Average Error: 0.1 → 0.0
Time: 1.2s
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 r906981 = 4.0;
        double r906982 = x;
        double r906983 = y;
        double r906984 = r906982 - r906983;
        double r906985 = z;
        double r906986 = 0.5;
        double r906987 = r906985 * r906986;
        double r906988 = r906984 - r906987;
        double r906989 = r906981 * r906988;
        double r906990 = r906989 / r906985;
        return r906990;
}
↓
double f(double x, double y, double z) {
        double r906991 = 4.0;
        double r906992 = x;
        double r906993 = y;
        double r906994 = r906992 - r906993;
        double r906995 = z;
        double r906996 = r906994 / r906995;
        double r906997 = 2.0;
        double r906998 = -r906997;
        double r906999 = fma(r906991, r906996, r906998);
        return r906999;
}

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 2020046 +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))