Average Error: 0.1 → 0.0
Time: 2.2s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)
double f(double x, double y, double z) {
        double r228108 = 1.0;
        double r228109 = 4.0;
        double r228110 = x;
        double r228111 = y;
        double r228112 = 0.25;
        double r228113 = r228111 * r228112;
        double r228114 = r228110 + r228113;
        double r228115 = z;
        double r228116 = r228114 - r228115;
        double r228117 = r228109 * r228116;
        double r228118 = r228117 / r228111;
        double r228119 = r228108 + r228118;
        return r228119;
}
↓
double f(double x, double y, double z) {
        double r228120 = 4.0;
        double r228121 = x;
        double r228122 = y;
        double r228123 = r228121 / r228122;
        double r228124 = z;
        double r228125 = r228124 / r228122;
        double r228126 = r228123 - r228125;
        double r228127 = 2.0;
        double r228128 = fma(r228120, r228126, r228127);
        return r228128;
}

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. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  3. Simplified0.0

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

  4. Taylor expanded around 0 0.0

    \[\leadsto \color{blue}{\left(4 \cdot \frac{x}{y} + 2\right) - 4 \cdot \frac{z}{y}}\]

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1 (/ (* 4 (- (+ x (* y 0.25)) z)) y)))