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

Percentage Accurate: 72.3% → 99.8%

Time: 14.7s

Precision: binary64

Cost: 7492

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999998)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

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)) <= 0.999999998) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	return 1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))));
}

↓

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999998) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	return 1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))

↓

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999998:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	return Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
end

↓

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999998)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999999998], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Target

Original	72.3%
Target	99.8%
Herbie	99.8%

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999999997999999946

   1. Initial program 99.6%

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

   2. Simplified 99.7%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \]
      Step-by-step derivation

      [Start] 99.6%	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 99.6%	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 99.7%	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      neg-sub0 [=>] 99.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{0 - \frac{x - y}{1 - y}}\right) \]
      div-sub [=>] 99.7%	\[ 1 - \mathsf{log1p}\left(0 - \color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      associate--r- [=>] 99.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(0 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}}\right) \]
      neg-sub0 [<=] 99.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\frac{x}{1 - y}\right)} + \frac{y}{1 - y}\right) \]
      +-commutative [=>] 99.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} + \left(-\frac{x}{1 - y}\right)}\right) \]
      sub-neg [<=] 99.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 99.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   if 0.999999997999999946 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 4.1%

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

   2. Simplified 4.1%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \]
      Step-by-step derivation

      [Start] 4.1%	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 4.1%	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 4.1%	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      neg-sub0 [=>] 4.1%	\[ 1 - \mathsf{log1p}\left(\color{blue}{0 - \frac{x - y}{1 - y}}\right) \]
      div-sub [=>] 4.2%	\[ 1 - \mathsf{log1p}\left(0 - \color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      associate--r- [=>] 4.2%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(0 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}}\right) \]
      neg-sub0 [<=] 4.2%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\frac{x}{1 - y}\right)} + \frac{y}{1 - y}\right) \]
      +-commutative [=>] 4.2%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} + \left(-\frac{x}{1 - y}\right)}\right) \]
      sub-neg [<=] 4.2%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 4.1%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Taylor expanded in y around -inf 85.8%

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

   4. Simplified 85.8%

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

      [Start] 85.8%	\[ 1 - \left(\log \left(-1 \cdot \left(x - 1\right)\right) + \log \left(\frac{-1}{y}\right)\right) \]
      sub-neg [=>] 85.8%	\[ 1 - \left(\log \left(-1 \cdot \color{blue}{\left(x + \left(-1\right)\right)}\right) + \log \left(\frac{-1}{y}\right)\right) \]
      metadata-eval [=>] 85.8%	\[ 1 - \left(\log \left(-1 \cdot \left(x + \color{blue}{-1}\right)\right) + \log \left(\frac{-1}{y}\right)\right) \]
      distribute-lft-in [=>] 85.8%	\[ 1 - \left(\log \color{blue}{\left(-1 \cdot x + -1 \cdot -1\right)} + \log \left(\frac{-1}{y}\right)\right) \]
      metadata-eval [=>] 85.8%	\[ 1 - \left(\log \left(-1 \cdot x + \color{blue}{1}\right) + \log \left(\frac{-1}{y}\right)\right) \]
      +-commutative [<=] 85.8%	\[ 1 - \left(\log \color{blue}{\left(1 + -1 \cdot x\right)} + \log \left(\frac{-1}{y}\right)\right) \]
      log1p-def [=>] 85.8%	\[ 1 - \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)} + \log \left(\frac{-1}{y}\right)\right) \]
      mul-1-neg [=>] 85.8%	\[ 1 - \left(\mathsf{log1p}\left(\color{blue}{-x}\right) + \log \left(\frac{-1}{y}\right)\right) \]

   5. Taylor expanded in y around 0 0.0%

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

   6. Simplified 99.9%

      \[\leadsto 1 - \color{blue}{\log \left(\frac{-\left(1 - x\right)}{y}\right)} \]
      Step-by-step derivation

      [Start] 0.0%	\[ 1 - \left(\log -1 + \left(\log \left(1 - x\right) + -1 \cdot \log y\right)\right) \]
      neg-mul-1 [<=] 0.0%	\[ 1 - \left(\log -1 + \left(\log \left(1 - x\right) + \color{blue}{\left(-\log y\right)}\right)\right) \]
      log-rec [<=] 0.0%	\[ 1 - \left(\log -1 + \left(\log \left(1 - x\right) + \color{blue}{\log \left(\frac{1}{y}\right)}\right)\right) \]
      +-commutative [=>] 0.0%	\[ 1 - \left(\log -1 + \color{blue}{\left(\log \left(\frac{1}{y}\right) + \log \left(1 - x\right)\right)}\right) \]
      associate-+r+ [=>] 0.0%	\[ 1 - \color{blue}{\left(\left(\log -1 + \log \left(\frac{1}{y}\right)\right) + \log \left(1 - x\right)\right)} \]
      log-rec [=>] 0.0%	\[ 1 - \left(\left(\log -1 + \color{blue}{\left(-\log y\right)}\right) + \log \left(1 - x\right)\right) \]
      sub-neg [<=] 0.0%	\[ 1 - \left(\color{blue}{\left(\log -1 - \log y\right)} + \log \left(1 - x\right)\right) \]
      log-div [<=] 85.8%	\[ 1 - \left(\color{blue}{\log \left(\frac{-1}{y}\right)} + \log \left(1 - x\right)\right) \]
      log-prod [<=] 99.9%	\[ 1 - \color{blue}{\log \left(\frac{-1}{y} \cdot \left(1 - x\right)\right)} \]
      associate-*l/ [=>] 99.9%	\[ 1 - \log \color{blue}{\left(\frac{-1 \cdot \left(1 - x\right)}{y}\right)} \]
      mul-1-neg [=>] 99.9%	\[ 1 - \log \left(\frac{\color{blue}{-\left(1 - x\right)}}{y}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternative 2
Accuracy	98.5%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;y \leq -3.85 \lor \neg \left(y \leq 350000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]

Alternative 3
Accuracy	98.7%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	79.7%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;y \leq -18:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	79.1%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -11:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 6
Accuracy	62.5%
Cost	6656

\[1 - \mathsf{log1p}\left(-x\right) \]

Alternative 7
Accuracy	5.6%
Cost	704

\[1 + \left(\frac{-1}{y} - \frac{0.5}{y \cdot y}\right) \]

Reproduce

herbie shell --seed 2023263 
(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))))))