Average Error: 0.2 → 0.0
Time: 15.1s
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 r202004 = 1.0;
        double r202005 = 4.0;
        double r202006 = x;
        double r202007 = y;
        double r202008 = 0.75;
        double r202009 = r202007 * r202008;
        double r202010 = r202006 + r202009;
        double r202011 = z;
        double r202012 = r202010 - r202011;
        double r202013 = r202005 * r202012;
        double r202014 = r202013 / r202007;
        double r202015 = r202004 + r202014;
        return r202015;
}
↓
double f(double x, double y, double z) {
        double r202016 = 4.0;
        double r202017 = x;
        double r202018 = z;
        double r202019 = r202017 - r202018;
        double r202020 = y;
        double r202021 = r202019 / r202020;
        double r202022 = fma(r202016, r202021, r202016);
        return r202022;
}

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 2019303 +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)))