Average Error: 0.1 → 0.1
Time: 21.5s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\left(\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) \cdot y + \log \left({\left(\sqrt{z}\right)}^{\frac{1}{3}}\right) \cdot y\right) + \log \left({\left(\sqrt{z}\right)}^{\frac{1}{3}}\right) \cdot y\right) + x \cdot 0.5\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\left(\left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) \cdot y + \log \left({\left(\sqrt{z}\right)}^{\frac{1}{3}}\right) \cdot y\right) + \log \left({\left(\sqrt{z}\right)}^{\frac{1}{3}}\right) \cdot y\right) + x \cdot 0.5
double f(double x, double y, double z) {
        double r12020357 = x;
        double r12020358 = 0.5;
        double r12020359 = r12020357 * r12020358;
        double r12020360 = y;
        double r12020361 = 1.0;
        double r12020362 = z;
        double r12020363 = r12020361 - r12020362;
        double r12020364 = log(r12020362);
        double r12020365 = r12020363 + r12020364;
        double r12020366 = r12020360 * r12020365;
        double r12020367 = r12020359 + r12020366;
        return r12020367;
}
↓
double f(double x, double y, double z) {
        double r12020368 = 2.0;
        double r12020369 = z;
        double r12020370 = cbrt(r12020369);
        double r12020371 = log(r12020370);
        double r12020372 = 1.0;
        double r12020373 = r12020372 - r12020369;
        double r12020374 = fma(r12020368, r12020371, r12020373);
        double r12020375 = y;
        double r12020376 = r12020374 * r12020375;
        double r12020377 = sqrt(r12020369);
        double r12020378 = 0.3333333333333333;
        double r12020379 = pow(r12020377, r12020378);
        double r12020380 = log(r12020379);
        double r12020381 = r12020380 * r12020375;
        double r12020382 = r12020376 + r12020381;
        double r12020383 = r12020382 + r12020381;
        double r12020384 = x;
        double r12020385 = 0.5;
        double r12020386 = r12020384 * r12020385;
        double r12020387 = r12020383 + r12020386;
        return r12020387;
}

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 distribute-lft-in0.1

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

  5. Applied add-cube-cbrt0.1

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

  6. Applied log-prod0.1

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

  7. Applied distribute-lft-in0.1

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

  8. Applied associate-+r+0.1

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

  9. Simplified0.1

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

  11. Applied pow1/30.1

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

  13. Applied add-sqr-sqrt0.1

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

  14. Applied unpow-prod-down0.1

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

  15. Applied log-prod0.1

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

  16. Applied distribute-lft-in0.1

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

  17. Applied associate-+r+0.1

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

  18. Final simplification0.1

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

Reproduce

herbie shell --seed 2019179 +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)))))