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

Percentage Accurate: 72.7% → 99.9%

Time: 14.0s

Alternatives: 10

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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y)
	return log(Float64(y * Float64(Float64(exp(1) / Float64(x + -1.0)) - Float64(exp(1) / Float64(y * Float64(x + -1.0))))))
end

MATLAB

function tmp = code(x, y)
	tmp = log((y * ((2.71828182845904523536 / (x + -1.0)) - (2.71828182845904523536 / (y * (x + -1.0))))));
end

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Taylor expanded in y around inf 37.7%

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

5. Simplified 37.7%

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

   1. add-log-exp 37.7%

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

   2. sub-neg 37.7%

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

   3. exp-sum 37.7%

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

   4. neg-log 38.1%

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

   5. clear-num 38.1%

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

   6. add-exp-log 38.1%

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

   7. associate-+l+ 38.1%

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

   8. +-commutative 38.1%

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

7. Applied egg-rr 38.1%

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

   1. exp-1-e 38.1%

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

   2. associate-+r+ 38.1%

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

   3. +-commutative 38.1%

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

9. Simplified 38.1%

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

10. Taylor expanded in y around inf 99.9%

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

    1. +-commutative 99.9%

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

    2. mul-1-neg 99.9%

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

    3. sub-neg 99.9%

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

    4. sub-neg 99.9%

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

    5. metadata-eval 99.9%

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

    6. sub-neg 99.9%

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

    7. metadata-eval 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

    \[\leadsto \log \left(y \cdot \left(\frac{e}{x + -1} - \frac{e}{y \cdot \left(x + -1\right)}\right)\right) \]
14. Add Preprocessing

Alternative 2: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y}{\left(x + -1\right) + \frac{x + -1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99999)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (log (* E (/ y (+ (+ x -1.0) (/ (+ x -1.0) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((((double) M_E) * (y / ((x + -1.0) + ((x + -1.0) / y)))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99999) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((Math.E * (y / ((x + -1.0) + ((x + -1.0) / y)))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.99999:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((math.e * (y / ((x + -1.0) + ((x + -1.0) / y)))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99999)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(exp(1) * Float64(y / Float64(Float64(x + -1.0) + 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], 0.99999], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E * N[(y / N[(N[(x + -1.0), $MachinePrecision] + N[(N[(x + -1.0), $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)) < 0.999990000000000046

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. log1p-define 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate--r- 99.8%

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

      6. metadata-eval 99.8%

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

      7. +-commutative 99.8%

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

   3. Simplified 99.8%

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

   if 0.999990000000000046 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.8%

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

   3. Taylor expanded in y around inf 99.8%

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

   5. Simplified 99.8%

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

      1. add-log-exp 99.8%

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

      2. sub-neg 99.8%

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

      3. exp-sum 99.8%

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

      4. neg-log 99.8%

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

      5. clear-num 99.8%

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

      6. add-exp-log 99.8%

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

      7. associate-+l+ 99.8%

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

      8. +-commutative 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. exp-1-e 99.8%

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

      2. associate-+r+ 99.8%

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

      3. +-commutative 99.8%

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

   9. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y}{\left(x + -1\right) + \frac{x + -1}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (log (/ (* y E) (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log(((y * ((double) M_E)) / (x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log(((y * Math.E) / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log(((y * math.e) / (x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(Float64(y * exp(1)) / Float64(x + -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999999], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(y * E), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.999998999999999971

   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 0.999998999999999971 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.7%

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

   3. Taylor expanded in y around inf 100.0%

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

   5. Simplified 100.0%

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

      1. add-log-exp 100.0%

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

      2. sub-neg 100.0%

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

      3. exp-sum 100.0%

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

      4. neg-log 100.0%

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

      5. clear-num 100.0%

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

      6. add-exp-log 100.0%

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

      7. associate-+l+ 100.0%

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

      8. +-commutative 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. exp-1-e 100.0%

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

      2. associate-+r+ 100.0%

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

      3. +-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in y around inf 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.8) (not (<= y 1.0)))
   (log (/ (* y E) (+ x -1.0)))
   (- (- 1.0 y) (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.8) || !(y <= 1.0)) {
		tmp = log(((y * ((double) M_E)) / (x + -1.0)));
	} else {
		tmp = (1.0 - y) - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.8) || !(y <= 1.0)) {
		tmp = Math.log(((y * Math.E) / (x + -1.0)));
	} else {
		tmp = (1.0 - y) - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.8) or not (y <= 1.0):
		tmp = math.log(((y * math.e) / (x + -1.0)))
	else:
		tmp = (1.0 - y) - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.8) || !(y <= 1.0))
		tmp = log(Float64(Float64(y * exp(1)) / Float64(x + -1.0)));
	else
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.8], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[Log[N[(N[(y * E), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000004 or 1 < y

   1. Initial program 33.4%

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

   3. Taylor expanded in y around inf 99.2%

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

   5. Simplified 99.2%

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

      1. add-log-exp 99.2%

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

      2. sub-neg 99.2%

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

      3. exp-sum 99.2%

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

      4. neg-log 99.2%

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

      5. clear-num 99.2%

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

      6. add-exp-log 99.2%

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

      7. associate-+l+ 99.2%

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

      8. +-commutative 99.2%

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

   7. Applied egg-rr 99.2%

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

      1. exp-1-e 99.2%

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

      2. associate-+r+ 99.2%

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

      3. +-commutative 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in y around inf 98.0%

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

   if -1.80000000000000004 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.2%

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

   5. Simplified 98.2%

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

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.8)
   (log (/ (* y E) (+ x -1.0)))
   (if (<= y 0.035)
     (- (- 1.0 y) (log1p (- x)))
     (- 1.0 (log (/ x (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.8) {
		tmp = log(((y * ((double) M_E)) / (x + -1.0)));
	} else if (y <= 0.035) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.8) {
		tmp = Math.log(((y * Math.E) / (x + -1.0)));
	} else if (y <= 0.035) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.8:
		tmp = math.log(((y * math.e) / (x + -1.0)))
	elif y <= 0.035:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.8)
		tmp = log(Float64(Float64(y * exp(1)) / Float64(x + -1.0)));
	elseif (y <= 0.035)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.8], N[Log[N[(N[(y * E), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 0.035], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.80000000000000004

   1. Initial program 20.9%

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

   3. Taylor expanded in y around inf 98.8%

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

   5. Simplified 98.8%

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

      1. add-log-exp 98.8%

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

      2. sub-neg 98.8%

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

      3. exp-sum 98.8%

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

      4. neg-log 98.8%

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

      5. clear-num 98.8%

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

      6. add-exp-log 98.9%

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

      7. associate-+l+ 98.9%

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

      8. +-commutative 98.9%

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

   7. Applied egg-rr 98.9%

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

      1. exp-1-e 98.9%

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

      2. associate-+r+ 98.9%

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

      3. +-commutative 98.9%

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

   9. Simplified 98.9%

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

   10. Taylor expanded in y around inf 97.6%

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

   if -1.80000000000000004 < y < 0.035000000000000003

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.2%

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

   5. Simplified 98.2%

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

   if 0.035000000000000003 < y

   1. Initial program 57.6%

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

   3. Taylor expanded in x around inf 99.8%

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

      1. mul-1-neg 99.8%

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

      2. distribute-neg-frac2 99.8%

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

      3. sub-neg 99.8%

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

      4. distribute-neg-in 99.8%

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

      5. metadata-eval 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8:\\ \;\;\;\;\log \left(\frac{y \cdot e}{x + -1}\right)\\ \mathbf{elif}\;y \leq 0.035:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -27:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -27.0)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) (+ 1.0 (log (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -27.0) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 + log((y / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -27.0) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 + Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -27.0:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 + math.log((y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -27.0)
		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(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -27.0], 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[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -27

   1. Initial program 20.9%

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

   3. Taylor expanded in y around inf 97.6%

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

      1. associate-*r/ 97.6%

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

      2. neg-mul-1 97.6%

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

      3. distribute-neg-in 97.6%

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

      4. metadata-eval 97.6%

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

      5. mul-1-neg 97.6%

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

      6. remove-double-neg 97.6%

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

   5. Simplified 97.6%

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

      1. sub-neg 97.6%

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

      2. neg-log 97.6%

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

      3. clear-num 97.6%

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

      4. +-commutative 97.6%

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

   7. Applied egg-rr 97.6%

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

   8. Taylor expanded in x around 0 65.3%

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

      1. neg-mul-1 65.3%

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

   10. Simplified 65.3%

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

   if -27 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.2%

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

   5. Simplified 98.2%

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

   if 1 < y

   1. Initial program 57.6%

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

   3. Taylor expanded in y around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. mul-1-neg 98.7%

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

      6. remove-double-neg 98.7%

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

   5. Simplified 98.7%

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

      1. sub-neg 98.7%

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

      2. neg-log 98.7%

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

      3. clear-num 98.7%

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

      4. +-commutative 98.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in x around inf 98.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -27:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05)
   (+ 1.0 (log (- y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (+ 1.0 (log (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05) {
		tmp = 1.0 + log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 + log((y / x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.05) {
		tmp = 1.0 + Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 + Math.log((y / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05:
		tmp = 1.0 + math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 + math.log((y / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.05)
		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(y / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.05], 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[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05000000000000004

   1. Initial program 22.1%

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

   3. Taylor expanded in y around inf 96.4%

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

      1. associate-*r/ 96.4%

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

      2. neg-mul-1 96.4%

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

      3. distribute-neg-in 96.4%

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

      4. metadata-eval 96.4%

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

      5. mul-1-neg 96.4%

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

      6. remove-double-neg 96.4%

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

   5. Simplified 96.4%

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

      1. sub-neg 96.4%

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

      2. neg-log 96.4%

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

      3. clear-num 96.4%

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

      4. +-commutative 96.4%

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

   7. Applied egg-rr 96.4%

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

   8. Taylor expanded in x around 0 64.6%

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

      1. neg-mul-1 64.6%

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

   10. Simplified 64.6%

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

   if -1.05000000000000004 < y < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 98.0%

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

      1. sub-neg 98.0%

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

      2. mul-1-neg 98.0%

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

      3. log1p-define 98.0%

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

      4. mul-1-neg 98.0%

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

   5. Simplified 98.0%

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

   if 1 < y

   1. Initial program 57.6%

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

   3. Taylor expanded in y around inf 98.7%

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

      1. associate-*r/ 98.7%

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

      2. neg-mul-1 98.7%

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

      3. distribute-neg-in 98.7%

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

      4. metadata-eval 98.7%

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

      5. mul-1-neg 98.7%

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

      6. remove-double-neg 98.7%

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

   5. Simplified 98.7%

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

      1. sub-neg 98.7%

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

      2. neg-log 98.7%

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

      3. clear-num 98.7%

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

      4. +-commutative 98.7%

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

   7. Applied egg-rr 98.7%

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

   8. Taylor expanded in x around inf 98.6%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.05) (+ 1.0 (log (- y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.05) {
		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 <= -1.05) {
		tmp = 1.0 + Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.05:
		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 <= -1.05)
		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, -1.05], 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 < -1.05000000000000004

   1. Initial program 22.1%

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

   3. Taylor expanded in y around inf 96.4%

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

      1. associate-*r/ 96.4%

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

      2. neg-mul-1 96.4%

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

      3. distribute-neg-in 96.4%

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

      4. metadata-eval 96.4%

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

      5. mul-1-neg 96.4%

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

      6. remove-double-neg 96.4%

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

   5. Simplified 96.4%

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

      1. sub-neg 96.4%

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

      2. neg-log 96.4%

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

      3. clear-num 96.4%

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

      4. +-commutative 96.4%

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

   7. Applied egg-rr 96.4%

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

   8. Taylor expanded in x around 0 64.6%

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

      1. neg-mul-1 64.6%

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

   10. Simplified 64.6%

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

   if -1.05000000000000004 < y

   1. Initial program 92.7%

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

   3. Taylor expanded in y around 0 81.1%

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

      1. sub-neg 81.1%

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

      2. mul-1-neg 81.1%

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

      3. log1p-define 81.1%

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

      4. mul-1-neg 81.1%

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

   5. Simplified 81.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05:\\ \;\;\;\;1 + \log \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 22.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.7%

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

3. Taylor expanded in y around inf 39.0%

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

   1. associate-*r/ 39.0%

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

   2. neg-mul-1 39.0%

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

   3. distribute-neg-in 39.0%

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

   4. metadata-eval 39.0%

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

   5. mul-1-neg 39.0%

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

   6. remove-double-neg 39.0%

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

5. Simplified 39.0%

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

   1. sub-neg 39.0%

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

   2. neg-log 39.1%

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

   3. clear-num 39.2%

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

   4. +-commutative 39.2%

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

7. Applied egg-rr 39.2%

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

8. Taylor expanded in x around 0 18.9%

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

   1. neg-mul-1 18.9%

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

10. Simplified 18.9%

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

11. Final simplification 18.9%

    \[\leadsto 1 + \log \left(-y\right) \]
12. Add Preprocessing

Alternative 10: 2.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{log1p}\left(-1\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log1p -1.0)))

C

double code(double x, double y) {
	return 1.0 - log1p(-1.0);
}

Java

public static double code(double x, double y) {
	return 1.0 - Math.log1p(-1.0);
}

Python

def code(x, y):
	return 1.0 - math.log1p(-1.0)

Julia

function code(x, y)
	return Float64(1.0 - log1p(-1.0))
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[1 + -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

   1. sub-neg 74.7%

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

   2. log1p-define 74.8%

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

   3. distribute-neg-frac2 74.8%

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

   4. neg-sub0 74.8%

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

   5. associate--r- 74.8%

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

   6. metadata-eval 74.8%

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

   7. +-commutative 74.8%

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

3. Simplified 74.8%

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

5. Taylor expanded in y around inf 2.4%

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

6. Final simplification 2.4%

   \[\leadsto 1 - \mathsf{log1p}\left(-1\right) \]
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 2024074 
(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))))))