Average Error: 0.1 → 0.0
Time: 14.1s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)
double f(double x, double y, double z) {
        double r180831 = 1.0;
        double r180832 = 4.0;
        double r180833 = x;
        double r180834 = y;
        double r180835 = 0.25;
        double r180836 = r180834 * r180835;
        double r180837 = r180833 + r180836;
        double r180838 = z;
        double r180839 = r180837 - r180838;
        double r180840 = r180832 * r180839;
        double r180841 = r180840 / r180834;
        double r180842 = r180831 + r180841;
        return r180842;
}
↓
double f(double x, double y, double z) {
        double r180843 = 4.0;
        double r180844 = x;
        double r180845 = z;
        double r180846 = r180844 - r180845;
        double r180847 = y;
        double r180848 = r180846 / r180847;
        double r180849 = 2.0;
        double r180850 = fma(r180843, r180848, r180849);
        return r180850;
}

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. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +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)))