Average Error: 0.2 → 0.0
Time: 18.1s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x - z}{y}, 4\right)
double f(double x, double y, double z) {
        double r206305 = 1.0;
        double r206306 = 4.0;
        double r206307 = x;
        double r206308 = y;
        double r206309 = 0.75;
        double r206310 = r206308 * r206309;
        double r206311 = r206307 + r206310;
        double r206312 = z;
        double r206313 = r206311 - r206312;
        double r206314 = r206306 * r206313;
        double r206315 = r206314 / r206308;
        double r206316 = r206305 + r206315;
        return r206316;
}
↓
double f(double x, double y, double z) {
        double r206317 = 4.0;
        double r206318 = x;
        double r206319 = z;
        double r206320 = r206318 - r206319;
        double r206321 = y;
        double r206322 = r206320 / r206321;
        double r206323 = fma(r206317, r206322, r206317);
        return r206323;
}

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

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

  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

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019347 +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)))