Average Error: 0.2 → 0.0
Time: 1.3s
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 r866582 = 4.0;
        double r866583 = x;
        double r866584 = y;
        double r866585 = r866583 - r866584;
        double r866586 = z;
        double r866587 = 0.5;
        double r866588 = r866586 * r866587;
        double r866589 = r866585 - r866588;
        double r866590 = r866582 * r866589;
        double r866591 = r866590 / r866586;
        return r866591;
}
↓
double f(double x, double y, double z) {
        double r866592 = 4.0;
        double r866593 = x;
        double r866594 = y;
        double r866595 = r866593 - r866594;
        double r866596 = z;
        double r866597 = r866595 / r866596;
        double r866598 = 2.0;
        double r866599 = -r866598;
        double r866600 = fma(r866592, r866597, r866599);
        return r866600;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020021 +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))