Average Error: 0.1 → 0.1
Time: 59.7s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1.0 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(\left(\log z + 1.0\right) - z, y, x \cdot 0.5\right)\]
x \cdot 0.5 + y \cdot \left(\left(1.0 - z\right) + \log z\right)
↓
\mathsf{fma}\left(\left(\log z + 1.0\right) - z, y, x \cdot 0.5\right)
double f(double x, double y, double z) {
        double r18173984 = x;
        double r18173985 = 0.5;
        double r18173986 = r18173984 * r18173985;
        double r18173987 = y;
        double r18173988 = 1.0;
        double r18173989 = z;
        double r18173990 = r18173988 - r18173989;
        double r18173991 = log(r18173989);
        double r18173992 = r18173990 + r18173991;
        double r18173993 = r18173987 * r18173992;
        double r18173994 = r18173986 + r18173993;
        return r18173994;
}
↓
double f(double x, double y, double z) {
        double r18173995 = z;
        double r18173996 = log(r18173995);
        double r18173997 = 1.0;
        double r18173998 = r18173996 + r18173997;
        double r18173999 = r18173998 - r18173995;
        double r18174000 = y;
        double r18174001 = x;
        double r18174002 = 0.5;
        double r18174003 = r18174001 * r18174002;
        double r18174004 = fma(r18173999, r18174000, r18174003);
        return r18174004;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right)\]

Derivation

  1. Initial program 0.1

    \[x \cdot 0.5 + y \cdot \left(\left(1.0 - z\right) + \log z\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \left(1.0 - \left(z - \log z\right)\right) \cdot y\right)}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(0.5 \cdot x + \left(1.0 \cdot y + y \cdot \log z\right)\right) - z \cdot y}\]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\log z + 1.0\right) - z, y, 0.5 \cdot x\right)}\]

  5. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\left(\log z + 1.0\right) - z, y, x \cdot 0.5\right)\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))