Average Error: 0.1 → 0.0
Time: 22.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)
double f(double x, double y, double z) {
        double r182674 = 1.0;
        double r182675 = 4.0;
        double r182676 = x;
        double r182677 = y;
        double r182678 = 0.25;
        double r182679 = r182677 * r182678;
        double r182680 = r182676 + r182679;
        double r182681 = z;
        double r182682 = r182680 - r182681;
        double r182683 = r182675 * r182682;
        double r182684 = r182683 / r182677;
        double r182685 = r182674 + r182684;
        return r182685;
}
↓
double f(double x, double y, double z) {
        double r182686 = x;
        double r182687 = y;
        double r182688 = r182686 / r182687;
        double r182689 = z;
        double r182690 = r182689 / r182687;
        double r182691 = r182688 - r182690;
        double r182692 = 4.0;
        double r182693 = 2.0;
        double r182694 = fma(r182691, r182692, r182693);
        return r182694;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  4. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - z}{y}, 4, 2\right)}\]
  5. Using strategy rm

  6. Applied div-sub0.0

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{x}{y} - \frac{z}{y}}, 4, 2\right)\]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 2\right)\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))