Average Error: 0.2 → 0.0
Time: 1.4s
Precision: 64
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{z}, -\mathsf{fma}\left(4, \frac{y}{z}, 2\right)\right)\]
\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}
↓
\mathsf{fma}\left(4, \frac{x}{z}, -\mathsf{fma}\left(4, \frac{y}{z}, 2\right)\right)
double f(double x, double y, double z) {
        double r793579 = 4.0;
        double r793580 = x;
        double r793581 = y;
        double r793582 = r793580 - r793581;
        double r793583 = z;
        double r793584 = 0.5;
        double r793585 = r793583 * r793584;
        double r793586 = r793582 - r793585;
        double r793587 = r793579 * r793586;
        double r793588 = r793587 / r793583;
        return r793588;
}
↓
double f(double x, double y, double z) {
        double r793589 = 4.0;
        double r793590 = x;
        double r793591 = z;
        double r793592 = r793590 / r793591;
        double r793593 = y;
        double r793594 = r793593 / r793591;
        double r793595 = 2.0;
        double r793596 = fma(r793589, r793594, r793595);
        double r793597 = -r793596;
        double r793598 = fma(r793589, r793592, r793597);
        return r793598;
}

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. Simplified0.3

    \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(x - \mathsf{fma}\left(0.5, z, y\right)\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{4 \cdot \frac{x}{z} - \left(4 \cdot \frac{y}{z} + 2\right)}\]

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020001 +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))