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

Average Error: 1.8 → 0.9

Time: 13.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -284079592.06856072 \lor \neg \left(y \le 3.38613032208109627 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}\right)}^{3}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\log \left(\frac{1}{a}\right) \cdot t + b} \cdot {\left(\frac{1}{z}\right)}^{y}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\log \left(\frac{1}{a}\right) \cdot t + b} \cdot {\left(\frac{1}{z}\right)}^{y}}}}{\sqrt[3]{y}}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((y <= -284079592.0685607) || !(y <= 3.386130322081096e-35))) {
		VAR = ((double) (((double) cbrt(((double) pow(((double) (x * ((double) (((double) pow(((double) (1.0 / a)), 1.0)) / ((double) exp(((double) (((double) (y * ((double) log(((double) (1.0 / z)))))) + ((double) (((double) (((double) log(((double) (1.0 / a)))) * t)) + b)))))))))), 3.0)))) / y));
	} else {
		VAR = ((double) (((double) (x * ((double) (((double) sqrt(((double) (((double) pow(((double) (1.0 / a)), 1.0)) / ((double) (((double) exp(((double) (((double) (((double) log(((double) (1.0 / a)))) * t)) + b)))) * ((double) pow(((double) (1.0 / z)), y)))))))) / ((double) (((double) cbrt(y)) * ((double) cbrt(y)))))))) * ((double) (((double) sqrt(((double) (((double) pow(((double) (1.0 / a)), 1.0)) / ((double) (((double) exp(((double) (((double) (((double) log(((double) (1.0 / a)))) * t)) + b)))) * ((double) pow(((double) (1.0 / z)), y)))))))) / ((double) cbrt(y))))));
	}
	return VAR;
}

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	0.9

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

Derivation

1. Split input into 2 regimes

2. if y < -284079592.0685607 or 3.386130322081096e-35 < y

   1. Initial program 0.1

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

   2. Taylor expanded around inf 0.1

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

   3. Simplified 0.0

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

   5. Applied add-cbrt-cube 0.1

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

   6. Applied add-cbrt-cube 21.7

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

   7. Applied cbrt-undiv 21.7

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

   8. Applied add-cbrt-cube 35.8

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

   9. Applied cbrt-unprod 35.9

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

   10. Simplified 0.8

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

   if -284079592.0685607 < y < 3.386130322081096e-35

   1. Initial program 3.5

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

   2. Taylor expanded around inf 3.5

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

   3. Simplified 2.2

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

   5. Applied *-un-lft-identity 2.2

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

   6. Applied times-frac 2.6

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

   7. Simplified 2.6

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

   8. Simplified 2.9

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

   10. Applied add-cube-cbrt 3.1

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

   11. Applied add-sqr-sqrt 3.2

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

   12. Applied times-frac 3.2

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

   13. Applied associate-*r* 0.9

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

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -284079592.06856072 \lor \neg \left(y \le 3.38613032208109627 \cdot 10^{-35}\right):\\ \;\;\;\;\frac{\sqrt[3]{{\left(x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}\right)}^{3}}}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot \frac{\sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\log \left(\frac{1}{a}\right) \cdot t + b} \cdot {\left(\frac{1}{z}\right)}^{y}}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) \cdot \frac{\sqrt{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{\log \left(\frac{1}{a}\right) \cdot t + b} \cdot {\left(\frac{1}{z}\right)}^{y}}}}{\sqrt[3]{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 
(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))