Average Error: 0.1 → 0.0
Time: 1.6s
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 r905389 = 4.0;
        double r905390 = x;
        double r905391 = y;
        double r905392 = r905390 - r905391;
        double r905393 = z;
        double r905394 = 0.5;
        double r905395 = r905393 * r905394;
        double r905396 = r905392 - r905395;
        double r905397 = r905389 * r905396;
        double r905398 = r905397 / r905393;
        return r905398;
}
↓
double f(double x, double y, double z) {
        double r905399 = 4.0;
        double r905400 = x;
        double r905401 = y;
        double r905402 = r905400 - r905401;
        double r905403 = z;
        double r905404 = r905402 / r905403;
        double r905405 = 2.0;
        double r905406 = -r905405;
        double r905407 = fma(r905399, r905404, r905406);
        return r905407;
}

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 2020060 +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))