Average Error: 0.2 → 0.0
Time: 13.0s
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 r191110 = 1.0;
        double r191111 = 4.0;
        double r191112 = x;
        double r191113 = y;
        double r191114 = 0.75;
        double r191115 = r191113 * r191114;
        double r191116 = r191112 + r191115;
        double r191117 = z;
        double r191118 = r191116 - r191117;
        double r191119 = r191111 * r191118;
        double r191120 = r191119 / r191113;
        double r191121 = r191110 + r191120;
        return r191121;
}
↓
double f(double x, double y, double z) {
        double r191122 = 4.0;
        double r191123 = x;
        double r191124 = z;
        double r191125 = r191123 - r191124;
        double r191126 = y;
        double r191127 = r191125 / r191126;
        double r191128 = fma(r191122, r191127, r191122);
        return r191128;
}

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 2019304 +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)))