Average Error: 0.2 → 0.0
Time: 2.3s
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 r314744 = 1.0;
        double r314745 = 4.0;
        double r314746 = x;
        double r314747 = y;
        double r314748 = 0.75;
        double r314749 = r314747 * r314748;
        double r314750 = r314746 + r314749;
        double r314751 = z;
        double r314752 = r314750 - r314751;
        double r314753 = r314745 * r314752;
        double r314754 = r314753 / r314747;
        double r314755 = r314744 + r314754;
        return r314755;
}
↓
double f(double x, double y, double z) {
        double r314756 = 4.0;
        double r314757 = x;
        double r314758 = y;
        double r314759 = r314757 / r314758;
        double r314760 = z;
        double r314761 = r314760 / r314758;
        double r314762 = r314756 * r314761;
        double r314763 = r314756 - r314762;
        double r314764 = fma(r314756, r314759, r314763);
        return r314764;
}

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 2020034 +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)))