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

Average Error: 11.3 → 1.6

Time: 3.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -15.5218136111124299 \lor \neg \left(x \le 2.533222095666427 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}{x}\\ \end{array}\]

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 (((x <= -15.52181361111243) || !(x <= 2.533222095666427e-13))) {
		VAR = ((double) (1.0 / ((double) (x * ((double) exp(y))))));
	} else {
		VAR = ((double) (((double) (((double) pow(((double) (((double) (((double) cbrt(x)) * ((double) cbrt(x)))) / 1.0)), x)) * ((double) pow(((double) (((double) cbrt(x)) / ((double) (x + y)))), x)))) / x));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	11.3
Target	7.8
Herbie	1.6

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

Derivation

1. Split input into 2 regimes

2. if x < -15.52181361111243 or 2.533222095666427e-13 < x

   1. Initial program 11.3

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

   2. Simplified 11.3

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

   3. Taylor expanded around inf 0.5

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

   4. Simplified 0.5

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

   6. Applied clear-num 0.5

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

   7. Simplified 0.5

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

   if -15.52181361111243 < x < 2.533222095666427e-13

   1. Initial program 11.3

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

   2. Simplified 11.3

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

   4. Applied *-un-lft-identity 11.3

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

   5. Applied add-cube-cbrt 11.3

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

   6. Applied times-frac 11.3

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

   7. Applied unpow-prod-down 3.0

      \[\leadsto \frac{\color{blue}{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}}{x}\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15.5218136111124299 \lor \neg \left(x \le 2.533222095666427 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{1}{x \cdot e^{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right)}^{x} \cdot {\left(\frac{\sqrt[3]{x}}{x + y}\right)}^{x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(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))