Average Error: 0.2 → 0.0
Time: 13.5s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 4\right)\]
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 r11296223 = 1.0;
        double r11296224 = 4.0;
        double r11296225 = x;
        double r11296226 = y;
        double r11296227 = 0.75;
        double r11296228 = r11296226 * r11296227;
        double r11296229 = r11296225 + r11296228;
        double r11296230 = z;
        double r11296231 = r11296229 - r11296230;
        double r11296232 = r11296224 * r11296231;
        double r11296233 = r11296232 / r11296226;
        double r11296234 = r11296223 + r11296233;
        return r11296234;
}
↓
double f(double x, double y, double z) {
        double r11296235 = 4.0;
        double r11296236 = x;
        double r11296237 = y;
        double r11296238 = r11296236 / r11296237;
        double r11296239 = z;
        double r11296240 = r11296239 / r11296237;
        double r11296241 = r11296238 - r11296240;
        double r11296242 = fma(r11296235, r11296241, r11296235);
        return r11296242;
}

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 2019192 +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)))