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

Average Error: 0.1 → 0.1

Time: 25.4s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z) {
        double r269329 = x;
        double r269330 = 0.5;
        double r269331 = r269329 * r269330;
        double r269332 = y;
        double r269333 = 1.0;
        double r269334 = z;
        double r269335 = r269333 - r269334;
        double r269336 = log(r269334);
        double r269337 = r269335 + r269336;
        double r269338 = r269332 * r269337;
        double r269339 = r269331 + r269338;
        return r269339;
}

↓

double f(double x, double y, double z) {
        double r269340 = x;
        double r269341 = 0.5;
        double r269342 = r269340 * r269341;
        double r269343 = 2.0;
        double r269344 = z;
        double r269345 = cbrt(r269344);
        double r269346 = log(r269345);
        double r269347 = 1.0;
        double r269348 = r269347 - r269344;
        double r269349 = fma(r269343, r269346, r269348);
        double r269350 = y;
        double r269351 = r269349 * r269350;
        double r269352 = 1.0;
        double r269353 = r269352 / r269344;
        double r269354 = -0.3333333333333333;
        double r269355 = pow(r269353, r269354);
        double r269356 = log(r269355);
        double r269357 = r269356 * r269350;
        double r269358 = r269351 + r269357;
        double r269359 = r269342 + r269358;
        return r269359;
}

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. Taylor expanded around inf 0.1

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

12. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019306 +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)))))