Average Error: 0.1 → 0.0
Time: 1.5s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r244346 = 1.0;
        double r244347 = 4.0;
        double r244348 = x;
        double r244349 = y;
        double r244350 = 0.75;
        double r244351 = r244349 * r244350;
        double r244352 = r244348 + r244351;
        double r244353 = z;
        double r244354 = r244352 - r244353;
        double r244355 = r244347 * r244354;
        double r244356 = r244355 / r244349;
        double r244357 = r244346 + r244356;
        return r244357;
}
↓
double f(double x, double y, double z) {
        double r244358 = 4.0;
        double r244359 = x;
        double r244360 = y;
        double r244361 = r244359 / r244360;
        double r244362 = z;
        double r244363 = r244362 / r244360;
        double r244364 = r244358 * r244363;
        double r244365 = r244358 - r244364;
        double r244366 = fma(r244358, r244361, r244365);
        return r244366;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

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

  2. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 +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)))