Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B

Average Error: 18.5 → 0.3

Time: 4.8s

Precision: 64

Math

\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -79025762686.63672:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{elif}\;y \le 4.14547202165829592 \cdot 10^{23}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \frac{\frac{-\frac{1}{y}}{y} + \frac{x}{\frac{{y}^{4}}{x}}}{\frac{x}{{y}^{2}} + \frac{1}{y}} + \frac{x}{y}\right)\\ \end{array}\]

C

double code(double x, double y) {
	return (1.0 - log((1.0 - ((x - y) / (1.0 - y)))));
}

↓

double code(double x, double y) {
	double temp;
	if ((y <= -79025762686.63672)) {
		temp = (1.0 - log(((1.0 * ((x / pow(y, 2.0)) - (1.0 / y))) + (x / y))));
	} else {
		double temp_1;
		if ((y <= 4.145472021658296e+23)) {
			temp_1 = (1.0 - (log(sqrt((1.0 - ((x - y) / (1.0 - y))))) + log(sqrt((1.0 - ((x - y) / (1.0 - y)))))));
		} else {
			temp_1 = (1.0 - log(((1.0 * (((-(1.0 / y) / y) + (x / (pow(y, 4.0) / x))) / ((x / pow(y, 2.0)) + (1.0 / y)))) + (x / y))));
		}
		temp = temp_1;
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.5
Target	0.1
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if y < -79025762686.63672

   1. Initial program 52.6

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

   2. Taylor expanded around inf 0.1

      \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)}\]

   3. Simplified 0.1

      \[\leadsto 1 - \log \color{blue}{\left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)}\]

   if -79025762686.63672 < y < 4.145472021658296e+23

   1. Initial program 0.2

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 0.2

      \[\leadsto 1 - \log \color{blue}{\left(\sqrt{1 - \frac{x - y}{1 - y}} \cdot \sqrt{1 - \frac{x - y}{1 - y}}\right)}\]

   4. Applied log-prod 0.2

      \[\leadsto 1 - \color{blue}{\left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)}\]

   if 4.145472021658296e+23 < y

   1. Initial program 32.5

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

   2. Taylor expanded around inf 0.0

      \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)}\]

   3. Simplified 0.0

      \[\leadsto 1 - \log \color{blue}{\left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)}\]
   4. Using strategy rm

   5. Applied flip-- 1.8

      \[\leadsto 1 - \log \left(1 \cdot \color{blue}{\frac{\frac{x}{{y}^{2}} \cdot \frac{x}{{y}^{2}} - \frac{1}{y} \cdot \frac{1}{y}}{\frac{x}{{y}^{2}} + \frac{1}{y}}} + \frac{x}{y}\right)\]

   6. Simplified 1.8

      \[\leadsto 1 - \log \left(1 \cdot \frac{\color{blue}{\frac{-\frac{1}{y}}{y} + \frac{x}{\frac{{y}^{4}}{x}}}}{\frac{x}{{y}^{2}} + \frac{1}{y}} + \frac{x}{y}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -79025762686.63672:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{elif}\;y \le 4.14547202165829592 \cdot 10^{23}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \frac{\frac{-\frac{1}{y}}{y} + \frac{x}{\frac{{y}^{4}}{x}}}{\frac{x}{{y}^{2}} + \frac{1}{y}} + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))