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

Average Error: 18.3 → 0.1

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -411945099.211768568 \lor \neg \left(y \le 36362977.003381863\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{e^{1}}{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{\frac{e^{1}}{1 - \frac{x - y}{1 - y}}}\right)\\ \end{array}\]

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 VAR;
	if (((y <= -411945099.21176857) || !(y <= 36362977.00338186))) {
		VAR = log((exp(1.0) / ((1.0 * ((x / pow(y, 2.0)) - (1.0 / y))) + (x / y))));
	} else {
		VAR = (log(sqrt((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))) + log(sqrt((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.3
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -411945099.21176857 or 36362977.00338186 < y

   1. Initial program 46.8

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

   3. Applied add-log-exp 46.8

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

   4. Applied diff-log 46.8

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

   5. Taylor expanded around inf 0.1

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

   6. Simplified 0.1

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

   if -411945099.21176857 < y < 36362977.00338186

   1. Initial program 0.1

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

   3. Applied add-log-exp 0.1

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

   4. Applied diff-log 0.1

      \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 0.1

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

   7. Applied log-prod 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -411945099.211768568 \lor \neg \left(y \le 36362977.003381863\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{\frac{e^{1}}{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{\frac{e^{1}}{1 - \frac{x - y}{1 - y}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 
(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))))))