Average Error: 0.1 → 0.0
Time: 10.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)
double f(double x, double y, double z) {
        double r9854315 = 1.0;
        double r9854316 = 4.0;
        double r9854317 = x;
        double r9854318 = y;
        double r9854319 = 0.25;
        double r9854320 = r9854318 * r9854319;
        double r9854321 = r9854317 + r9854320;
        double r9854322 = z;
        double r9854323 = r9854321 - r9854322;
        double r9854324 = r9854316 * r9854323;
        double r9854325 = r9854324 / r9854318;
        double r9854326 = r9854315 + r9854325;
        return r9854326;
}
↓
double f(double x, double y, double z) {
        double r9854327 = 4.0;
        double r9854328 = x;
        double r9854329 = y;
        double r9854330 = r9854328 / r9854329;
        double r9854331 = z;
        double r9854332 = r9854331 / r9854329;
        double r9854333 = r9854330 - r9854332;
        double r9854334 = 2.0;
        double r9854335 = fma(r9854327, r9854333, r9854334);
        return r9854335;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Simplified0.0

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

  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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))