Average Error: 0.2 → 0.0
Time: 4.5s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y}, 4 - 4 \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r387117 = 1.0;
        double r387118 = 4.0;
        double r387119 = x;
        double r387120 = y;
        double r387121 = 0.75;
        double r387122 = r387120 * r387121;
        double r387123 = r387119 + r387122;
        double r387124 = z;
        double r387125 = r387123 - r387124;
        double r387126 = r387118 * r387125;
        double r387127 = r387126 / r387120;
        double r387128 = r387117 + r387127;
        return r387128;
}
↓
double f(double x, double y, double z) {
        double r387129 = 4.0;
        double r387130 = x;
        double r387131 = y;
        double r387132 = r387130 / r387131;
        double r387133 = z;
        double r387134 = r387133 / r387131;
        double r387135 = r387129 * r387134;
        double r387136 = r387129 - r387135;
        double r387137 = fma(r387129, r387132, r387136);
        return r387137;
}

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. Taylor expanded around 0 0.0

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

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020100 +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)))