Average Error: 0.1 → 0.1
Time: 17.1s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[x \cdot 0.5 + \mathsf{fma}\left(-1, -\log z, 1 - z\right) \cdot y\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
x \cdot 0.5 + \mathsf{fma}\left(-1, -\log z, 1 - z\right) \cdot y
double f(double x, double y, double z) {
        double r163972 = x;
        double r163973 = 0.5;
        double r163974 = r163972 * r163973;
        double r163975 = y;
        double r163976 = 1.0;
        double r163977 = z;
        double r163978 = r163976 - r163977;
        double r163979 = log(r163977);
        double r163980 = r163978 + r163979;
        double r163981 = r163975 * r163980;
        double r163982 = r163974 + r163981;
        return r163982;
}
↓
double f(double x, double y, double z) {
        double r163983 = x;
        double r163984 = 0.5;
        double r163985 = r163983 * r163984;
        double r163986 = -1.0;
        double r163987 = z;
        double r163988 = log(r163987);
        double r163989 = -r163988;
        double r163990 = 1.0;
        double r163991 = r163990 - r163987;
        double r163992 = fma(r163986, r163989, r163991);
        double r163993 = y;
        double r163994 = r163992 * r163993;
        double r163995 = r163985 + r163994;
        return r163995;
}

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

    \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
  2. Using strategy rm

  3. Applied add-cube-cbrt0.1

    \[\leadsto x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right)\]

  4. Applied log-prod0.1

    \[\leadsto x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right)\]

  5. Applied associate-+r+0.1

    \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(\left(\left(1 - z\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \log \left(\sqrt[3]{z}\right)\right)}\]

  6. Simplified0.1

    \[\leadsto x \cdot 0.5 + y \cdot \left(\color{blue}{\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right)} + \log \left(\sqrt[3]{z}\right)\right)\]

  7. Taylor expanded around inf 0.2

    \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\left(\left(3 \cdot \log \left({\left(\frac{1}{z}\right)}^{\frac{-1}{3}}\right) + 1\right) - z\right)}\]

  8. Simplified0.1

    \[\leadsto x \cdot 0.5 + y \cdot \color{blue}{\mathsf{fma}\left(-1, -\log z, 1 - z\right)}\]

  9. Final simplification0.1

    \[\leadsto x \cdot 0.5 + \mathsf{fma}\left(-1, -\log z, 1 - z\right) \cdot y\]

Reproduce

herbie shell --seed 2019194 +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)))))