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

Average Error: 18.7 → 0.1

Time: 7.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999272517870863:\\ \;\;\;\;\log \left(\sqrt{e^{1}}\right) - \log \left(\frac{1 - \frac{x - y}{1 - y}}{\sqrt{e^{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]

C

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

↓

double code(double x, double y) {
	double VAR;
	if ((((double) (((double) (x - y)) / ((double) (1.0 - y)))) <= 0.9999927251787086)) {
		VAR = ((double) (((double) log(((double) sqrt(((double) exp(1.0)))))) - ((double) log(((double) (((double) (1.0 - ((double) (((double) (x - y)) / ((double) (1.0 - y)))))) / ((double) sqrt(((double) exp(1.0))))))))));
	} else {
		VAR = ((double) (1.0 - ((double) log(((double) fma(1.0, ((double) (((double) (x / ((double) pow(y, 2.0)))) - ((double) (1.0 / y)))), ((double) (x / y))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.7
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 (/ (- x y) (- 1.0 y)) < 0.9999927251787086

   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

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

   7. Applied associate-/l* 0.1

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

   8. Applied log-div 0.1

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

   if 0.9999927251787086 < (/ (- x y) (- 1.0 y))

   1. Initial program 62.2

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

   2. Taylor expanded around inf 0.3

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

   3. Simplified 0.3

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999272517870863:\\ \;\;\;\;\log \left(\sqrt{e^{1}}\right) - \log \left(\frac{1 - \frac{x - y}{1 - y}}{\sqrt{e^{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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))))))