Average Error: 0.1 → 0.0
Time: 1.8s
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 r826890 = 4.0;
        double r826891 = x;
        double r826892 = y;
        double r826893 = r826891 - r826892;
        double r826894 = z;
        double r826895 = 0.5;
        double r826896 = r826894 * r826895;
        double r826897 = r826893 - r826896;
        double r826898 = r826890 * r826897;
        double r826899 = r826898 / r826894;
        return r826899;
}
↓
double f(double x, double y, double z) {
        double r826900 = 4.0;
        double r826901 = x;
        double r826902 = y;
        double r826903 = r826901 - r826902;
        double r826904 = z;
        double r826905 = r826903 / r826904;
        double r826906 = 2.0;
        double r826907 = -r826906;
        double r826908 = fma(r826900, r826905, r826907);
        return r826908;
}

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 2020083 +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))