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

Percentage Accurate: 71.8% → 99.9%

Time: 12.7s

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: 71.8% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. sub-neg 70.5%

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

   2. log1p-define 70.6%

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

   3. distribute-neg-frac2 70.6%

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

   4. neg-sub0 70.6%

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

   5. associate--r- 70.6%

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

   6. metadata-eval 70.6%

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

   7. +-commutative 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in y around -inf 34.8%

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

7. Simplified 34.8%

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

   1. associate--r+ 34.8%

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

   2. add-log-exp 34.4%

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

   3. exp-diff 34.4%

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

   4. +-commutative 34.4%

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

   5. exp-sum 34.4%

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

   6. add-exp-log 34.8%

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

9. Applied egg-rr 34.8%

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

    1. exp-diff 34.8%

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

    2. exp-1-e 34.8%

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

    3. log1p-undefine 34.8%

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

    4. neg-mul-1 34.8%

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

    5. rem-exp-log 45.6%

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

    6. neg-mul-1 45.6%

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

    7. sub-neg 45.6%

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

    8. associate-*l/ 45.6%

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

    9. mul-1-neg 45.6%

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

11. Simplified 45.6%

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

12. Taylor expanded in y around inf 99.9%

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

    1. +-commutative 99.9%

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

    2. neg-mul-1 99.9%

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

    3. unsub-neg 99.9%

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

14. Simplified 99.9%

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

15. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.99998)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (log (* E (/ y (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99998) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log((((double) M_E) * (y / (x + -1.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.99998) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log((Math.E * (y / (x + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.99998:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log((math.e * (y / (x + -1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.99998)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(exp(1) * Float64(y / 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], 0.99998], 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[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99997999999999998

   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 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate--r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   if 0.99997999999999998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.1%

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

      1. sub-neg 6.1%

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

      2. log1p-define 6.1%

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

      3. distribute-neg-frac2 6.1%

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

      4. neg-sub0 6.1%

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

      5. associate--r- 6.1%

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

      6. metadata-eval 6.1%

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

      7. +-commutative 6.1%

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

   3. Simplified 6.1%

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

   5. Taylor expanded in y around inf 6.1%

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

      1. +-commutative 6.1%

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

      2. associate--r+ 6.1%

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

      3. sub-neg 6.1%

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

      4. div-sub 6.1%

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

      5. sub-neg 6.1%

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

      6. metadata-eval 6.1%

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

      7. metadata-eval 6.1%

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

   7. Simplified 6.1%

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

      1. *-un-lft-identity 6.1%

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

      2. +-commutative 6.1%

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

   9. Applied egg-rr 6.1%

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

       1. *-lft-identity 6.1%

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

       2. log1p-define 6.1%

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

       3. associate-+r+ 99.2%

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

       4. metadata-eval 99.2%

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

       5. +-lft-identity 99.2%

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

       6. +-commutative 99.2%

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

   11. Simplified 99.2%

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

       1. add-log-exp 99.2%

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

       2. sub-neg 99.2%

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

       3. exp-sum 99.2%

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

       4. e-exp-1 99.2%

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

       5. neg-log 99.2%

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

       6. clear-num 99.2%

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

       7. add-exp-log 99.2%

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

       8. +-commutative 99.2%

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

   13. Applied egg-rr 99.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.99998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.75) (not (<= y 1.0)))
   (log (* E (/ y (+ x -1.0))))
   (- (- 1.0 y) (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.75) || !(y <= 1.0)) {
		tmp = log((((double) M_E) * (y / (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 <= -1.75) || !(y <= 1.0)) {
		tmp = Math.log((Math.E * (y / (x + -1.0))));
	} else {
		tmp = (1.0 - y) - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.75) or not (y <= 1.0):
		tmp = math.log((math.e * (y / (x + -1.0))))
	else:
		tmp = (1.0 - y) - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.75) || !(y <= 1.0))
		tmp = log(Float64(exp(1) * Float64(y / 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, -1.75], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[Log[N[(E * N[(y / N[(x + -1.0), $MachinePrecision]), $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 < -1.75 or 1 < y

   1. Initial program 35.1%

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

      1. sub-neg 35.1%

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

      2. log1p-define 35.1%

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

      3. distribute-neg-frac2 35.1%

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

      4. neg-sub0 35.1%

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

      5. associate--r- 35.1%

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

      6. metadata-eval 35.1%

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

      7. +-commutative 35.1%

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

   3. Simplified 35.1%

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

   5. Taylor expanded in y around inf 32.5%

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

      1. +-commutative 32.5%

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

      2. associate--r+ 32.5%

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

      3. sub-neg 32.5%

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

      4. div-sub 32.5%

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

      5. sub-neg 32.5%

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

      6. metadata-eval 32.5%

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

      7. metadata-eval 32.5%

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

   7. Simplified 32.5%

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

      1. *-un-lft-identity 32.5%

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

      2. +-commutative 32.5%

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

   9. Applied egg-rr 32.5%

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

       1. *-lft-identity 32.5%

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

       2. log1p-define 32.5%

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

       3. associate-+r+ 96.7%

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

       4. metadata-eval 96.7%

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

       5. +-lft-identity 96.7%

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

       6. +-commutative 96.7%

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

   11. Simplified 96.7%

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

       1. add-log-exp 96.7%

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

       2. sub-neg 96.7%

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

       3. exp-sum 96.7%

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

       4. e-exp-1 96.7%

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

       5. neg-log 96.7%

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

       6. clear-num 96.7%

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

       7. add-exp-log 96.7%

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

       8. +-commutative 96.7%

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

   13. Applied egg-rr 96.7%

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

   if -1.75 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      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

   5. Taylor expanded in y around 0 98.4%

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

   7. Simplified 98.4%

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

4. Final simplification 97.6%

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

Alternative 4: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.85)
   (log (* E (/ y (+ x -1.0))))
   (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) (- 1.0 (log (/ (+ x -1.0) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.85) {
		tmp = log((((double) M_E) * (y / (x + -1.0))));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.85) {
		tmp = Math.log((Math.E * (y / (x + -1.0))));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.85:
		tmp = math.log((math.e * (y / (x + -1.0))))
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.85)
		tmp = log(Float64(exp(1) * Float64(y / Float64(x + -1.0))));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.85], N[Log[N[(E * N[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $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[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8500000000000001

   1. Initial program 25.4%

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

      1. sub-neg 25.4%

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

      2. log1p-define 25.4%

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

      3. distribute-neg-frac2 25.4%

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

      4. neg-sub0 25.4%

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

      5. associate--r- 25.4%

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

      6. metadata-eval 25.4%

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

      7. +-commutative 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in y around inf 22.8%

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

      1. +-commutative 22.8%

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

      2. associate--r+ 22.8%

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

      3. sub-neg 22.8%

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

      4. div-sub 22.8%

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

      5. sub-neg 22.8%

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

      6. metadata-eval 22.8%

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

      7. metadata-eval 22.8%

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

   7. Simplified 22.8%

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

      1. *-un-lft-identity 22.8%

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

      2. +-commutative 22.8%

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

   9. Applied egg-rr 22.8%

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

       1. *-lft-identity 22.8%

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

       2. log1p-define 22.8%

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

       3. associate-+r+ 96.4%

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

       4. metadata-eval 96.4%

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

       5. +-lft-identity 96.4%

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

       6. +-commutative 96.4%

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

   11. Simplified 96.4%

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

       1. add-log-exp 96.4%

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

       2. sub-neg 96.4%

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

       3. exp-sum 96.4%

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

       4. e-exp-1 96.4%

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

       5. neg-log 96.4%

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

       6. clear-num 96.4%

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

       7. add-exp-log 96.4%

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

       8. +-commutative 96.4%

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

   13. Applied egg-rr 96.4%

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

   if -1.8500000000000001 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      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

   5. Taylor expanded in y around 0 98.4%

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

   7. Simplified 98.4%

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

   if 1 < y

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. log1p-define 65.4%

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

      3. distribute-neg-frac2 65.4%

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

      4. neg-sub0 65.4%

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

      5. associate--r- 65.4%

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

      6. metadata-eval 65.4%

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

      7. +-commutative 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in y around inf 63.2%

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

      1. +-commutative 63.2%

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

      2. associate--r+ 63.2%

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

      3. sub-neg 63.2%

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

      4. div-sub 63.2%

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

      5. sub-neg 63.2%

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

      6. metadata-eval 63.2%

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

      7. metadata-eval 63.2%

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

   7. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. +-commutative 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. *-lft-identity 63.2%

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

       2. log1p-define 63.2%

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

       3. associate-+r+ 97.8%

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

       4. metadata-eval 97.8%

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

       5. +-lft-identity 97.8%

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

       6. +-commutative 97.8%

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

   11. Simplified 97.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85:\\ \;\;\;\;\log \left(e \cdot \frac{y}{x + -1}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.7)
   (log (* y (/ E (+ x -1.0))))
   (if (<= y 1.0) (- (- 1.0 y) (log1p (- x))) (- 1.0 (log (/ (+ x -1.0) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.7) {
		tmp = log((y * (((double) M_E) / (x + -1.0))));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - log1p(-x);
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.7) {
		tmp = Math.log((y * (Math.E / (x + -1.0))));
	} else if (y <= 1.0) {
		tmp = (1.0 - y) - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.7:
		tmp = math.log((y * (math.e / (x + -1.0))))
	elif y <= 1.0:
		tmp = (1.0 - y) - math.log1p(-x)
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.7)
		tmp = log(Float64(y * Float64(exp(1) / Float64(x + -1.0))));
	elseif (y <= 1.0)
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.7], N[Log[N[(y * N[(E / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $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[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.69999999999999996

   1. Initial program 25.4%

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

      1. sub-neg 25.4%

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

      2. log1p-define 25.4%

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

      3. distribute-neg-frac2 25.4%

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

      4. neg-sub0 25.4%

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

      5. associate--r- 25.4%

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

      6. metadata-eval 25.4%

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

      7. +-commutative 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in y around -inf 97.8%

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

   7. Simplified 97.8%

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

      1. associate--r+ 97.8%

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

      2. add-log-exp 97.8%

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

      3. exp-diff 97.8%

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

      4. +-commutative 97.8%

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

      5. exp-sum 97.8%

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

      6. add-exp-log 97.8%

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

   9. Applied egg-rr 97.8%

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

       1. exp-diff 97.8%

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

       2. exp-1-e 97.8%

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

       3. log1p-undefine 97.8%

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

       4. neg-mul-1 97.8%

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

       5. rem-exp-log 98.1%

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

       6. neg-mul-1 98.1%

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

       7. sub-neg 98.1%

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

       8. associate-*l/ 98.1%

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

       9. mul-1-neg 98.1%

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

   11. Simplified 98.1%

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

   12. Taylor expanded in y around inf 96.4%

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

       1. mul-1-neg 96.4%

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

       2. associate-/l* 96.5%

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

       3. distribute-rgt-neg-in 96.5%

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

       4. distribute-neg-frac2 96.5%

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

       5. neg-sub0 96.5%

          \[\leadsto \log \left(y \cdot \frac{e}{\color{blue}{0 - \left(1 - x\right)}}\right) \]

       6. sub-neg 96.5%

          \[\leadsto \log \left(y \cdot \frac{e}{0 - \color{blue}{\left(1 + \left(-x\right)\right)}}\right) \]

       7. associate--r+ 96.5%

          \[\leadsto \log \left(y \cdot \frac{e}{\color{blue}{\left(0 - 1\right) - \left(-x\right)}}\right) \]

       8. metadata-eval 96.5%

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

   14. Simplified 96.5%

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

   if -1.69999999999999996 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      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

   5. Taylor expanded in y around 0 98.4%

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

   7. Simplified 98.4%

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

   if 1 < y

   1. Initial program 65.4%

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

      1. sub-neg 65.4%

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

      2. log1p-define 65.4%

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

      3. distribute-neg-frac2 65.4%

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

      4. neg-sub0 65.4%

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

      5. associate--r- 65.4%

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

      6. metadata-eval 65.4%

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

      7. +-commutative 65.4%

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

   3. Simplified 65.4%

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

   5. Taylor expanded in y around inf 63.2%

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

      1. +-commutative 63.2%

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

      2. associate--r+ 63.2%

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

      3. sub-neg 63.2%

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

      4. div-sub 63.2%

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

      5. sub-neg 63.2%

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

      6. metadata-eval 63.2%

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

      7. metadata-eval 63.2%

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

   7. Simplified 63.2%

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

      1. *-un-lft-identity 63.2%

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

      2. +-commutative 63.2%

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

   9. Applied egg-rr 63.2%

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

       1. *-lft-identity 63.2%

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

       2. log1p-define 63.2%

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

       3. associate-+r+ 97.8%

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

       4. metadata-eval 97.8%

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

       5. +-lft-identity 97.8%

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

       6. +-commutative 97.8%

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

   11. Simplified 97.8%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.6) (- 1.0 (log (/ -1.0 y))) (- (- 1.0 y) (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.6) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = (1.0 - y) - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.6) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = (1.0 - y) - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.6:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = (1.0 - y) - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.6)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.6], N[(1.0 - N[Log[N[(-1.0 / y), $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 < -4.5999999999999996

   1. Initial program 25.4%

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

      1. sub-neg 25.4%

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

      2. log1p-define 25.4%

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

      3. distribute-neg-frac2 25.4%

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

      4. neg-sub0 25.4%

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

      5. associate--r- 25.4%

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

      6. metadata-eval 25.4%

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

      7. +-commutative 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in y around inf 22.8%

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

      1. +-commutative 22.8%

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

      2. associate--r+ 22.8%

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

      3. sub-neg 22.8%

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

      4. div-sub 22.8%

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

      5. sub-neg 22.8%

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

      6. metadata-eval 22.8%

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

      7. metadata-eval 22.8%

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

   7. Simplified 22.8%

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

   8. Taylor expanded in x around 0 71.4%

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

      1. distribute-neg-frac 71.4%

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

      2. metadata-eval 71.4%

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

   10. Simplified 71.4%

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

   if -4.5999999999999996 < y

   1. Initial program 94.2%

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

      1. sub-neg 94.2%

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

      2. log1p-define 94.2%

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

      3. distribute-neg-frac2 94.2%

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

      4. neg-sub0 94.2%

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

      5. associate--r- 94.2%

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

      6. metadata-eval 94.2%

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

      7. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in y around 0 82.0%

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

   7. Simplified 82.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5

   1. Initial program 25.4%

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

      1. sub-neg 25.4%

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

      2. log1p-define 25.4%

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

      3. distribute-neg-frac2 25.4%

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

      4. neg-sub0 25.4%

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

      5. associate--r- 25.4%

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

      6. metadata-eval 25.4%

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

      7. +-commutative 25.4%

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

   3. Simplified 25.4%

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

   5. Taylor expanded in y around inf 22.8%

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

      1. +-commutative 22.8%

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

      2. associate--r+ 22.8%

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

      3. sub-neg 22.8%

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

      4. div-sub 22.8%

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

      5. sub-neg 22.8%

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

      6. metadata-eval 22.8%

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

      7. metadata-eval 22.8%

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

   7. Simplified 22.8%

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

   8. Taylor expanded in x around 0 71.4%

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

      1. distribute-neg-frac 71.4%

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

      2. metadata-eval 71.4%

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

   10. Simplified 71.4%

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

   if -4.5 < y

   1. Initial program 94.2%

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

      1. sub-neg 94.2%

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

      2. log1p-define 94.2%

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

      3. distribute-neg-frac2 94.2%

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

      4. neg-sub0 94.2%

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

      5. associate--r- 94.2%

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

      6. metadata-eval 94.2%

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

      7. +-commutative 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in y around 0 81.0%

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

      1. log1p-define 81.1%

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

      2. mul-1-neg 81.1%

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

   7. Simplified 81.1%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. sub-neg 70.5%

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

   2. log1p-define 70.6%

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

   3. distribute-neg-frac2 70.6%

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

   4. neg-sub0 70.6%

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

   5. associate--r- 70.6%

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

   6. metadata-eval 70.6%

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

   7. +-commutative 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in y around 0 57.9%

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

   1. log1p-define 57.9%

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

   2. mul-1-neg 57.9%

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

7. Simplified 57.9%

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

8. Final simplification 57.9%

   \[\leadsto 1 - \mathsf{log1p}\left(-x\right) \]
9. Add Preprocessing

Alternative 9: 43.8% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.5%

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

   1. sub-neg 70.5%

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

   2. log1p-define 70.6%

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

   3. distribute-neg-frac2 70.6%

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

   4. neg-sub0 70.6%

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

   5. associate--r- 70.6%

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

   6. metadata-eval 70.6%

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

   7. +-commutative 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in y around 0 57.9%

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

   1. log1p-define 57.9%

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

   2. mul-1-neg 57.9%

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

7. Simplified 57.9%

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

8. Taylor expanded in x around 0 40.2%

   \[\leadsto \color{blue}{1 + x} \]

9. Final simplification 40.2%

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

Alternative 10: 43.6% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 70.5%

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

   1. sub-neg 70.5%

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

   2. log1p-define 70.6%

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

   3. distribute-neg-frac2 70.6%

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

   4. neg-sub0 70.6%

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

   5. associate--r- 70.6%

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

   6. metadata-eval 70.6%

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

   7. +-commutative 70.6%

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

3. Simplified 70.6%

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

5. Taylor expanded in y around 0 57.9%

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

   1. log1p-define 57.9%

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

   2. mul-1-neg 57.9%

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

7. Simplified 57.9%

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

8. Taylor expanded in x around 0 39.7%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 39.7%

   \[\leadsto 1 \]
10. 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 2024115 
(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))))))