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

Average Error: 18.0 → 0.2

Time: 7.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -954400778077.866821 \lor \neg \left(y \le 0.99315470732115574\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, 1, -\frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt{\sqrt[3]{1 - y}}}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt{\sqrt[3]{1 - y}}}\right)\right)\\ \end{array}\]

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 <= -954400778077.8668) || !(y <= 0.9931547073211557))) {
		VAR = (1.0 - log(fma(1.0, ((x / pow(y, 2.0)) - (1.0 / y)), (x / y))));
	} else {
		VAR = (1.0 - log((fma(1.0, 1.0, -((cbrt((1.0 + y)) / sqrt(cbrt((1.0 - y)))) * (((x - y) / (cbrt(((1.0 * 1.0) - (y * y))) * cbrt((1.0 - y)))) / sqrt(cbrt((1.0 - y)))))) + fma(-(cbrt((1.0 + y)) / sqrt(cbrt((1.0 - y)))), (((x - y) / (cbrt(((1.0 * 1.0) - (y * y))) * cbrt((1.0 - y)))) / sqrt(cbrt((1.0 - y)))), ((cbrt((1.0 + y)) / sqrt(cbrt((1.0 - y)))) * (((x - y) / (cbrt(((1.0 * 1.0) - (y * y))) * cbrt((1.0 - y)))) / sqrt(cbrt((1.0 - y)))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.0
Target	0.1
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if y < -954400778077.8668 or 0.9931547073211557 < y

   1. Initial program 47.0

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

   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

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

   if -954400778077.8668 < y < 0.9931547073211557

   1. Initial program 0.2

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

   3. Applied add-cube-cbrt 0.3

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

   4. Applied associate-/r* 0.3

      \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 0.3

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

   7. Applied flip-- 0.3

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

   8. Applied cbrt-div 0.2

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

   9. Applied associate-*r/ 0.2

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

   10. Applied associate-/r/ 0.2

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

   11. Applied times-frac 0.2

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

   12. Applied add-sqr-sqrt 0.2

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

   13. Applied prod-diff 0.2

       \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(\sqrt{1}, \sqrt{1}, -\frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 \cdot 1 - y \cdot y}}}{\sqrt{\sqrt[3]{1 - y}}}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 \cdot 1 - y \cdot y}}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 \cdot 1 - y \cdot y}}}{\sqrt{\sqrt[3]{1 - y}}}\right)\right)}\]

   14. Simplified 0.2

       \[\leadsto 1 - \log \left(\color{blue}{\mathsf{fma}\left(1, 1, -\frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt{\sqrt[3]{1 - y}}}\right)} + \mathsf{fma}\left(-\frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 \cdot 1 - y \cdot y}}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 \cdot 1 - y \cdot y}}}{\sqrt{\sqrt[3]{1 - y}}}\right)\right)\]

   15. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -954400778077.866821 \lor \neg \left(y \le 0.99315470732115574\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, 1, -\frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt{\sqrt[3]{1 - y}}}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt{\sqrt[3]{1 - y}}}, \frac{\sqrt[3]{1 + y}}{\sqrt{\sqrt[3]{1 - y}}} \cdot \frac{\frac{x - y}{\sqrt[3]{1 \cdot 1 - y \cdot y} \cdot \sqrt[3]{1 - y}}}{\sqrt{\sqrt[3]{1 - y}}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 +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))))))