Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F

Average Error: 10.2 → 0.7

Time: 31.1s

Precision: 64

Math

\[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.0215350936667226 \cdot 10^{149}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{elif}\;x \le 6.2316967628408393 \cdot 10^{-6}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x + y} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right) \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot y} \cdot \frac{1}{x}\\ \end{array}\]

C

double code(double x, double y) {
	return (exp((x * log((x / (x + y))))) / x);
}

↓

double code(double x, double y) {
	double temp;
	if ((x <= -1.0215350936667226e+149)) {
		temp = (1.0 / (x * exp(y)));
	} else {
		double temp_1;
		if ((x <= 6.231696762840839e-06)) {
			temp_1 = ((pow(((cbrt(x) * cbrt((cbrt(x) * cbrt(x)))) / (cbrt((x + y)) * cbrt((cbrt((x + y)) * cbrt((x + y)))))), x) * pow((cbrt(cbrt(x)) / cbrt(cbrt((x + y)))), x)) * (pow((cbrt(x) / cbrt((x + y))), x) / x));
		} else {
			temp_1 = (exp((-1.0 * y)) * (1.0 / x));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.2
Target	7.4
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;y \lt -3.73118442066479561 \cdot 10^{94}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \mathbf{elif}\;y \lt 2.81795924272828789 \cdot 10^{37}:\\ \;\;\;\;\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}\\ \mathbf{elif}\;y \lt 2.347387415166998 \cdot 10^{178}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{y + x}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\frac{-1}{y}}}{x}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if x < -1.0215350936667226e+149

   1. Initial program 15.6

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

   2. Simplified 15.6

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]

   3. Taylor expanded around inf 0.0

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]

   4. Simplified 0.0

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x}\]
   5. Using strategy rm

   6. Applied clear-num 0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{e^{-1 \cdot y}}}}\]

   7. Simplified 0.0

      \[\leadsto \frac{1}{\color{blue}{x \cdot e^{y}}}\]

   if -1.0215350936667226e+149 < x < 6.231696762840839e-06

   1. Initial program 9.5

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

   2. Simplified 9.5

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 9.5

      \[\leadsto \frac{{\left(\frac{x}{x + y}\right)}^{x}}{\color{blue}{1 \cdot x}}\]

   5. Applied add-cube-cbrt 17.1

      \[\leadsto \frac{{\left(\frac{x}{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}\right)}^{x}}{1 \cdot x}\]

   6. Applied add-cube-cbrt 9.5

      \[\leadsto \frac{{\left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}\right)}^{x}}{1 \cdot x}\]

   7. Applied times-frac 9.5

      \[\leadsto \frac{{\color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}^{x}}{1 \cdot x}\]

   8. Applied unpow-prod-down 2.6

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}}{1 \cdot x}\]

   9. Applied times-frac 2.6

      \[\leadsto \color{blue}{\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x}}{1} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}}\]

   10. Simplified 2.6

       \[\leadsto \color{blue}{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x}} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]
   11. Using strategy rm

   12. Applied add-cube-cbrt 5.2

       \[\leadsto {\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}\right) \cdot \sqrt[3]{x + y}}}}\right)}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]

   13. Applied cbrt-prod 8.2

       \[\leadsto {\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y}}\right)}}\right)}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]

   14. Applied associate-*r* 8.7

       \[\leadsto {\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\color{blue}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) \cdot \sqrt[3]{\sqrt[3]{x + y}}}}\right)}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]

   15. Applied add-cube-cbrt 9.1

       \[\leadsto {\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]

   16. Applied cbrt-prod 5.8

       \[\leadsto {\left(\frac{\sqrt[3]{x} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]

   17. Applied associate-*r* 2.6

       \[\leadsto {\left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}}{\left(\sqrt[3]{x + y} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right) \cdot \sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]

   18. Applied times-frac 2.6

       \[\leadsto {\color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x + y} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}} \cdot \frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}}^{x} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]

   19. Applied unpow-prod-down 1.1

       \[\leadsto \color{blue}{\left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x + y} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right)} \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\]

   if 6.231696762840839e-06 < x

   1. Initial program 9.1

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]

   2. Simplified 9.1

      \[\leadsto \color{blue}{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\]

   3. Taylor expanded around inf 0.3

      \[\leadsto \frac{\color{blue}{e^{-y}}}{x}\]

   4. Simplified 0.3

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot y}}}{x}\]
   5. Using strategy rm

   6. Applied div-inv 0.3

      \[\leadsto \color{blue}{e^{-1 \cdot y} \cdot \frac{1}{x}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0215350936667226 \cdot 10^{149}:\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{elif}\;x \le 6.2316967628408393 \cdot 10^{-6}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x + y} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y}}}\right)}^{x}\right) \cdot \frac{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot y} \cdot \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, F"
  :precision binary64

  :herbie-target
  (if (< y -3.7311844206647956e+94) (/ (exp (/ -1 y)) x) (if (< y 2.817959242728288e+37) (/ (pow (/ x (+ y x)) x) x) (if (< y 2.347387415166998e+178) (log (exp (/ (pow (/ x (+ y x)) x) x))) (/ (exp (/ -1 y)) x))))

  (/ (exp (* x (log (/ x (+ x y))))) x))