Average Error: 0.2 → 0.0
Time: 13.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)
double f(double x, double y, double z) {
        double r164521 = 1.0;
        double r164522 = 4.0;
        double r164523 = x;
        double r164524 = y;
        double r164525 = 0.75;
        double r164526 = r164524 * r164525;
        double r164527 = r164523 + r164526;
        double r164528 = z;
        double r164529 = r164527 - r164528;
        double r164530 = r164522 * r164529;
        double r164531 = r164530 / r164524;
        double r164532 = r164521 + r164531;
        return r164532;
}
↓
double f(double x, double y, double z) {
        double r164533 = 4.0;
        double r164534 = x;
        double r164535 = y;
        double r164536 = r164534 / r164535;
        double r164537 = z;
        double r164538 = r164537 / r164535;
        double r164539 = r164536 - r164538;
        double r164540 = fma(r164533, r164539, r164533);
        return r164540;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4, 0.75 - \frac{z - x}{y}, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))