Average Error: 0.1 → 0.0
Time: 10.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(\frac{x}{y} - \frac{z}{y}, 4, 4\right)
double f(double x, double y, double z) {
        double r230270 = 1.0;
        double r230271 = 4.0;
        double r230272 = x;
        double r230273 = y;
        double r230274 = 0.75;
        double r230275 = r230273 * r230274;
        double r230276 = r230272 + r230275;
        double r230277 = z;
        double r230278 = r230276 - r230277;
        double r230279 = r230271 * r230278;
        double r230280 = r230279 / r230273;
        double r230281 = r230270 + r230280;
        return r230281;
}
↓
double f(double x, double y, double z) {
        double r230282 = x;
        double r230283 = y;
        double r230284 = r230282 / r230283;
        double r230285 = z;
        double r230286 = r230285 / r230283;
        double r230287 = r230284 - r230286;
        double r230288 = 4.0;
        double r230289 = fma(r230287, r230288, r230288);
        return r230289;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 0.1

    \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]

  2. Simplified0.0

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

  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

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

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +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)))