Average Error: 0.2 → 0.0
Time: 21.0s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 2\right)
double f(double x, double y, double z) {
        double r186326 = 1.0;
        double r186327 = 4.0;
        double r186328 = x;
        double r186329 = y;
        double r186330 = 0.25;
        double r186331 = r186329 * r186330;
        double r186332 = r186328 + r186331;
        double r186333 = z;
        double r186334 = r186332 - r186333;
        double r186335 = r186327 * r186334;
        double r186336 = r186335 / r186329;
        double r186337 = r186326 + r186336;
        return r186337;
}
↓
double f(double x, double y, double z) {
        double r186338 = 4.0;
        double r186339 = x;
        double r186340 = z;
        double r186341 = r186339 - r186340;
        double r186342 = y;
        double r186343 = r186341 / r186342;
        double r186344 = 2.0;
        double r186345 = fma(r186338, r186343, r186344);
        return r186345;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.2

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(0.25 + \frac{x - z}{y}, 4, 1\right)}\]

  3. Taylor expanded around 0 0.0

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 +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)))