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

↓

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

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 r227051 = 1.0;
        double r227052 = 4.0;
        double r227053 = x;
        double r227054 = y;
        double r227055 = 0.25;
        double r227056 = r227054 * r227055;
        double r227057 = r227053 + r227056;
        double r227058 = z;
        double r227059 = r227057 - r227058;
        double r227060 = r227052 * r227059;
        double r227061 = r227060 / r227054;
        double r227062 = r227051 + r227061;
        return r227062;
}

↓

double f(double x, double y, double z) {
        double r227063 = 4.0;
        double r227064 = x;
        double r227065 = y;
        double r227066 = r227064 / r227065;
        double r227067 = 2.0;
        double r227068 = z;
        double r227069 = r227068 / r227065;
        double r227070 = r227063 * r227069;
        double r227071 = r227067 - r227070;
        double r227072 = fma(r227063, r227066, r227071);
        return r227072;
}

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 2020047 +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