Average Error: 0.2 → 0.0
Time: 1.7s
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 r205569 = 1.0;
        double r205570 = 4.0;
        double r205571 = x;
        double r205572 = y;
        double r205573 = 0.75;
        double r205574 = r205572 * r205573;
        double r205575 = r205571 + r205574;
        double r205576 = z;
        double r205577 = r205575 - r205576;
        double r205578 = r205570 * r205577;
        double r205579 = r205578 / r205572;
        double r205580 = r205569 + r205579;
        return r205580;
}
↓
double f(double x, double y, double z) {
        double r205581 = 4.0;
        double r205582 = x;
        double r205583 = y;
        double r205584 = r205582 / r205583;
        double r205585 = z;
        double r205586 = r205585 / r205583;
        double r205587 = r205581 * r205586;
        double r205588 = r205581 - r205587;
        double r205589 = fma(r205581, r205584, r205588);
        return r205589;
}

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

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

  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 2020056 +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)))