Average Error: 0.1 → 0.2
Time: 2.1s
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 r262084 = 1.0;
        double r262085 = 4.0;
        double r262086 = x;
        double r262087 = y;
        double r262088 = 0.25;
        double r262089 = r262087 * r262088;
        double r262090 = r262086 + r262089;
        double r262091 = z;
        double r262092 = r262090 - r262091;
        double r262093 = r262085 * r262092;
        double r262094 = r262093 / r262087;
        double r262095 = r262084 + r262094;
        return r262095;
}
↓
double f(double x, double y, double z) {
        double r262096 = 4.0;
        double r262097 = y;
        double r262098 = r262096 / r262097;
        double r262099 = x;
        double r262100 = z;
        double r262101 = r262099 - r262100;
        double r262102 = 2.0;
        double r262103 = fma(r262098, r262101, r262102);
        return r262103;
}

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 2020033 +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)))