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

Average Error: 18.7 → 0.1

Time: 9.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5153583551608314:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right) \cdot \left(x + -1\right)\right)\\ \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))

↓

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.5153583551608314)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (-
    1.0
    (log
     (*
      (+ (/ 1.0 y) (+ (/ 1.0 (pow y 3.0)) (/ 1.0 (pow y 2.0))))
      (+ x -1.0))))))

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 tmp;
	if (((x - y) / (1.0 - y)) <= 0.5153583551608314) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((((1.0 / y) + ((1.0 / pow(y, 3.0)) + (1.0 / pow(y, 2.0)))) * (x + -1.0)));
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.7
Target	0.1
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.5153583551608314

   1. Initial program 0.0

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

   2. Simplified 0.0

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]

   if 0.5153583551608314 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 60.7

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

   2. Taylor expanded in y around inf 0.4

      \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right)} \]

   3. Applied *-un-lft-identity_binary64 0.4

      \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{\color{blue}{1 \cdot {y}^{2}}}\right)\right)\right) \]

   4. Applied *-un-lft-identity_binary64 0.4

      \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{\color{blue}{1 \cdot 1}}{1 \cdot {y}^{2}}\right)\right)\right) \]

   5. Applied times-frac_binary64 0.4

      \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \color{blue}{\frac{1}{1} \cdot \frac{1}{{y}^{2}}}\right)\right)\right) \]

   6. Applied *-un-lft-identity_binary64 0.4

      \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{y} + \left(\frac{1}{\color{blue}{1 \cdot {y}^{3}}} + \frac{1}{1} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

   7. Applied *-un-lft-identity_binary64 0.4

      \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{y} + \left(\frac{\color{blue}{1 \cdot 1}}{1 \cdot {y}^{3}} + \frac{1}{1} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

   8. Applied times-frac_binary64 0.4

      \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{y} + \left(\color{blue}{\frac{1}{1} \cdot \frac{1}{{y}^{3}}} + \frac{1}{1} \cdot \frac{1}{{y}^{2}}\right)\right)\right) \]

   9. Applied distribute-lft-out_binary64 0.4

      \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{y} + \color{blue}{\frac{1}{1} \cdot \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)}\right)\right) \]

   10. Applied *-un-lft-identity_binary64 0.4

       \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{1}{\color{blue}{1 \cdot y}} + \frac{1}{1} \cdot \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right) \]

   11. Applied *-un-lft-identity_binary64 0.4

       \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\frac{\color{blue}{1 \cdot 1}}{1 \cdot y} + \frac{1}{1} \cdot \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right) \]

   12. Applied times-frac_binary64 0.4

       \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \left(\color{blue}{\frac{1}{1} \cdot \frac{1}{y}} + \frac{1}{1} \cdot \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right) \]

   13. Applied distribute-lft-out_binary64 0.4

       \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \frac{x}{{y}^{2}}\right)\right) - \color{blue}{\frac{1}{1} \cdot \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)}\right) \]

   14. Applied div-inv_binary64 0.4

       \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\frac{x}{{y}^{3}} + \color{blue}{x \cdot \frac{1}{{y}^{2}}}\right)\right) - \frac{1}{1} \cdot \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right) \]

   15. Applied div-inv_binary64 0.4

       \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \left(\color{blue}{x \cdot \frac{1}{{y}^{3}}} + x \cdot \frac{1}{{y}^{2}}\right)\right) - \frac{1}{1} \cdot \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right) \]

   16. Applied distribute-lft-out_binary64 0.4

       \[\leadsto 1 - \log \left(\left(\frac{x}{y} + \color{blue}{x \cdot \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)}\right) - \frac{1}{1} \cdot \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right) \]

   17. Applied div-inv_binary64 0.4

       \[\leadsto 1 - \log \left(\left(\color{blue}{x \cdot \frac{1}{y}} + x \cdot \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right) - \frac{1}{1} \cdot \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right) \]

   18. Applied distribute-lft-out_binary64 0.4

       \[\leadsto 1 - \log \left(\color{blue}{x \cdot \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)} - \frac{1}{1} \cdot \left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right)\right) \]

   19. Applied distribute-rgt-out--_binary64 0.4

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5153583551608314:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{1}{y} + \left(\frac{1}{{y}^{3}} + \frac{1}{{y}^{2}}\right)\right) \cdot \left(x + -1\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))