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

Average Error: 18.7 → 0.1

Time: 6.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -160923004.823839575 \lor \neg \left(y \le 292197028.99763989\right):\\ \;\;\;\;\log \left(\sqrt{e^{1}}\right) - \log \left(\frac{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}{\sqrt{e^{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\sqrt{1 - \frac{x - y}{1 - y}}}\right) - \log \left(\sqrt{\frac{\sqrt{1 - \frac{x - y}{1 - y}}}{\sqrt{e^{1}}}} \cdot \sqrt{\frac{\sqrt{1 - \frac{x - y}{1 - y}}}{\sqrt{e^{1}}}}\right)\\ \end{array}\]

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 (((y <= -160923004.82383958) || !(y <= 292197028.9976399))) {
		VAR = ((double) (((double) log(((double) sqrt(((double) exp(1.0)))))) - ((double) log(((double) (((double) (((double) (1.0 * ((double) (((double) (x / ((double) pow(y, 2.0)))) - ((double) (1.0 / y)))))) + ((double) (x / y)))) / ((double) sqrt(((double) exp(1.0))))))))));
	} else {
		VAR = ((double) (((double) log(((double) (((double) sqrt(((double) exp(1.0)))) / ((double) sqrt(((double) (1.0 - ((double) (((double) (x - y)) / ((double) (1.0 - y)))))))))))) - ((double) log(((double) (((double) sqrt(((double) (((double) sqrt(((double) (1.0 - ((double) (((double) (x - y)) / ((double) (1.0 - y)))))))) / ((double) sqrt(((double) exp(1.0)))))))) * ((double) sqrt(((double) (((double) sqrt(((double) (1.0 - ((double) (((double) (x - y)) / ((double) (1.0 - y)))))))) / ((double) sqrt(((double) exp(1.0))))))))))))));
	}
	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 y < -160923004.82383958 or 292197028.9976399 < y

   1. Initial program 46.7

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

   3. Applied add-log-exp 46.7

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

   4. Applied diff-log 46.7

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

   6. Applied add-sqr-sqrt 46.7

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

   7. Applied associate-/l* 46.7

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

   8. Applied log-div 46.7

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

   9. Taylor expanded around inf 0.1

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

   10. Simplified 0.1

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

   if -160923004.82383958 < y < 292197028.9976399

   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

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

   10. Applied *-un-lft-identity 0.1

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

   11. Applied sqrt-prod 0.1

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

   12. Applied add-sqr-sqrt 0.1

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

   13. Applied times-frac 0.1

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

   14. Applied log-prod 0.1

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

   15. Applied associate--r+ 0.1

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

   16. Simplified 0.1

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

   18. Applied add-sqr-sqrt 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -160923004.823839575 \lor \neg \left(y \le 292197028.99763989\right):\\ \;\;\;\;\log \left(\sqrt{e^{1}}\right) - \log \left(\frac{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}{\sqrt{e^{1}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\sqrt{1 - \frac{x - y}{1 - y}}}\right) - \log \left(\sqrt{\frac{\sqrt{1 - \frac{x - y}{1 - y}}}{\sqrt{e^{1}}}} \cdot \sqrt{\frac{\sqrt{1 - \frac{x - y}{1 - y}}}{\sqrt{e^{1}}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(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))))))