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

Average Error: 11.3 → 5.6

Time: 5.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 5489.1021751240096:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{elif}\;y \le 8.5693806117251377 \cdot 10^{46}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{x}}{\sqrt{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt{x}}{\sqrt{x + y}}\right)}^{x}}{x}\\ \end{array}\]

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 ((y <= 5489.10217512401)) {
		VAR = ((double) (((double) (((double) pow(((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / ((double) (((double) cbrt(((double) (x + y)))) * ((double) cbrt(((double) (x + y)))))))), x)) * ((double) pow(((double) (((double) cbrt(x)) / ((double) cbrt(((double) (x + y)))))), x)))) / x));
	} else {
		double VAR_1;
		if ((y <= 8.569380611725138e+46)) {
			VAR_1 = ((double) log(((double) exp(((double) (((double) pow(((double) (x / ((double) (x + y)))), x)) / x))))));
		} else {
			VAR_1 = ((double) (((double) (((double) pow(((double) (((double) sqrt(x)) / ((double) sqrt(((double) (x + y)))))), x)) * ((double) pow(((double) (((double) sqrt(x)) / ((double) sqrt(((double) (x + y)))))), x)))) / x));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	8.0
Herbie	5.6

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

Derivation

1. Split input into 3 regimes

2. if y < 5489.1021751240096

   1. Initial program 4.7

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

   2. Simplified 4.7

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

   4. Applied add-cube-cbrt 28.5

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

   5. Applied add-cube-cbrt 4.7

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

   6. Applied times-frac 4.7

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

   7. Applied unpow-prod-down 1.6

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

   if 5489.1021751240096 < y < 8.5693806117251377e46

   1. Initial program 38.4

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

   2. Simplified 38.4

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

   4. Applied add-log-exp 18.6

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

   if 8.5693806117251377e46 < y

   1. Initial program 31.6

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

   2. Simplified 31.6

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

   4. Applied add-sqr-sqrt 32.0

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

   5. Applied add-sqr-sqrt 33.4

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

   6. Applied times-frac 33.4

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

   7. Applied unpow-prod-down 18.9

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

4. Final simplification 5.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 5489.1021751240096:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}^{x}}{x}\\ \mathbf{elif}\;y \le 8.5693806117251377 \cdot 10^{46}:\\ \;\;\;\;\log \left(e^{\frac{{\left(\frac{x}{x + y}\right)}^{x}}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt{x}}{\sqrt{x + y}}\right)}^{x} \cdot {\left(\frac{\sqrt{x}}{\sqrt{x + y}}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(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 (/ -1.0 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 (/ -1.0 y)) x))))

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