Average Error: 0.1 → 0.2
Time: 2.4s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{4}{y}, x - z, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{4}{y}, x - z, 2\right)
double f(double x, double y, double z) {
        double r294311 = 1.0;
        double r294312 = 4.0;
        double r294313 = x;
        double r294314 = y;
        double r294315 = 0.25;
        double r294316 = r294314 * r294315;
        double r294317 = r294313 + r294316;
        double r294318 = z;
        double r294319 = r294317 - r294318;
        double r294320 = r294312 * r294319;
        double r294321 = r294320 / r294314;
        double r294322 = r294311 + r294321;
        return r294322;
}
↓
double f(double x, double y, double z) {
        double r294323 = 4.0;
        double r294324 = y;
        double r294325 = r294323 / r294324;
        double r294326 = x;
        double r294327 = z;
        double r294328 = r294326 - r294327;
        double r294329 = 2.0;
        double r294330 = fma(r294325, r294328, r294329);
        return r294330;
}

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. Simplified0.2

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

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2020100 +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)))