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

↓

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

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

↓

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

double f(double x, double y, double z) {
        double r221879 = 1.0;
        double r221880 = 4.0;
        double r221881 = x;
        double r221882 = y;
        double r221883 = 0.25;
        double r221884 = r221882 * r221883;
        double r221885 = r221881 + r221884;
        double r221886 = z;
        double r221887 = r221885 - r221886;
        double r221888 = r221880 * r221887;
        double r221889 = r221888 / r221882;
        double r221890 = r221879 + r221889;
        return r221890;
}

↓

double f(double x, double y, double z) {
        double r221891 = 4.0;
        double r221892 = x;
        double r221893 = y;
        double r221894 = r221892 / r221893;
        double r221895 = 2.0;
        double r221896 = z;
        double r221897 = r221896 / r221893;
        double r221898 = r221891 * r221897;
        double r221899 = r221895 - r221898;
        double r221900 = fma(r221891, r221894, r221899);
        return r221900;
}

Derivation

Initial program 0.1
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
Taylor expanded around 0 0.0
\[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]
Simplified0.0
\[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{x}{y}, 2 - 4 \cdot \frac{z}{y}\right)}\]
Final simplification0.0
\[\leadsto \mathsf{fma}\left(4, \frac{x}{y}, 2 - 4 \cdot \frac{z}{y}\right)\]

Reproduce

herbie shell --seed 2020018 +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)))

Error

Derivation

Reproduce