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

Average Error: 2.0 → 0.7

Time: 14.1s

Precision: 64

Math

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

↓

\[\left(\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y}}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r519446 = x;
        double r519447 = y;
        double r519448 = z;
        double r519449 = log(r519448);
        double r519450 = r519447 * r519449;
        double r519451 = t;
        double r519452 = 1.0;
        double r519453 = r519451 - r519452;
        double r519454 = a;
        double r519455 = log(r519454);
        double r519456 = r519453 * r519455;
        double r519457 = r519450 + r519456;
        double r519458 = b;
        double r519459 = r519457 - r519458;
        double r519460 = exp(r519459);
        double r519461 = r519446 * r519460;
        double r519462 = r519461 / r519447;
        return r519462;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r519463 = x;
        double r519464 = 1.0;
        double r519465 = a;
        double r519466 = r519464 / r519465;
        double r519467 = 1.0;
        double r519468 = 2.0;
        double r519469 = r519467 / r519468;
        double r519470 = pow(r519466, r519469);
        double r519471 = y;
        double r519472 = z;
        double r519473 = r519464 / r519472;
        double r519474 = log(r519473);
        double r519475 = r519471 * r519474;
        double r519476 = log(r519466);
        double r519477 = t;
        double r519478 = r519476 * r519477;
        double r519479 = b;
        double r519480 = r519478 + r519479;
        double r519481 = r519475 + r519480;
        double r519482 = exp(r519481);
        double r519483 = sqrt(r519482);
        double r519484 = r519470 / r519483;
        double r519485 = r519463 * r519484;
        double r519486 = cbrt(r519470);
        double r519487 = r519486 * r519486;
        double r519488 = cbrt(r519483);
        double r519489 = r519488 * r519488;
        double r519490 = r519487 / r519489;
        double r519491 = cbrt(r519471);
        double r519492 = r519491 * r519491;
        double r519493 = r519490 / r519492;
        double r519494 = r519485 * r519493;
        double r519495 = r519486 / r519488;
        double r519496 = r519495 / r519491;
        double r519497 = r519494 * r519496;
        return r519497;
}

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	2.0
Target	11.4
Herbie	0.7

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

Derivation

1. Initial program 2.0

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

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 1.3

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

5. Applied add-sqr-sqrt 1.3

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

6. Applied sqr-pow 1.3

   \[\leadsto \frac{x \cdot \frac{\color{blue}{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot \sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]

7. Applied times-frac 1.3

   \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right)}}{y}\]

8. Applied associate-*r* 1.3

   \[\leadsto \frac{\color{blue}{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{y}\]
9. Using strategy rm

10. Applied *-un-lft-identity 1.3

    \[\leadsto \frac{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{\color{blue}{1 \cdot y}}\]

11. Applied times-frac 1.1

    \[\leadsto \color{blue}{\frac{x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{1} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}}\]

12. Simplified 1.1

    \[\leadsto \color{blue}{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right)} \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{y}\]
13. Using strategy rm

14. Applied add-cube-cbrt 1.3

    \[\leadsto \left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

15. Applied add-cube-cbrt 1.3

    \[\leadsto \left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\color{blue}{\left(\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

16. Applied add-cube-cbrt 1.3

    \[\leadsto \left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}\right) \cdot \sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}}{\left(\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

17. Applied times-frac 1.3

    \[\leadsto \left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}} \cdot \frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

18. Applied times-frac 1.3

    \[\leadsto \left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y}}\right)}\]

19. Applied associate-*r* 0.7

    \[\leadsto \color{blue}{\left(\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y}}}\]

20. Final simplification 0.7

    \[\leadsto \left(\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}} \cdot \sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\frac{\sqrt[3]{{\left(\frac{1}{a}\right)}^{\left(\frac{1}{2}\right)}}}{\sqrt[3]{\sqrt{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}}}{\sqrt[3]{y}}\]

Reproduce

herbie shell --seed 2020060 
(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))