Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2

Average Error: 1.8 → 0.2

Time: 35.7s

Precision: 64

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -6.12033557721357572 \cdot 10^{-38} \lor \neg \left(x \le 1.2266431991864999 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{1}{y} \cdot \left(\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}{\frac{y}{x}}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((x <= -6.120335577213576e-38) || !(x <= 1.2266431991864999e-73))) {
		VAR = ((1.0 / y) * ((pow((1.0 / a), 1.0) / exp(((y * log((1.0 / z))) + ((log((1.0 / a)) * t) + b)))) * x));
	} else {
		VAR = ((pow((1.0 / a), 1.0) / exp(((y * log((1.0 / z))) + ((log((1.0 / a)) * t) + b)))) / (y / x));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

1. Split input into 2 regimes

2. if x < -6.120335577213576e-38 or 1.2266431991864999e-73 < x

   1. Initial program 0.9

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   2. Taylor expanded around inf 0.9

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

   3. Simplified 10.7

      \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}{\frac{y}{x}}}\]
   4. Using strategy rm

   5. Applied div-inv 10.7

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

   6. Applied *-un-lft-identity 10.7

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

   7. Applied *-un-lft-identity 10.7

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

   8. Applied *-un-lft-identity 10.7

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

   9. Applied times-frac 10.7

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

   10. Applied unpow-prod-down 10.7

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

   11. Applied times-frac 10.7

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

   12. Applied times-frac 0.2

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

   13. Simplified 0.2

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

   14. Simplified 0.2

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

   if -6.120335577213576e-38 < x < 1.2266431991864999e-73

   1. Initial program 3.0

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

   2. Taylor expanded around inf 3.1

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}{\frac{y}{x}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.12033557721357572 \cdot 10^{-38} \lor \neg \left(x \le 1.2266431991864999 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{1}{y} \cdot \left(\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)}}}{\frac{y}{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020078 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))