Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 1.9 → 0.6

Time: 16.4s

Precision: 64

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

\[\left(\left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}} \cdot \sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y}}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r442508 = x;
        double r442509 = y;
        double r442510 = z;
        double r442511 = log(r442510);
        double r442512 = r442509 * r442511;
        double r442513 = t;
        double r442514 = 1.0;
        double r442515 = r442513 - r442514;
        double r442516 = a;
        double r442517 = log(r442516);
        double r442518 = r442515 * r442517;
        double r442519 = r442512 + r442518;
        double r442520 = b;
        double r442521 = r442519 - r442520;
        double r442522 = exp(r442521);
        double r442523 = r442508 * r442522;
        double r442524 = r442523 / r442509;
        return r442524;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r442525 = x;
        double r442526 = 1.0;
        double r442527 = cbrt(r442526);
        double r442528 = r442527 * r442527;
        double r442529 = r442528 / r442526;
        double r442530 = 1.0;
        double r442531 = pow(r442529, r442530);
        double r442532 = y;
        double r442533 = z;
        double r442534 = r442526 / r442533;
        double r442535 = log(r442534);
        double r442536 = r442532 * r442535;
        double r442537 = a;
        double r442538 = r442526 / r442537;
        double r442539 = log(r442538);
        double r442540 = t;
        double r442541 = r442539 * r442540;
        double r442542 = b;
        double r442543 = r442541 + r442542;
        double r442544 = r442536 + r442543;
        double r442545 = exp(r442544);
        double r442546 = sqrt(r442545);
        double r442547 = r442531 / r442546;
        double r442548 = r442525 * r442547;
        double r442549 = r442527 / r442537;
        double r442550 = pow(r442549, r442530);
        double r442551 = cbrt(r442550);
        double r442552 = r442551 * r442551;
        double r442553 = cbrt(r442546);
        double r442554 = r442553 * r442553;
        double r442555 = r442552 / r442554;
        double r442556 = cbrt(r442532);
        double r442557 = r442556 * r442556;
        double r442558 = r442555 / r442557;
        double r442559 = r442548 * r442558;
        double r442560 = r442551 / r442553;
        double r442561 = r442560 / r442556;
        double r442562 = r442559 * r442561;
        return r442562;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	1.9
Target	11.0
Herbie	0.6

\[\begin{array}{l} \mathbf{if}\;t \lt -0.88458485041274715:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.22883740731:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

1. Initial program 1.9

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

2. Taylor expanded around inf 2.0

   \[\leadsto \frac{x \cdot \color{blue}{e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}}{y}\]

3. Simplified 1.2

   \[\leadsto \frac{x \cdot \color{blue}{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]
4. Using strategy rm

5. Applied add-sqr-sqrt 1.2

   \[\leadsto \frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{\color{blue}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{y}\]

6. Applied *-un-lft-identity 1.2

   \[\leadsto \frac{x \cdot \frac{{\left(\frac{1}{\color{blue}{1 \cdot a}}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

7. Applied add-cube-cbrt 1.2

   \[\leadsto \frac{x \cdot \frac{{\left(\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

8. Applied times-frac 1.2

   \[\leadsto \frac{x \cdot \frac{{\color{blue}{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{a}\right)}}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

9. Applied unpow-prod-down 1.2

   \[\leadsto \frac{x \cdot \frac{\color{blue}{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1} \cdot {\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

10. Applied times-frac 1.2

    \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right)}}{y}\]

11. Applied associate-*r* 1.2

    \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{y}\]
12. Using strategy rm

13. Applied *-un-lft-identity 1.2

    \[\leadsto \frac{\left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{\color{blue}{1 \cdot y}}\]

14. Applied times-frac 1.5

    \[\leadsto \color{blue}{\frac{x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{1} \cdot \frac{\frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}}\]

15. Simplified 1.5

    \[\leadsto \color{blue}{\left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right)} \cdot \frac{\frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]
16. Using strategy rm

17. Applied add-cube-cbrt 1.6

    \[\leadsto \left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

18. Applied add-cube-cbrt 1.6

    \[\leadsto \left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}{\color{blue}{\left(\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

19. Applied add-cube-cbrt 1.7

    \[\leadsto \left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}} \cdot \sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}\right) \cdot \sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}}{\left(\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

20. Applied times-frac 1.7

    \[\leadsto \left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}} \cdot \sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}} \cdot \frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

21. Applied times-frac 1.7

    \[\leadsto \left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}} \cdot \sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y}}\right)}\]

22. Applied associate-*r* 0.6

    \[\leadsto \color{blue}{\left(\left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}} \cdot \sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y}}}\]

23. Final simplification 0.6

    \[\leadsto \left(\left(x \cdot \frac{{\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}\right)}^{1}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}} \cdot \sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{\sqrt[3]{1}}{a}\right)}^{1}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y}}\]

Reproduce

herbie shell --seed 2020062 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))