Average Error: 0.2 → 0.0
Time: 1.8s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r232338 = 1.0;
        double r232339 = 4.0;
        double r232340 = x;
        double r232341 = y;
        double r232342 = 0.75;
        double r232343 = r232341 * r232342;
        double r232344 = r232340 + r232343;
        double r232345 = z;
        double r232346 = r232344 - r232345;
        double r232347 = r232339 * r232346;
        double r232348 = r232347 / r232341;
        double r232349 = r232338 + r232348;
        return r232349;
}
↓
double f(double x, double y, double z) {
        double r232350 = 1.0;
        double r232351 = 4.0;
        double r232352 = x;
        double r232353 = y;
        double r232354 = r232352 / r232353;
        double r232355 = 3.0;
        double r232356 = z;
        double r232357 = r232356 / r232353;
        double r232358 = r232351 * r232357;
        double r232359 = r232355 - r232358;
        double r232360 = fma(r232351, r232354, r232359);
        double r232361 = r232350 + r232360;
        return r232361;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Taylor expanded around 0 0.0

    \[\leadsto 1 + \color{blue}{\left(\left(4 \cdot \frac{x}{y} + 3\right) - 4 \cdot \frac{z}{y}\right)}\]

  3. Simplified0.0

    \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)}\]

  4. Final simplification0.0

    \[\leadsto 1 + \mathsf{fma}\left(4, \frac{x}{y}, 3 - 4 \cdot \frac{z}{y}\right)\]

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, A"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.75)) z)) y)))