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

Average Error: 18.8 → 0.6

Time: 5.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.89532822582097144 \cdot 10^{-6}:\\ \;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\frac{1 - \frac{x - y}{1 - y}}{\sqrt{e^{1}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

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)))) <= 3.8953282258209714e-06)) {
		VAR = ((double) log(((double) (((double) sqrt(((double) exp(1.0)))) / ((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) (((double) (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	18.8
Target	0.1
Herbie	0.6

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

Derivation

1. Split input into 2 regimes

2. if (/ (- x y) (- 1.0 y)) < 3.8953282258209714e-06

   1. Initial program 0.0

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

   3. Applied add-log-exp 0.0

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

   4. Applied diff-log 0.0

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

   6. Applied add-sqr-sqrt 0.0

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

   7. Applied associate-/l* 0.0

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

   if 3.8953282258209714e-06 < (/ (- x y) (- 1.0 y))

   1. Initial program 60.0

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

   2. Taylor expanded around inf 1.8

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

   3. Simplified 1.8

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.89532822582097144 \cdot 10^{-6}:\\ \;\;\;\;\log \left(\frac{\sqrt{e^{1}}}{\frac{1 - \frac{x - y}{1 - y}}{\sqrt{e^{1}}}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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))))))