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

Average Error: 11.3 → 0.2

Time: 8.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -503.575504887666682 \lor \neg \left(x \le 9.87489366393525115 \cdot 10^{-8}\right):\\ \;\;\;\;e^{-1 \cdot y} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}}\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if (((x <= -503.5755048876667) || !(x <= 9.874893663935251e-08))) {
		VAR = ((double) (((double) exp(((double) (-1.0 * y)))) * ((double) (1.0 / x))));
	} else {
		VAR = ((double) (((double) (((double) pow(((double) (((double) (((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) * ((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))))) / ((double) (((double) cbrt(((double) (((double) cbrt(((double) (x + y)))) * ((double) cbrt(((double) (x + y)))))))) * ((double) cbrt(((double) (((double) cbrt(((double) (x + y)))) * ((double) cbrt(((double) (x + y)))))))))))), x)) * ((double) pow(((double) (((double) cbrt(((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) / ((double) cbrt(((double) (((double) cbrt(((double) (x + y)))) * ((double) cbrt(((double) (x + y)))))))))), x)))) / ((double) (x / ((double) pow(((double) (((double) cbrt(x)) / ((double) cbrt(((double) (x + y)))))), x))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	7.9
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if x < -503.5755048876667 or 9.874893663935251e-08 < x

   1. Initial program 11.0

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

   2. Simplified 11.0

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

   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

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

   if -503.5755048876667 < x < 9.874893663935251e-08

   1. Initial program 11.7

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

   2. Simplified 11.7

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

   4. Applied add-cube-cbrt 11.7

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

   5. Applied add-cube-cbrt 11.7

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

   6. Applied times-frac 11.7

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

   7. Applied unpow-prod-down 2.8

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

   8. Applied associate-/l* 2.8

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

   10. Applied add-cube-cbrt 2.8

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

   11. Applied add-cube-cbrt 2.8

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

   12. Applied times-frac 2.8

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

   13. Applied unpow-prod-down 0.0

       \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x}}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -503.575504887666682 \lor \neg \left(x \le 9.87489366393525115 \cdot 10^{-8}\right):\\ \;\;\;\;e^{-1 \cdot y} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}\right)}^{x}}{\frac{x}{{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +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))