Average Error: 0.0 → 0.0
Time: 22.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)
double f(double x, double y, double z) {
        double r119511 = 1.0;
        double r119512 = 4.0;
        double r119513 = x;
        double r119514 = y;
        double r119515 = 0.25;
        double r119516 = r119514 * r119515;
        double r119517 = r119513 + r119516;
        double r119518 = z;
        double r119519 = r119517 - r119518;
        double r119520 = r119512 * r119519;
        double r119521 = r119520 / r119514;
        double r119522 = r119511 + r119521;
        return r119522;
}
↓
double f(double x, double y, double z) {
        double r119523 = 4.0;
        double r119524 = x;
        double r119525 = z;
        double r119526 = r119524 - r119525;
        double r119527 = y;
        double r119528 = r119526 / r119527;
        double r119529 = 2.0;
        double r119530 = fma(r119523, r119528, r119529);
        return r119530;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.0

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))