Average Error: 0.1 → 0.1
Time: 6.2s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right), \log \left({z}^{\frac{1}{3}}\right) \cdot y\right)\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right), \log \left({z}^{\frac{1}{3}}\right) \cdot y\right)\right)
double f(double x, double y, double z) {
        double r262322 = x;
        double r262323 = 0.5;
        double r262324 = r262322 * r262323;
        double r262325 = y;
        double r262326 = 1.0;
        double r262327 = z;
        double r262328 = r262326 - r262327;
        double r262329 = log(r262327);
        double r262330 = r262328 + r262329;
        double r262331 = r262325 * r262330;
        double r262332 = r262324 + r262331;
        return r262332;
}
↓
double f(double x, double y, double z) {
        double r262333 = x;
        double r262334 = 0.5;
        double r262335 = y;
        double r262336 = 2.0;
        double r262337 = z;
        double r262338 = cbrt(r262337);
        double r262339 = log(r262338);
        double r262340 = 1.0;
        double r262341 = r262340 - r262337;
        double r262342 = fma(r262336, r262339, r262341);
        double r262343 = 0.3333333333333333;
        double r262344 = pow(r262337, r262343);
        double r262345 = log(r262344);
        double r262346 = r262345 * r262335;
        double r262347 = fma(r262335, r262342, r262346);
        double r262348 = fma(r262333, r262334, r262347);
        return r262348;
}

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. Simplified0.1

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

  4. Applied add-cube-cbrt0.1

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

  5. Applied log-prod0.1

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

  6. Applied associate-+r+0.1

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

  7. Simplified0.1

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

  9. Applied distribute-lft-in0.1

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

  10. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, y \cdot \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) + \color{blue}{\log \left({z}^{\frac{1}{3}}\right) \cdot y}\right)\]
  11. Using strategy rm

  12. Applied fma-def0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

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

  (+ (* x 0.5) (* y (+ (- 1 z) (log z)))))