Average Error: 0.2 → 0.0
Time: 2.2s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[1 + \mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 3\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
1 + \mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 3\right)
double f(double x, double y, double z) {
        double r265625 = 1.0;
        double r265626 = 4.0;
        double r265627 = x;
        double r265628 = y;
        double r265629 = 0.75;
        double r265630 = r265628 * r265629;
        double r265631 = r265627 + r265630;
        double r265632 = z;
        double r265633 = r265631 - r265632;
        double r265634 = r265626 * r265633;
        double r265635 = r265634 / r265628;
        double r265636 = r265625 + r265635;
        return r265636;
}
↓
double f(double x, double y, double z) {
        double r265637 = 1.0;
        double r265638 = 4.0;
        double r265639 = x;
        double r265640 = y;
        double r265641 = r265639 / r265640;
        double r265642 = z;
        double r265643 = r265642 / r265640;
        double r265644 = r265641 - r265643;
        double r265645 = 3.0;
        double r265646 = fma(r265638, r265644, r265645);
        double r265647 = r265637 + r265646;
        return r265647;
}

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. Taylor expanded around 0 0.0

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 +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)))