Average Error: 0.2 → 0.0
Time: 6.2s
Precision: 64
\[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4.0, \frac{x}{y} - \frac{z}{y}, 4.0\right)\]
1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4.0, \frac{x}{y} - \frac{z}{y}, 4.0\right)
double f(double x, double y, double z) {
        double r4156681 = 1.0;
        double r4156682 = 4.0;
        double r4156683 = x;
        double r4156684 = y;
        double r4156685 = 0.75;
        double r4156686 = r4156684 * r4156685;
        double r4156687 = r4156683 + r4156686;
        double r4156688 = z;
        double r4156689 = r4156687 - r4156688;
        double r4156690 = r4156682 * r4156689;
        double r4156691 = r4156690 / r4156684;
        double r4156692 = r4156681 + r4156691;
        return r4156692;
}
↓
double f(double x, double y, double z) {
        double r4156693 = 4.0;
        double r4156694 = x;
        double r4156695 = y;
        double r4156696 = r4156694 / r4156695;
        double r4156697 = z;
        double r4156698 = r4156697 / r4156695;
        double r4156699 = r4156696 - r4156698;
        double r4156700 = fma(r4156693, r4156699, r4156693);
        return r4156700;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1.0 + \frac{4.0 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75 + \frac{x - z}{y}, 4.0, 1.0\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4.0 \cdot \frac{x}{y} + 4.0\right) - 4.0 \cdot \frac{z}{y}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(4.0, \frac{x}{y} - \frac{z}{y}, 4.0\right)}\]

  5. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(4.0, \frac{x}{y} - \frac{z}{y}, 4.0\right)\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))