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

Average Error: 2.0 → 1.0

Time: 17.8s

Precision: 64

Math

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

↓

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

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 ((x * (pow(sqrt((1.0 / a)), 1.0) / (cbrt(exp(fma(y, log((1.0 / z)), fma(log((1.0 / a)), t, b)))) * cbrt(exp(fma(y, log((1.0 / z)), fma(log((1.0 / a)), t, b))))))) / (y / (pow(pow(a, -0.5), 1.0) / cbrt(exp(fma(y, log((1.0 / z)), fma(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	2.0
Target	11.3
Herbie	1.0

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

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

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

5. Applied add-cube-cbrt 1.3

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

6. Applied add-sqr-sqrt 1.3

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

7. Applied unpow-prod-down 1.3

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

8. Applied times-frac 1.3

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

9. Applied associate-*r* 1.3

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

11. Applied associate-/l* 1.0

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

13. Applied inv-pow 1.0

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

14. Applied sqrt-pow1 1.0

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

15. Simplified 1.0

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

16. Final simplification 1.0

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

Reproduce

herbie shell --seed 2020079 +o rules:numerics
(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))