System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2

Average Error: 0.1 → 0.1

Time: 26.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r195067 = x;
        double r195068 = 0.5;
        double r195069 = r195067 * r195068;
        double r195070 = y;
        double r195071 = 1.0;
        double r195072 = z;
        double r195073 = r195071 - r195072;
        double r195074 = log(r195072);
        double r195075 = r195073 + r195074;
        double r195076 = r195070 * r195075;
        double r195077 = r195069 + r195076;
        return r195077;
}

↓

double f(double x, double y, double z) {
        double r195078 = x;
        double r195079 = 0.5;
        double r195080 = r195078 * r195079;
        double r195081 = 2.0;
        double r195082 = z;
        double r195083 = cbrt(r195082);
        double r195084 = log(r195083);
        double r195085 = 1.0;
        double r195086 = r195085 - r195082;
        double r195087 = fma(r195081, r195084, r195086);
        double r195088 = y;
        double r195089 = r195087 * r195088;
        double r195090 = 0.6666666666666666;
        double r195091 = pow(r195082, r195090);
        double r195092 = pow(r195091, r195090);
        double r195093 = cbrt(r195091);
        double r195094 = r195092 * r195093;
        double r195095 = cbrt(r195094);
        double r195096 = cbrt(r195083);
        double r195097 = r195095 * r195096;
        double r195098 = log(r195097);
        double r195099 = r195098 * r195088;
        double r195100 = r195089 + r195099;
        double r195101 = r195080 + r195100;
        return r195101;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.1

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

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-in 0.1

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

4. Simplified 0.1

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

6. Applied add-cube-cbrt 0.1

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

7. Applied log-prod 0.1

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

8. Applied distribute-rgt-in 0.1

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

9. Applied associate-+r+ 0.1

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

10. Simplified 0.1

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

12. Applied add-cube-cbrt 0.1

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

13. Applied cbrt-prod 0.1

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

14. Simplified 0.1

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

16. Applied add-cube-cbrt 0.1

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019326 +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)))))