Average Error: 0.1 → 0.0
Time: 8.8s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)
double f(double x, double y, double z) {
        double r14129705 = 1.0;
        double r14129706 = 4.0;
        double r14129707 = x;
        double r14129708 = y;
        double r14129709 = 0.25;
        double r14129710 = r14129708 * r14129709;
        double r14129711 = r14129707 + r14129710;
        double r14129712 = z;
        double r14129713 = r14129711 - r14129712;
        double r14129714 = r14129706 * r14129713;
        double r14129715 = r14129714 / r14129708;
        double r14129716 = r14129705 + r14129715;
        return r14129716;
}
↓
double f(double x, double y, double z) {
        double r14129717 = 4.0;
        double r14129718 = x;
        double r14129719 = z;
        double r14129720 = r14129718 - r14129719;
        double r14129721 = y;
        double r14129722 = r14129720 / r14129721;
        double r14129723 = 2.0;
        double r14129724 = fma(r14129717, r14129722, r14129723);
        return r14129724;
}

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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))