Average Error: 0.2 → 0.0
Time: 16.9s
Precision: 64
\[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}\]
↓
\[\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)\]
1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y}
↓
\mathsf{fma}\left(4, \frac{x}{y} - \frac{z}{y}, 2\right)
double f(double x, double y, double z) {
        double r22975 = 1.0;
        double r22976 = 4.0;
        double r22977 = x;
        double r22978 = y;
        double r22979 = 0.25;
        double r22980 = r22978 * r22979;
        double r22981 = r22977 + r22980;
        double r22982 = z;
        double r22983 = r22981 - r22982;
        double r22984 = r22976 * r22983;
        double r22985 = r22984 / r22978;
        double r22986 = r22975 + r22985;
        return r22986;
}
↓
double f(double x, double y, double z) {
        double r22987 = 4.0;
        double r22988 = x;
        double r22989 = y;
        double r22990 = r22988 / r22989;
        double r22991 = z;
        double r22992 = r22991 / r22989;
        double r22993 = r22990 - r22992;
        double r22994 = 2.0;
        double r22995 = fma(r22987, r22993, r22994);
        return r22995;
}

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. Using strategy rm

  6. Applied div-sub0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019310 +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)))