Average Error: 0.2 → 0.0
Time: 4.7s
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 r279104 = 1.0;
        double r279105 = 4.0;
        double r279106 = x;
        double r279107 = y;
        double r279108 = 0.75;
        double r279109 = r279107 * r279108;
        double r279110 = r279106 + r279109;
        double r279111 = z;
        double r279112 = r279110 - r279111;
        double r279113 = r279105 * r279112;
        double r279114 = r279113 / r279107;
        double r279115 = r279104 + r279114;
        return r279115;
}
↓
double f(double x, double y, double z) {
        double r279116 = 4.0;
        double r279117 = x;
        double r279118 = y;
        double r279119 = r279117 / r279118;
        double r279120 = z;
        double r279121 = r279120 / r279118;
        double r279122 = r279116 * r279121;
        double r279123 = r279116 - r279122;
        double r279124 = fma(r279116, r279119, r279123);
        return r279124;
}

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 2020036 +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)))