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

Average Error: 18.4 → 0.1

Time: 3.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -44196708.016604125 \lor \neg \left(y \leq 55739289.00260231\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 + \frac{\frac{y - x}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (or (<= y -44196708.016604125) (not (<= y 55739289.00260231)))
   (- 1.0 (log (+ (/ x y) (* (/ 1.0 y) (+ (/ x y) -1.0)))))
   (-
    1.0
    (log
     (+
      1.0
      (/
       (/ (- y x) (* (cbrt (- 1.0 y)) (cbrt (- 1.0 y))))
       (cbrt (- 1.0 y))))))))

C

double code(double x, double y) {
	return ((double) (1.0 - ((double) log(((double) (1.0 - (((double) (x - y)) / ((double) (1.0 - y)))))))));
}

↓

double code(double x, double y) {
	double VAR;
	if (((y <= -44196708.016604125) || !(y <= 55739289.00260231))) {
		VAR = ((double) (1.0 - ((double) log(((double) ((x / y) + ((double) ((1.0 / y) * ((double) ((x / y) + -1.0))))))))));
	} else {
		VAR = ((double) (1.0 - ((double) log(((double) (1.0 + ((((double) (y - x)) / ((double) (((double) cbrt(((double) (1.0 - y)))) * ((double) cbrt(((double) (1.0 - y))))))) / ((double) cbrt(((double) (1.0 - y)))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.4
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -44196708.016604125 or 55739289.002602309 < y

   1. Initial program 46.3

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

   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

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

   if -44196708.016604125 < y < 55739289.002602309

   1. Initial program 0.1

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

   3. Applied add-cube-cbrt 0.1

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

   4. Applied associate-/r* 0.1

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -44196708.016604125 \lor \neg \left(y \leq 55739289.00260231\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \frac{1}{y} \cdot \left(\frac{x}{y} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 + \frac{\frac{y - x}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020198 
(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))))))