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

Average Error: 1.8 → 1.1

Time: 30.5s

Precision: 64

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]

↓

\[\frac{\sqrt[3]{e^{\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}} \cdot \left(x \cdot \frac{\sqrt[3]{{e}^{\left(\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{e^{\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r9557368 = x;
        double r9557369 = y;
        double r9557370 = z;
        double r9557371 = log(r9557370);
        double r9557372 = r9557369 * r9557371;
        double r9557373 = t;
        double r9557374 = 1.0;
        double r9557375 = r9557373 - r9557374;
        double r9557376 = a;
        double r9557377 = log(r9557376);
        double r9557378 = r9557375 * r9557377;
        double r9557379 = r9557372 + r9557378;
        double r9557380 = b;
        double r9557381 = r9557379 - r9557380;
        double r9557382 = exp(r9557381);
        double r9557383 = r9557368 * r9557382;
        double r9557384 = r9557383 / r9557369;
        return r9557384;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r9557385 = z;
        double r9557386 = log(r9557385);
        double r9557387 = y;
        double r9557388 = r9557386 * r9557387;
        double r9557389 = t;
        double r9557390 = 1.0;
        double r9557391 = r9557389 - r9557390;
        double r9557392 = a;
        double r9557393 = log(r9557392);
        double r9557394 = r9557391 * r9557393;
        double r9557395 = r9557388 + r9557394;
        double r9557396 = b;
        double r9557397 = r9557395 - r9557396;
        double r9557398 = exp(r9557397);
        double r9557399 = cbrt(r9557398);
        double r9557400 = cbrt(r9557387);
        double r9557401 = r9557400 * r9557400;
        double r9557402 = cbrt(r9557401);
        double r9557403 = cbrt(r9557400);
        double r9557404 = r9557402 * r9557403;
        double r9557405 = r9557399 / r9557404;
        double r9557406 = x;
        double r9557407 = exp(1.0);
        double r9557408 = pow(r9557407, r9557397);
        double r9557409 = cbrt(r9557408);
        double r9557410 = r9557409 * r9557399;
        double r9557411 = r9557410 / r9557401;
        double r9557412 = r9557406 * r9557411;
        double r9557413 = r9557405 * r9557412;
        return r9557413;
}

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	1.8
Target	11.2
Herbie	1.1

\[\begin{array}{l} \mathbf{if}\;t \lt -0.8845848504127471:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.2288374073:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1.0\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1.0\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

1. Initial program 1.8

   \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
2. Using strategy rm

3. Applied *-un-lft-identity 1.8

   \[\leadsto \frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{\color{blue}{1 \cdot y}}\]

4. Applied times-frac 2.0

   \[\leadsto \color{blue}{\frac{x}{1} \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}}\]

5. Simplified 2.0

   \[\leadsto \color{blue}{x} \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{y}\]
6. Using strategy rm

7. Applied add-cube-cbrt 2.0

   \[\leadsto x \cdot \frac{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\]

8. Applied add-cube-cbrt 2.0

   \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}\right) \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\]

9. Applied times-frac 2.0

   \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y}}\right)}\]

10. Applied associate-*r* 1.1

    \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y}}}\]
11. Using strategy rm

12. Applied *-un-lft-identity 1.1

    \[\leadsto \left(x \cdot \frac{\sqrt[3]{e^{\color{blue}{1 \cdot \left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y}}\]

13. Applied exp-prod 1.1

    \[\leadsto \left(x \cdot \frac{\sqrt[3]{\color{blue}{{\left(e^{1}\right)}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y}}\]

14. Simplified 1.1

    \[\leadsto \left(x \cdot \frac{\sqrt[3]{{\color{blue}{e}}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y}}\]
15. Using strategy rm

16. Applied add-cube-cbrt 1.1

    \[\leadsto \left(x \cdot \frac{\sqrt[3]{{e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\]

17. Applied cbrt-prod 1.1

    \[\leadsto \left(x \cdot \frac{\sqrt[3]{{e}^{\left(\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt[3]{e^{\left(y \cdot \log z + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\color{blue}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}\]

18. Final simplification 1.1

    \[\leadsto \frac{\sqrt[3]{e^{\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}} \cdot \left(x \cdot \frac{\sqrt[3]{{e}^{\left(\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b\right)}} \cdot \sqrt[3]{e^{\left(\log z \cdot y + \left(t - 1.0\right) \cdot \log a\right) - b}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right)\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))