Average Error: 0.2 → 0.0
Time: 12.7s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)
double f(double x, double y, double z) {
        double r11291279 = 1.0;
        double r11291280 = 4.0;
        double r11291281 = x;
        double r11291282 = y;
        double r11291283 = 0.75;
        double r11291284 = r11291282 * r11291283;
        double r11291285 = r11291281 + r11291284;
        double r11291286 = z;
        double r11291287 = r11291285 - r11291286;
        double r11291288 = r11291280 * r11291287;
        double r11291289 = r11291288 / r11291282;
        double r11291290 = r11291279 + r11291289;
        return r11291290;
}
↓
double f(double x, double y, double z) {
        double r11291291 = 4.0;
        double r11291292 = x;
        double r11291293 = y;
        double r11291294 = r11291292 / r11291293;
        double r11291295 = z;
        double r11291296 = r11291295 / r11291293;
        double r11291297 = r11291294 - r11291296;
        double r11291298 = fma(r11291291, r11291297, r11291291);
        return r11291298;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.75 - \frac{z - x}{y}, 4, 1\right)}\]

  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

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

  5. Taylor expanded around 0 0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))