Average Error: 0.1 → 0.0
Time: 10.5s
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 r235898 = 1.0;
        double r235899 = 4.0;
        double r235900 = x;
        double r235901 = y;
        double r235902 = 0.25;
        double r235903 = r235901 * r235902;
        double r235904 = r235900 + r235903;
        double r235905 = z;
        double r235906 = r235904 - r235905;
        double r235907 = r235899 * r235906;
        double r235908 = r235907 / r235901;
        double r235909 = r235898 + r235908;
        return r235909;
}
↓
double f(double x, double y, double z) {
        double r235910 = 4.0;
        double r235911 = x;
        double r235912 = y;
        double r235913 = r235911 / r235912;
        double r235914 = z;
        double r235915 = r235914 / r235912;
        double r235916 = r235913 - r235915;
        double r235917 = 2.0;
        double r235918 = fma(r235910, r235916, r235917);
        return r235918;
}

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

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

  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. Using strategy rm

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020045 +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)))