0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +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))