Average Error: 0.1 → 0.0
Time: 3.2s
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 r881960 = 4.0;
        double r881961 = x;
        double r881962 = y;
        double r881963 = r881961 - r881962;
        double r881964 = z;
        double r881965 = 0.5;
        double r881966 = r881964 * r881965;
        double r881967 = r881963 - r881966;
        double r881968 = r881960 * r881967;
        double r881969 = r881968 / r881964;
        return r881969;
}
↓
double f(double x, double y, double z) {
        double r881970 = 4.0;
        double r881971 = x;
        double r881972 = z;
        double r881973 = r881971 / r881972;
        double r881974 = y;
        double r881975 = r881974 / r881972;
        double r881976 = r881973 - r881975;
        double r881977 = 2.0;
        double r881978 = -r881977;
        double r881979 = fma(r881970, r881976, r881978);
        return r881979;
}

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. 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 2020056 +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))