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

↓

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

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 r319246 = 1.0;
        double r319247 = 4.0;
        double r319248 = x;
        double r319249 = y;
        double r319250 = 0.25;
        double r319251 = r319249 * r319250;
        double r319252 = r319248 + r319251;
        double r319253 = z;
        double r319254 = r319252 - r319253;
        double r319255 = r319247 * r319254;
        double r319256 = r319255 / r319249;
        double r319257 = r319246 + r319256;
        return r319257;
}

↓

double f(double x, double y, double z) {
        double r319258 = 4.0;
        double r319259 = x;
        double r319260 = y;
        double r319261 = r319259 / r319260;
        double r319262 = 2.0;
        double r319263 = z;
        double r319264 = r319263 / r319260;
        double r319265 = r319258 * r319264;
        double r319266 = r319262 - r319265;
        double r319267 = fma(r319258, r319261, r319266);
        return r319267;
}

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 2020060 +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