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

Percentage Accurate: 72.7% → 99.8%

Time: 11.1s

Alternatives: 14

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
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]

Initial Program: 72.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - \log \left(1 - \frac{x - y}{1 - y}\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - log((1.0d0 - ((x - y) / (1.0d0 - y))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
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]

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{\left(x + -1\right) + \frac{-1 + \left(x - \frac{1 - x}{y}\right)}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.5)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (+ 1.0 (log (/ y (+ (+ x -1.0) (/ (+ -1.0 (- x (/ (- 1.0 x) y))) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.5) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + log((y / ((x + -1.0) + ((-1.0 + (x - ((1.0 - x) / y))) / y))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.5) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + Math.log((y / ((x + -1.0) + ((-1.0 + (x - ((1.0 - x) / y))) / y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.5:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + math.log((y / ((x + -1.0) + ((-1.0 + (x - ((1.0 - x) / y))) / y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.5)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + log(Float64(y / Float64(Float64(x + -1.0) + Float64(Float64(-1.0 + Float64(x - Float64(Float64(1.0 - x) / y))) / y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.5], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(y / N[(N[(x + -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(x - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.5

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 8.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 100.0%

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

   5. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. log-rec 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{\left(x + -1\right) + \frac{-1 + \left(x - \frac{1 - x}{y}\right)}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + \left(-1 + \frac{-1 + \frac{-1}{y}}{y}\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.5)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ x (+ -1.0 (/ (+ -1.0 (/ -1.0 y)) y))) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.5) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((x + (-1.0 + ((-1.0 + (-1.0 / y)) / y))) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.5) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((x + (-1.0 + ((-1.0 + (-1.0 / y)) / y))) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.5:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((x + (-1.0 + ((-1.0 + (-1.0 / y)) / y))) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.5)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + Float64(-1.0 + Float64(Float64(-1.0 + Float64(-1.0 / y)) / y))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.5], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + N[(-1.0 + N[(N[(-1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.5

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 8.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 100.0%

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

      1. associate-*r/ 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate-*l/ 100.0%

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

      5. metadata-eval 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + \left(-1 + \frac{-1 + \frac{-1}{y}}{y}\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{\left(x + -1\right) - \frac{1 - x}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.5)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (+ 1.0 (log (/ y (- (+ x -1.0) (/ (- 1.0 x) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.5) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + log((y / ((x + -1.0) - ((1.0 - x) / y))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.5) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + Math.log((y / ((x + -1.0) - ((1.0 - x) / y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.5:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + math.log((y / ((x + -1.0) - ((1.0 - x) / y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.5)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + log(Float64(y / Float64(Float64(x + -1.0) - Float64(Float64(1.0 - x) / y)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.5], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(y / N[(N[(x + -1.0), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.5

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 8.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf 100.0%

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

   5. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. log-rec 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-+r+ 100.0%

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

      5. associate-+l+ 100.0%

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

      6. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around inf 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{\left(x + -1\right) - \frac{1 - x}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) - \frac{1 - x}{y}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.5)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (- (+ x -1.0) (/ (- 1.0 x) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.5) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((((x + -1.0) - ((1.0 - x) / y)) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.5) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((((x + -1.0) - ((1.0 - x) / y)) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.5:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((((x + -1.0) - ((1.0 - x) / y)) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.5)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x + -1.0) - Float64(Float64(1.0 - x) / y)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.5], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x + -1.0), $MachinePrecision] - N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.5

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 100.0%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 0.5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 8.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.5:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) - \frac{1 - x}{y}}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999996:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9999996)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9999996) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9999996) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.9999996:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.9999996)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.9999996], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99999959999999999

   1. Initial program 99.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Step-by-step derivation

      1. sub-neg 99.7%

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

      2. log1p-define 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. neg-sub0 99.7%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{0 - \left(1 - y\right)}}\right) \]

      5. associate--r- 99.7%

         \[\leadsto 1 - \mathsf{log1p}\left(\frac{x - y}{\color{blue}{\left(0 - 1\right) + y}}\right) \]

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   if 0.99999959999999999 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.9%

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

      1. associate-*r/ 99.9%

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

      2. neg-mul-1 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. metadata-eval 99.9%

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

      5. mul-1-neg 99.9%

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

      6. remove-double-neg 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999996:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y + -1}{x} \cdot e\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.75)
   (- 1.0 (log (/ (+ x -1.0) y)))
   (if (<= y 1.75e-5)
     (- (- 1.0 y) (log1p (- x)))
     (log (* (/ (+ y -1.0) x) E)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else if (y <= 1.75e-5) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = log((((y + -1.0) / x) * ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.75) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else if (y <= 1.75e-5) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = Math.log((((y + -1.0) / x) * Math.E));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	elif y <= 1.75e-5:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = math.log((((y + -1.0) / x) * math.e))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75)
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	elseif (y <= 1.75e-5)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(Float64(y + -1.0) / x) * exp(1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-5], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision] * E), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.75

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 98.9%

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

      1. associate-*r/ 98.9%

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

      2. neg-mul-1 98.9%

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

      3. distribute-neg-in 98.9%

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

      4. metadata-eval 98.9%

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

      5. mul-1-neg 98.9%

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

      6. remove-double-neg 98.9%

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

   5. Simplified 98.9%

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

   if -1.75 < y < 1.7499999999999998e-5

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.1%

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

   5. Simplified 99.2%

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

   if 1.7499999999999998e-5 < y

   1. Initial program 48.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-neg-frac2 99.3%

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

      3. sub-neg 99.3%

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

      4. distribute-neg-in 99.3%

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

      5. metadata-eval 99.3%

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

      6. remove-double-neg 99.3%

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

      7. +-commutative 99.3%

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

   5. Simplified 99.3%

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

      1. add-log-exp 99.3%

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

      2. sub-neg 99.3%

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

      3. exp-sum 99.3%

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

      4. neg-log 99.2%

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

      5. clear-num 99.3%

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

      6. add-exp-log 99.3%

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

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\log \left(e^{1} \cdot \frac{y + -1}{x}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. exp-1-e 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y + -1}{x} \cdot e\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12.5:\\ \;\;\;\;-1 + \left(2 - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y + -1}{x} \cdot e\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -12.5)
   (+ -1.0 (- 2.0 (log (/ -1.0 y))))
   (if (<= y 1.75e-5)
     (- (- 1.0 y) (log1p (- x)))
     (log (* (/ (+ y -1.0) x) E)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -12.5) {
		tmp = -1.0 + (2.0 - log((-1.0 / y)));
	} else if (y <= 1.75e-5) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = log((((y + -1.0) / x) * ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -12.5) {
		tmp = -1.0 + (2.0 - Math.log((-1.0 / y)));
	} else if (y <= 1.75e-5) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = Math.log((((y + -1.0) / x) * Math.E));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -12.5:
		tmp = -1.0 + (2.0 - math.log((-1.0 / y)))
	elif y <= 1.75e-5:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = math.log((((y + -1.0) / x) * math.e))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -12.5)
		tmp = Float64(-1.0 + Float64(2.0 - log(Float64(-1.0 / y))));
	elseif (y <= 1.75e-5)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(Float64(y + -1.0) / x) * exp(1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -12.5], N[(-1.0 + N[(2.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-5], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision] * E), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -12.5

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.3%

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

      1. log1p-define 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in y around -inf 67.6%

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

      1. expm1-log1p-u 66.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \log \left(\frac{-1}{y}\right)\right)\right)} \]

   8. Applied egg-rr 66.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \log \left(\frac{-1}{y}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-undefine 66.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(1 - \log \left(\frac{-1}{y}\right)\right)} - 1} \]

      2. sub-neg 66.5%

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

      3. log1p-undefine 66.5%

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

      4. rem-exp-log 67.6%

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

      5. associate-+r- 67.6%

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

      6. metadata-eval 67.6%

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

      7. metadata-eval 67.6%

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

   10. Simplified 67.6%

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

   if -12.5 < y < 1.7499999999999998e-5

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.1%

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

   5. Simplified 99.2%

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

   if 1.7499999999999998e-5 < y

   1. Initial program 48.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-neg-frac2 99.3%

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

      3. sub-neg 99.3%

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

      4. distribute-neg-in 99.3%

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

      5. metadata-eval 99.3%

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

      6. remove-double-neg 99.3%

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

      7. +-commutative 99.3%

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

   5. Simplified 99.3%

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

      1. add-log-exp 99.3%

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

      2. sub-neg 99.3%

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

      3. exp-sum 99.3%

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

      4. neg-log 99.2%

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

      5. clear-num 99.3%

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

      6. add-exp-log 99.3%

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

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\log \left(e^{1} \cdot \frac{y + -1}{x}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. exp-1-e 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12.5:\\ \;\;\;\;-1 + \left(2 - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y + -1}{x} \cdot e\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y + -1}{x} \cdot e\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -14.5)
   (+ 1.0 (log (- y)))
   (if (<= y 1.75e-5)
     (- (- 1.0 y) (log1p (- x)))
     (log (* (/ (+ y -1.0) x) E)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -14.5) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.75e-5) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = log((((y + -1.0) / x) * ((double) M_E)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -14.5) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.75e-5) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = Math.log((((y + -1.0) / x) * Math.E));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -14.5:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.75e-5:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = math.log((((y + -1.0) / x) * math.e))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -14.5)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.75e-5)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = log(Float64(Float64(Float64(y + -1.0) / x) * exp(1)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -14.5], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.75e-5], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision] * E), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -14.5

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.3%

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

      1. log1p-define 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in y around -inf 67.6%

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

      1. sub-neg 67.6%

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

      2. neg-log 67.6%

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

      3. clear-num 67.6%

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

      4. div-inv 67.6%

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

      5. metadata-eval 67.6%

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

   8. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{1 + \log \left(y \cdot -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative 67.6%

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

      2. neg-mul-1 67.6%

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

   10. Simplified 67.6%

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

   if -14.5 < y < 1.7499999999999998e-5

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 99.1%

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

   5. Simplified 99.2%

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

   if 1.7499999999999998e-5 < y

   1. Initial program 48.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.3%

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

      1. mul-1-neg 99.3%

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

      2. distribute-neg-frac2 99.3%

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

      3. sub-neg 99.3%

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

      4. distribute-neg-in 99.3%

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

      5. metadata-eval 99.3%

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

      6. remove-double-neg 99.3%

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

      7. +-commutative 99.3%

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

   5. Simplified 99.3%

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

      1. add-log-exp 99.3%

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

      2. sub-neg 99.3%

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

      3. exp-sum 99.3%

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

      4. neg-log 99.2%

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

      5. clear-num 99.3%

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

      6. add-exp-log 99.3%

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

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\log \left(e^{1} \cdot \frac{y + -1}{x}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.3%

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

      2. +-commutative 99.3%

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

      3. exp-1-e 99.3%

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

   9. Simplified 99.3%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14.5:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-5}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y + -1}{x} \cdot e\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -20:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -20.0)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -20.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -20.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -20.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -20.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -20.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -20

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.3%

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

      1. log1p-define 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in y around -inf 67.6%

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

      1. sub-neg 67.6%

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

      2. neg-log 67.6%

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

      3. clear-num 67.6%

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

      4. div-inv 67.6%

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

      5. metadata-eval 67.6%

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

   8. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{1 + \log \left(y \cdot -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative 67.6%

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

      2. neg-mul-1 67.6%

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

   10. Simplified 67.6%

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

   if -20 < y < 1

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 98.9%

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

   5. Simplified 98.9%

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

   if 1 < y

   1. Initial program 41.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-neg-frac2 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. metadata-eval 99.2%

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

      6. remove-double-neg 99.2%

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

      7. +-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf 96.5%

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

Alternative 10: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13.0)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -13.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -13.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -13.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -13.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -13

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.3%

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

      1. log1p-define 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in y around -inf 67.6%

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

      1. sub-neg 67.6%

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

      2. neg-log 67.6%

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

      3. clear-num 67.6%

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

      4. div-inv 67.6%

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

      5. metadata-eval 67.6%

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

   8. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{1 + \log \left(y \cdot -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative 67.6%

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

      2. neg-mul-1 67.6%

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

   10. Simplified 67.6%

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

   if -13 < y < 1

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 97.7%

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

      1. sub-neg 97.7%

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

      2. mul-1-neg 97.7%

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

      3. log1p-define 97.7%

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

      4. mul-1-neg 97.7%

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

   5. Simplified 97.7%

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

   if 1 < y

   1. Initial program 41.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 99.2%

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

      1. mul-1-neg 99.2%

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

      2. distribute-neg-frac2 99.2%

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

      3. sub-neg 99.2%

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

      4. distribute-neg-in 99.2%

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

      5. metadata-eval 99.2%

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

      6. remove-double-neg 99.2%

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

      7. +-commutative 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in y around inf 96.5%

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

Alternative 11: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -340:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -340.0) (+ 1.0 (log (- y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -340.0) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -340.0) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -340.0:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -340.0)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -340.0], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -340

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.3%

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

      1. log1p-define 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in y around -inf 67.6%

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

      1. sub-neg 67.6%

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

      2. neg-log 67.6%

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

      3. clear-num 67.6%

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

      4. div-inv 67.6%

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

      5. metadata-eval 67.6%

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

   8. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{1 + \log \left(y \cdot -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative 67.6%

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

      2. neg-mul-1 67.6%

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

   10. Simplified 67.6%

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

   if -340 < y

   1. Initial program 94.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 88.9%

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

      1. sub-neg 88.9%

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

      2. mul-1-neg 88.9%

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

      3. log1p-define 88.9%

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

      4. mul-1-neg 88.9%

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

   5. Simplified 88.9%

      \[\leadsto 1 - \color{blue}{\mathsf{log1p}\left(-x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 - x \cdot -0.3333333333333333\right) - -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4)
   (+ 1.0 (log (- y)))
   (+ 1.0 (* x (- (* x (- 0.5 (* x -0.3333333333333333))) -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4) {
		tmp = 1.0 + log(-y);
	} else {
		tmp = 1.0 + (x * ((x * (0.5 - (x * -0.3333333333333333))) - -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d0)) then
        tmp = 1.0d0 + log(-y)
    else
        tmp = 1.0d0 + (x * ((x * (0.5d0 - (x * (-0.3333333333333333d0)))) - (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 + (x * ((x * (0.5 - (x * -0.3333333333333333))) - -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4:
		tmp = 1.0 + math.log(-y)
	else:
		tmp = 1.0 + (x * ((x * (0.5 - (x * -0.3333333333333333))) - -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4)
		tmp = Float64(1.0 + log(Float64(-y)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(Float64(x * Float64(0.5 - Float64(x * -0.3333333333333333))) - -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4)
		tmp = 1.0 + log(-y);
	else
		tmp = 1.0 + (x * ((x * (0.5 - (x * -0.3333333333333333))) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4], N[(1.0 + N[Log[(-y)], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(N[(x * N[(0.5 - N[(x * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 5.3%

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

      1. log1p-define 5.3%

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

   5. Simplified 5.3%

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

   6. Taylor expanded in y around -inf 67.6%

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

      1. sub-neg 67.6%

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

      2. neg-log 67.6%

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

      3. clear-num 67.6%

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

      4. div-inv 67.6%

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

      5. metadata-eval 67.6%

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

   8. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{1 + \log \left(y \cdot -1\right)} \]
   9. Step-by-step derivation

      1. *-commutative 67.6%

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

      2. neg-mul-1 67.6%

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

   10. Simplified 67.6%

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

   if -1.3999999999999999 < y

   1. Initial program 94.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 88.9%

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

   4. Taylor expanded in x around 0 58.9%

      \[\leadsto 1 - \color{blue}{x \cdot \left(x \cdot \left(-0.3333333333333333 \cdot x - 0.5\right) - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \left(0.5 - x \cdot -0.3333333333333333\right) - -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.2% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ x + 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ x 1.0))

C

double code(double x, double y) {
	return x + 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x + 1.0d0
end function

Java

public static double code(double x, double y) {
	return x + 1.0;
}

Python

def code(x, y):
	return x + 1.0

Julia

function code(x, y)
	return Float64(x + 1.0)
end

MATLAB

function tmp = code(x, y)
	tmp = x + 1.0;
end

Wolfram

code[x_, y_] := N[(x + 1.0), $MachinePrecision]

Derivation

1. Initial program 69.2%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0 62.1%

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

4. Taylor expanded in x around 0 41.8%

   \[\leadsto 1 - \color{blue}{-1 \cdot x} \]
5. Step-by-step derivation

   1. neg-mul-1 41.8%

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

6. Simplified 41.8%

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

7. Final simplification 41.8%

   \[\leadsto x + 1 \]
8. Add Preprocessing

Alternative 14: 40.6% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ 1 - y \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 - y
end function

Java

public static double code(double x, double y) {
	return 1.0 - y;
}

Python

def code(x, y):
	return 1.0 - y

Julia

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

MATLAB

function tmp = code(x, y)
	tmp = 1.0 - y;
end

Wolfram

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

Derivation

1. Initial program 69.2%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 39.2%

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

   1. log1p-define 39.2%

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

5. Simplified 39.2%

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

6. Taylor expanded in y around 0 38.6%

   \[\leadsto 1 - \color{blue}{y} \]
7. Add Preprocessing

Developer target: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))
   (if (< y -81284752.61947241)
     t_0
     (if (< y 3.0094271212461764e+25)
       (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log(((x / (y * y)) - ((1.0d0 / y) - (x / y))))
    if (y < (-81284752.61947241d0)) then
        tmp = t_0
    else if (y < 3.0094271212461764d+25) then
        tmp = log((exp(1.0d0) / (1.0d0 - ((x - y) / (1.0d0 - y)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = Math.log((Math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x / (y * y)) - ((1.0 / y) - (x / y))))
	tmp = 0
	if y < -81284752.61947241:
		tmp = t_0
	elif y < 3.0094271212461764e+25:
		tmp = math.log((math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(x / Float64(y * y)) - Float64(Float64(1.0 / y) - Float64(x / y)))))
	tmp = 0.0
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log(Float64(exp(1.0) / Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	tmp = 0.0;
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / y), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -81284752.61947241], t$95$0, If[Less[y, 3.0094271212461764e+25], N[Log[N[(N[Exp[1.0], $MachinePrecision] / N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024085 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (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))))))