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

Average Error: 18.6 → 0.7

Time: 11.1s

Precision: binary64

Cost: 7176

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.78:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 0.055:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-x}{1 - y}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y)
 :precision binary64
 (if (<= y -1.78)
   (- 1.0 (log (/ (+ x -1.0) y)))
   (if (<= y 0.055)
     (- 1.0 (+ y (log1p (- x))))
     (- 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 (y <= -1.78) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else if (y <= 0.055) {
		tmp = 1.0 - (y + log1p(-x));
	} 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 (y <= -1.78) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else if (y <= 0.055) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} 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 y <= -1.78:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	elif y <= 0.055:
		tmp = 1.0 - (y + math.log1p(-x))
	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 (y <= -1.78)
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	elseif (y <= 0.055)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(-x) / Float64(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[y, -1.78], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.055], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Target

Original	18.6
Target	0.1
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if y < -1.78000000000000003

   1. Initial program 50.5

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

   2. Simplified 50.5

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

      [Start] 50.5	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 50.5	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 50.5	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 50.5	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 50.5	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 50.5	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 50.5	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 50.5	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 50.5	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 50.5	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Taylor expanded in y around inf 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ 1 - \left(\log \left(\frac{1}{y}\right) + \log \left(x - 1\right)\right) \]
      +-commutative [=>] 64.0	\[ 1 - \color{blue}{\left(\log \left(x - 1\right) + \log \left(\frac{1}{y}\right)\right)} \]
      log-rec [=>] 64.0	\[ 1 - \left(\log \left(x - 1\right) + \color{blue}{\left(-\log y\right)}\right) \]
      unsub-neg [=>] 64.0	\[ 1 - \color{blue}{\left(\log \left(x - 1\right) - \log y\right)} \]
      sub-neg [=>] 64.0	\[ 1 - \left(\log \color{blue}{\left(x + \left(-1\right)\right)} - \log y\right) \]
      metadata-eval [=>] 64.0	\[ 1 - \left(\log \left(x + \color{blue}{-1}\right) - \log y\right) \]
      +-commutative [=>] 64.0	\[ 1 - \left(\log \color{blue}{\left(-1 + x\right)} - \log y\right) \]

   5. Taylor expanded in y around 0 64.0

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

   6. Simplified 1.0

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

      [Start] 64.0	\[ 1 - \left(\log \left(x - 1\right) - \log y\right) \]
      sub-neg [=>] 64.0	\[ 1 - \left(\log \color{blue}{\left(x + \left(-1\right)\right)} - \log y\right) \]
      metadata-eval [=>] 64.0	\[ 1 - \left(\log \left(x + \color{blue}{-1}\right) - \log y\right) \]
      +-commutative [<=] 64.0	\[ 1 - \left(\log \color{blue}{\left(-1 + x\right)} - \log y\right) \]
      log-div [<=] 1.0	\[ 1 - \color{blue}{\log \left(\frac{-1 + x}{y}\right)} \]
      +-commutative [=>] 1.0	\[ 1 - \log \left(\frac{\color{blue}{x + -1}}{y}\right) \]
      metadata-eval [<=] 1.0	\[ 1 - \log \left(\frac{x + \color{blue}{\left(-1\right)}}{y}\right) \]
      sub-neg [<=] 1.0	\[ 1 - \log \left(\frac{\color{blue}{x - 1}}{y}\right) \]

   if -1.78000000000000003 < y < 0.0550000000000000003

   1. Initial program 0.0

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

   2. Simplified 0.0

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

      [Start] 0.0	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 0.0	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 0.0	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 0.0	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 0.0	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 0.0	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 0.0	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 0.0	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 0.0	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 0.0	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Taylor expanded in y around 0 0.4

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

   4. Simplified 0.4

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

      [Start] 0.4	\[ 1 - \left(y \cdot \left(\frac{1}{1 + -1 \cdot x} - \frac{x}{1 + -1 \cdot x}\right) + \log \left(1 + -1 \cdot x\right)\right) \]
      +-commutative [=>] 0.4	\[ 1 - \color{blue}{\left(\log \left(1 + -1 \cdot x\right) + y \cdot \left(\frac{1}{1 + -1 \cdot x} - \frac{x}{1 + -1 \cdot x}\right)\right)} \]
      log1p-def [=>] 0.4	\[ 1 - \left(\color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)} + y \cdot \left(\frac{1}{1 + -1 \cdot x} - \frac{x}{1 + -1 \cdot x}\right)\right) \]
      mul-1-neg [=>] 0.4	\[ 1 - \left(\mathsf{log1p}\left(\color{blue}{-x}\right) + y \cdot \left(\frac{1}{1 + -1 \cdot x} - \frac{x}{1 + -1 \cdot x}\right)\right) \]
      *-commutative [=>] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \color{blue}{\left(\frac{1}{1 + -1 \cdot x} - \frac{x}{1 + -1 \cdot x}\right) \cdot y}\right) \]
      mul-1-neg [=>] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\frac{1}{1 + \color{blue}{\left(-x\right)}} - \frac{x}{1 + -1 \cdot x}\right) \cdot y\right) \]
      sub-neg [<=] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\frac{1}{\color{blue}{1 - x}} - \frac{x}{1 + -1 \cdot x}\right) \cdot y\right) \]
      mul-1-neg [=>] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\frac{1}{1 - x} - \frac{x}{1 + \color{blue}{\left(-x\right)}}\right) \cdot y\right) \]
      sub-neg [<=] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(\frac{1}{1 - x} - \frac{x}{\color{blue}{1 - x}}\right) \cdot y\right) \]
      div-sub [<=] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \color{blue}{\frac{1 - x}{1 - x}} \cdot y\right) \]
      *-inverses [=>] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \color{blue}{1} \cdot y\right) \]
      metadata-eval [<=] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \color{blue}{\left(--1\right)} \cdot y\right) \]
      distribute-lft-neg-in [<=] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \color{blue}{\left(--1 \cdot y\right)}\right) \]
      neg-mul-1 [<=] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \left(-\color{blue}{\left(-y\right)}\right)\right) \]
      remove-double-neg [=>] 0.4	\[ 1 - \left(\mathsf{log1p}\left(-x\right) + \color{blue}{y}\right) \]

   if 0.0550000000000000003 < y

   1. Initial program 31.2

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

   2. Simplified 31.2

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

      [Start] 31.2	\[ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
      sub-neg [=>] 31.2	\[ 1 - \log \color{blue}{\left(1 + \left(-\frac{x - y}{1 - y}\right)\right)} \]
      log1p-def [=>] 31.2	\[ 1 - \color{blue}{\mathsf{log1p}\left(-\frac{x - y}{1 - y}\right)} \]
      div-sub [=>] 31.2	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} - \frac{y}{1 - y}\right)}\right) \]
      sub-neg [=>] 31.2	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\frac{x}{1 - y} + \left(-\frac{y}{1 - y}\right)\right)}\right) \]
      +-commutative [=>] 31.2	\[ 1 - \mathsf{log1p}\left(-\color{blue}{\left(\left(-\frac{y}{1 - y}\right) + \frac{x}{1 - y}\right)}\right) \]
      distribute-neg-in [=>] 31.2	\[ 1 - \mathsf{log1p}\left(\color{blue}{\left(-\left(-\frac{y}{1 - y}\right)\right) + \left(-\frac{x}{1 - y}\right)}\right) \]
      remove-double-neg [=>] 31.2	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y}} + \left(-\frac{x}{1 - y}\right)\right) \]
      sub-neg [<=] 31.2	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y}{1 - y} - \frac{x}{1 - y}}\right) \]
      div-sub [<=] 31.2	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{y - x}{1 - y}}\right) \]

   3. Taylor expanded in x around inf 30.3

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

   4. Simplified 30.3

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

      [Start] 30.3	\[ 1 - \mathsf{log1p}\left(-1 \cdot \frac{x}{1 - y}\right) \]
      mul-1-neg [=>] 30.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{-\frac{x}{1 - y}}\right) \]
      distribute-neg-frac [=>] 30.3	\[ 1 - \mathsf{log1p}\left(\color{blue}{\frac{-x}{1 - y}}\right) \]

   5. Taylor expanded in x around inf 3.0

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

   6. Simplified 1.7

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

      [Start] 3.0	\[ 1 - \left(\log \left(-\frac{1}{1 - y}\right) + -1 \cdot \log \left(\frac{1}{x}\right)\right) \]
      +-commutative [=>] 3.0	\[ 1 - \color{blue}{\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log \left(-\frac{1}{1 - y}\right)\right)} \]
      distribute-neg-frac [=>] 3.0	\[ 1 - \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log \color{blue}{\left(\frac{-1}{1 - y}\right)}\right) \]
      metadata-eval [=>] 3.0	\[ 1 - \left(-1 \cdot \log \left(\frac{1}{x}\right) + \log \left(\frac{\color{blue}{-1}}{1 - y}\right)\right) \]
      log-div [=>] 64.0	\[ 1 - \left(-1 \cdot \log \left(\frac{1}{x}\right) + \color{blue}{\left(\log -1 - \log \left(1 - y\right)\right)}\right) \]
      associate-+r- [=>] 64.0	\[ 1 - \color{blue}{\left(\left(-1 \cdot \log \left(\frac{1}{x}\right) + \log -1\right) - \log \left(1 - y\right)\right)} \]
      mul-1-neg [=>] 64.0	\[ 1 - \left(\left(\color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} + \log -1\right) - \log \left(1 - y\right)\right) \]
      log-rec [=>] 64.0	\[ 1 - \left(\left(\left(-\color{blue}{\left(-\log x\right)}\right) + \log -1\right) - \log \left(1 - y\right)\right) \]
      remove-double-neg [=>] 64.0	\[ 1 - \left(\left(\color{blue}{\log x} + \log -1\right) - \log \left(1 - y\right)\right) \]
      log-prod [<=] 63.5	\[ 1 - \left(\color{blue}{\log \left(x \cdot -1\right)} - \log \left(1 - y\right)\right) \]
      *-commutative [<=] 63.5	\[ 1 - \left(\log \color{blue}{\left(-1 \cdot x\right)} - \log \left(1 - y\right)\right) \]
      log-div [<=] 1.7	\[ 1 - \color{blue}{\log \left(\frac{-1 \cdot x}{1 - y}\right)} \]
      mul-1-neg [=>] 1.7	\[ 1 - \log \left(\frac{\color{blue}{-x}}{1 - y}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.78:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 0.055:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-x}{1 - y}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	18.1
Cost	7620

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

Alternative 2
Error	18.6
Cost	7492

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

Alternative 3
Error	0.7
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -1.62 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 4
Error	7.0
Cost	7048

\[\begin{array}{l} \mathbf{if}\;y \leq -35:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 5
Error	7.4
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 6
Error	13.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;y \leq -335:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 7
Error	24.0
Cost	6656

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

Alternative 8
Error	35.2
Cost	448

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

Alternative 9
Error	36.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023031 
(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))))))