Average Error: 0.2 → 0.0
Time: 2.5s
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 r296721 = 1.0;
        double r296722 = 4.0;
        double r296723 = x;
        double r296724 = y;
        double r296725 = 0.75;
        double r296726 = r296724 * r296725;
        double r296727 = r296723 + r296726;
        double r296728 = z;
        double r296729 = r296727 - r296728;
        double r296730 = r296722 * r296729;
        double r296731 = r296730 / r296724;
        double r296732 = r296721 + r296731;
        return r296732;
}
↓
double f(double x, double y, double z) {
        double r296733 = 4.0;
        double r296734 = x;
        double r296735 = y;
        double r296736 = r296734 / r296735;
        double r296737 = z;
        double r296738 = r296737 / r296735;
        double r296739 = r296733 * r296738;
        double r296740 = r296733 - r296739;
        double r296741 = fma(r296733, r296736, r296740);
        return r296741;
}

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 2019362 +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)))