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

Average Error: 10.8 → 5.1

Time: 5.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 1.20102239621656768 \cdot 10^{23} \lor \neg \left(y \le 1.9614730774488089 \cdot 10^{177}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + y}\right)}}{x}\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if (((y <= 1.2010223962165677e+23) || !(y <= 1.961473077448809e+177))) {
		VAR = (exp((x * ((2.0 * log((cbrt(x) / cbrt((x + y))))) + log((cbrt(x) / cbrt((x + y))))))) / x);
	} else {
		VAR = (exp((x * log(((cbrt(x) * cbrt(x)) * (cbrt(x) / (x + y)))))) / x);
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	10.8
Target	7.9
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 < 1.2010223962165677e+23 or 1.961473077448809e+177 < y

   1. Initial program 7.5

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

   3. Applied add-cube-cbrt 28.6

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

   4. Applied add-cube-cbrt 7.5

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

   5. Applied times-frac 7.5

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

   6. Applied log-prod 3.5

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

   7. Simplified 2.4

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

   if 1.2010223962165677e+23 < y < 1.961473077448809e+177

   1. Initial program 33.9

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

   3. Applied *-un-lft-identity 33.9

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

   4. Applied add-cube-cbrt 23.8

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

   5. Applied times-frac 24.0

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

   6. Simplified 24.0

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

4. Final simplification 5.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 1.20102239621656768 \cdot 10^{23} \lor \neg \left(y \le 1.9614730774488089 \cdot 10^{177}\right):\\ \;\;\;\;\frac{e^{x \cdot \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{x + y}}\right)\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x \cdot \log \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{\sqrt[3]{x}}{x + y}\right)}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(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))