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

Average Error: 18.4 → 0.1

Time: 5.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -73256728.0171860307 \lor \neg \left(y \le 73789365.95162582\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right) + \log \left(\sqrt{1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right)\right)\\ \end{array}\]

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 temp;
	if (((y <= -73256728.01718603) || !(y <= 73789365.95162582))) {
		temp = (1.0 - log(fma(1.0, ((x / pow(y, 2.0)) - (1.0 / y)), (x / y))));
	} else {
		temp = (1.0 - (log(sqrt((1.0 - ((cbrt((x - y)) * cbrt((x - y))) * (cbrt((x - y)) / (1.0 - y)))))) + log(sqrt((1.0 - ((cbrt((x - y)) * cbrt((x - y))) * (cbrt((x - y)) / (1.0 - y))))))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.4
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 < -73256728.01718603 or 73789365.95162582 < y

   1. Initial program 46.8

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

   2. Taylor expanded around inf 0.2

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

   3. Simplified 0.2

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

   if -73256728.01718603 < y < 73789365.95162582

   1. Initial program 0.1

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

   3. Applied *-un-lft-identity 0.1

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

   4. Applied add-cube-cbrt 0.1

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

   5. Applied times-frac 0.1

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

   6. Simplified 0.1

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

   8. Applied add-sqr-sqrt 0.1

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

   9. Applied log-prod 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -73256728.0171860307 \lor \neg \left(y \le 73789365.95162582\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right) + \log \left(\sqrt{1 - \left(\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}\right) \cdot \frac{\sqrt[3]{x - y}}{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))))