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

Percentage Accurate: 72.0% → 99.8%

Time: 10.9s

Precision: binary64

Cost: 7492

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

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)) 0.99999998)
   (- 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)) <= 0.99999998) {
		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)) <= 0.99999998) {
		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)) <= 0.99999998:
		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)) <= 0.99999998)
		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], 0.99999998], 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	72.0%
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)) < 0.999999980000000011

   1. Initial program 99.8%

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

   2. Simplified 99.8%

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

      [Start] 99.8%	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 99.8%	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 99.8%	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      neg-sub0 [=>] 99.8%	\[ 1 - \mathsf{log1p}\left(\color{blue}{0 - \frac{x - y}{1 - y}}\right) \]
      div-sub [=>] 99.8%	\[ 1 - \mathsf{log1p}\left(0 - \color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      associate--r- [=>] 99.8%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(0 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}}\right) \]
      neg-sub0 [<=] 99.8%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\frac{x}{1 - y}\right)} + \frac{y}{1 - y}\right) \]
      +-commutative [=>] 99.8%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} + \left(-\frac{x}{1 - y}\right)}\right) \]
      sub-neg [<=] 99.8%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 99.8%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   if 0.999999980000000011 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 4.7%

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

   2. Simplified 4.7%

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

      [Start] 4.7%	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 4.7%	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 4.7%	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      neg-sub0 [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{0 - \frac{x - y}{1 - y}}\right) \]
      div-sub [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(0 - \color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      associate--r- [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(0 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}}\right) \]
      neg-sub0 [<=] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\frac{x}{1 - y}\right)} + \frac{y}{1 - y}\right) \]
      +-commutative [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} + \left(-\frac{x}{1 - y}\right)}\right) \]
      sub-neg [<=] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Applied egg-rr 6.9%

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

      [Start] 4.7%	\[ 1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right) \]
      clear-num [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{1}{\frac{1 - y}{y - x}}}\right) \]
      associate-/r/ [=>] 6.9%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{1}{1 - y} \cdot \left(y - x\right)}\right) \]

   4. Taylor expanded in y around inf 4.7%

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

   5. Simplified 4.7%

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

      [Start] 4.7%	\[ 1 - \mathsf{log1p}\left(\frac{x}{y} - \left(\frac{1}{y} + 1\right)\right) \]
      associate--r+ [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(\frac{x}{y} - \frac{1}{y}\right) - 1}\right) \]
      div-sub [<=] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - 1}{y}} - 1\right) \]
      sub-neg [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{x - 1}{y} + \left(-1\right)}\right) \]
      sub-neg [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\frac{\color{blue}{x + \left(-1\right)}}{y} + \left(-1\right)\right) \]
      metadata-eval [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\frac{x + \color{blue}{-1}}{y} + \left(-1\right)\right) \]
      metadata-eval [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\frac{x + -1}{y} + \color{blue}{-1}\right) \]

   6. Applied egg-rr 100.0%

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

      [Start] 4.7%	\[ 1 - \mathsf{log1p}\left(\frac{x + -1}{y} + -1\right) \]
      div-inv [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(x + -1\right) \cdot \frac{1}{y}} + -1\right) \]
      fma-def [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(x + -1, \frac{1}{y}, -1\right)}\right) \]
      metadata-eval [<=] 4.7%	\[ 1 - \mathsf{log1p}\left(\mathsf{fma}\left(x + -1, \frac{1}{y}, \color{blue}{-1}\right)\right) \]
      fma-neg [<=] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(x + -1\right) \cdot \frac{1}{y} - 1}\right) \]
      div-inv [<=] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{x + -1}{y}} - 1\right) \]
      add-exp-log [=>] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{x + -1}{y}\right)}} - 1\right) \]
      expm1-udef [<=] 4.7%	\[ 1 - \mathsf{log1p}\left(\color{blue}{\mathsf{expm1}\left(\log \left(\frac{x + -1}{y}\right)\right)}\right) \]
      log1p-expm1-u [<=] 100.0%	\[ 1 - \color{blue}{\log \left(\frac{x + -1}{y}\right)} \]
      add-sqr-sqrt [=>] 60.0%	\[ 1 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} + -1}{y}\right) \]
      fma-def [=>] 60.0%	\[ 1 - \log \left(\frac{\color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -1\right)}}{y}\right) \]
      metadata-eval [<=] 60.0%	\[ 1 - \log \left(\frac{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, \color{blue}{-1}\right)}{y}\right) \]
      fma-neg [<=] 60.0%	\[ 1 - \log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x} - 1}}{y}\right) \]
      add-sqr-sqrt [<=] 100.0%	\[ 1 - \log \left(\frac{\color{blue}{x} - 1}{y}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.8%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternative 2
Accuracy	83.3%
Cost	7181

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+68}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -2.3 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	82.7%
Cost	7117

\[\begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+68}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1 \lor \neg \left(y \leq 0.0128\right):\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \end{array} \]

Alternative 4
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 5
Accuracy	98.7%
Cost	7113

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

Alternative 6
Accuracy	78.8%
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -28000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - x\right)\\ \end{array} \]

Alternative 7
Accuracy	62.1%
Cost	6720

\[1 - \log \left(1 - x\right) \]

Alternative 8
Accuracy	62.1%
Cost	6656

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

Alternative 9
Accuracy	43.8%
Cost	448

\[1 + \frac{x}{1 - y} \]

Alternative 10
Accuracy	42.4%
Cost	192

\[x + 1 \]

Alternative 11
Accuracy	42.1%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023165 
(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))))))