Average Error: 0.2 → 0.0
Time: 21.5s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)
double f(double x, double y, double z) {
        double r127618 = 1.0;
        double r127619 = 4.0;
        double r127620 = x;
        double r127621 = y;
        double r127622 = 0.75;
        double r127623 = r127621 * r127622;
        double r127624 = r127620 + r127623;
        double r127625 = z;
        double r127626 = r127624 - r127625;
        double r127627 = r127619 * r127626;
        double r127628 = r127627 / r127621;
        double r127629 = r127618 + r127628;
        return r127629;
}
↓
double f(double x, double y, double z) {
        double r127630 = 4.0;
        double r127631 = x;
        double r127632 = z;
        double r127633 = r127631 - r127632;
        double r127634 = y;
        double r127635 = r127633 / r127634;
        double r127636 = fma(r127630, r127635, r127630);
        return r127636;
}

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. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019326 +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)))