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

Average Error: 11.6 → 5.1

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 130.06933248317358 \lor \neg \left(y \le 1.514187256420666 \cdot 10^{102}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double temp;
	if (((y <= 130.06933248317358) || !(y <= 1.514187256420666e+102))) {
		temp = (exp(((x * (2.0 * log((cbrt(x) / cbrt((x + y)))))) + (x * log((cbrt(x) / cbrt((x + y))))))) / x);
	} else {
		temp = (exp((x * log(((x / (cbrt((x + y)) * cbrt((x + y)))) / cbrt((x + y)))))) / x);
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.6
Target	8.1
Herbie	5.1

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

Derivation

1. Split input into 2 regimes

2. if y < 130.06933248317358 or 1.514187256420666e+102 < y

   1. Initial program 9.0

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 28.3

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

   4. Applied add-cube-cbrt 9.0

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

   5. Applied times-frac 9.0

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

   6. Applied log-prod 4.5

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

   7. Applied distribute-lft-in 4.5

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

   8. Simplified 3.3

      \[\leadsto \frac{e^{\color{blue}{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\]

   if 130.06933248317358 < y < 1.514187256420666e+102

   1. Initial program 36.4

      \[\frac{e^{x \cdot \log \left(\frac{x}{x + y}\right)}}{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 22.5

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

   4. Applied associate-/r* 22.9

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 130.06933248317358 \lor \neg \left(y \le 1.514187256420666 \cdot 10^{102}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\frac{\frac{x}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}}}{\sqrt[3]{x + y}}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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))