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

Average Error: 1.9 → 0.5

Time: 16.6s

Precision: 64

Math

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

↓

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	return ((1.0 / ((cbrt(y) * cbrt(y)) / (pow(((cbrt(1.0) * cbrt(1.0)) / (cbrt(a) * cbrt(a))), 1.0) / sqrt(exp((((cbrt((y * log((1.0 / z)))) * cbrt((y * log((1.0 / z))))) * cbrt((y * log((1.0 / z))))) + ((log((1.0 / a)) * t) + b))))))) * (x / (cbrt(y) / (pow((cbrt(1.0) / cbrt(a)), 1.0) / sqrt(exp((((cbrt((y * log((1.0 / z)))) * cbrt((y * log((1.0 / z))))) * cbrt((y * log((1.0 / z))))) + ((log((1.0 / a)) * t) + b))))))));
}

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	10.9
Herbie	0.5

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

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 1.9

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

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-cube-cbrt 1.2

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

7. Applied associate-/l* 1.4

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

9. Applied add-sqr-sqrt 1.4

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

10. Applied add-cube-cbrt 1.5

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

11. Applied add-cube-cbrt 1.5

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

12. Applied times-frac 1.5

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

13. Applied unpow-prod-down 1.5

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

14. Applied times-frac 1.5

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

15. Applied add-cube-cbrt 1.6

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

16. Applied times-frac 1.6

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

17. Applied *-un-lft-identity 1.6

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

18. Applied times-frac 0.5

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

19. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020058 
(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))