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

Percentage Accurate: 72.2% → 99.9%

Time: 12.9s

Alternatives: 11

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: 72.2% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{e}{x + -1}}{\frac{e^{\frac{1}{y}}}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9995)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (log (/ (/ E (+ x -1.0)) (/ (exp (/ 1.0 y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9995) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = log(((((double) M_E) / (x + -1.0)) / (exp((1.0 / y)) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9995) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = Math.log(((Math.E / (x + -1.0)) / (Math.exp((1.0 / y)) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.9995:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = math.log(((math.e / (x + -1.0)) / (math.exp((1.0 / y)) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.9995)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(Float64(exp(1) / Float64(x + -1.0)) / Float64(exp(Float64(1.0 / y)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.9995], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(N[(E / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(1.0 / y), $MachinePrecision]], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99950000000000006

   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 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 0.99950000000000006 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.1%

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

      1. sub-neg 5.1%

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

      2. log1p-define 5.1%

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

      3. distribute-neg-frac2 5.1%

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

      4. neg-sub0 5.1%

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

      5. associate--r- 5.1%

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

      6. metadata-eval 5.1%

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

      7. +-commutative 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in y around -inf 83.8%

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

   7. Simplified 83.8%

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

      1. add-log-exp 83.8%

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

      2. associate--r+ 83.8%

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

      3. exp-diff 83.8%

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

      4. +-commutative 83.8%

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

      5. exp-sum 83.8%

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

      6. add-exp-log 84.0%

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

   9. Applied egg-rr 84.0%

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

       1. exp-diff 84.0%

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

       2. exp-1-e 84.0%

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

       3. neg-mul-1 84.0%

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

       4. log1p-define 84.0%

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

       5. rem-exp-log 100.0%

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

       6. neg-mul-1 100.0%

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

       7. sub-neg 100.0%

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

       8. associate-*l/ 100.0%

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

       9. mul-1-neg 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{e}{x + -1}}{\frac{e^{\frac{1}{y}}}{y}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9995)
   (- 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)) <= 0.9995) {
		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)) <= 0.9995) {
		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)) <= 0.9995:
		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)) <= 0.9995)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = log(Float64(y * Float64(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], 0.9995], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(y * N[(E / 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)) < 0.99950000000000006

   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 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 0.99950000000000006 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 5.1%

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

      1. sub-neg 5.1%

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

      2. log1p-define 5.1%

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

      3. distribute-neg-frac2 5.1%

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

      4. neg-sub0 5.1%

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

      5. associate--r- 5.1%

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

      6. metadata-eval 5.1%

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

      7. +-commutative 5.1%

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

   3. Simplified 5.1%

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

   5. Taylor expanded in y around inf 15.8%

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

      1. log-rec 15.8%

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

      2. unsub-neg 15.8%

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

      3. sub-neg 15.8%

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

      4. metadata-eval 15.8%

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

      5. +-commutative 15.8%

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

   7. Simplified 15.8%

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

      1. add-log-exp 15.8%

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

      2. exp-diff 15.8%

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

      3. diff-log 99.5%

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

      4. add-exp-log 99.5%

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

      5. +-commutative 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. associate-/r/ 99.5%

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

       2. exp-1-e 99.5%

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

       3. +-commutative 99.5%

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

   11. Simplified 99.5%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -215 \lor \neg \left(y \leq 11200000000000\right):\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -215.0) (not (<= y 11200000000000.0)))
   (log (* y (/ E (+ x -1.0))))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -215.0) || !(y <= 11200000000000.0)) {
		tmp = log((y * (((double) M_E) / (x + -1.0))));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -215.0) || !(y <= 11200000000000.0)) {
		tmp = Math.log((y * (Math.E / (x + -1.0))));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -215.0) or not (y <= 11200000000000.0):
		tmp = math.log((y * (math.e / (x + -1.0))))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -215.0) || !(y <= 11200000000000.0))
		tmp = log(Float64(y * Float64(exp(1) / Float64(x + -1.0))));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -215.0], N[Not[LessEqual[y, 11200000000000.0]], $MachinePrecision]], N[Log[N[(y * N[(E / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -215 or 1.12e13 < y

   1. Initial program 36.9%

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

      1. sub-neg 36.9%

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

      2. log1p-define 36.9%

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

      3. distribute-neg-frac2 36.9%

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

      4. neg-sub0 36.9%

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

      5. associate--r- 36.9%

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

      6. metadata-eval 36.9%

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

      7. +-commutative 36.9%

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

   3. Simplified 36.9%

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

   5. Taylor expanded in y around inf 32.3%

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

      1. log-rec 32.3%

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

      2. unsub-neg 32.3%

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

      3. sub-neg 32.3%

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

      4. metadata-eval 32.3%

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

      5. +-commutative 32.3%

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

   7. Simplified 32.3%

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

      1. add-log-exp 32.3%

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

      2. exp-diff 32.3%

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

      3. diff-log 98.4%

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

      4. add-exp-log 98.4%

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

      5. +-commutative 98.4%

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

   9. Applied egg-rr 98.4%

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

       1. associate-/r/ 98.5%

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

       2. exp-1-e 98.5%

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

       3. +-commutative 98.5%

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

   11. Simplified 98.5%

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

   if -215 < y < 1.12e13

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 99.1%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -215 \lor \neg \left(y \leq 11200000000000\right):\\ \;\;\;\;\log \left(y \cdot \frac{e}{x + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.7) (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 <= -1.7) || !(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 <= -1.7) || !(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 <= -1.7) 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 <= -1.7) || !(y <= 1.0))
		tmp = log(Float64(y * Float64(exp(1) / Float64(x + -1.0))));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.7], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[Log[N[(y * N[(E / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.69999999999999996 or 1 < y

   1. Initial program 37.5%

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

      1. sub-neg 37.5%

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

      2. log1p-define 37.5%

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

      3. distribute-neg-frac2 37.5%

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

      4. neg-sub0 37.5%

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

      5. associate--r- 37.5%

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

      6. metadata-eval 37.5%

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

      7. +-commutative 37.5%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in y around inf 32.0%

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

      1. log-rec 32.0%

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

      2. unsub-neg 32.0%

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

      3. sub-neg 32.0%

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

      4. metadata-eval 32.0%

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

      5. +-commutative 32.0%

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

   7. Simplified 32.0%

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

      1. add-log-exp 32.0%

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

      2. exp-diff 32.0%

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

      3. diff-log 97.8%

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

      4. add-exp-log 97.9%

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

      5. +-commutative 97.9%

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

   9. Applied egg-rr 97.9%

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

       1. associate-/r/ 97.9%

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

       2. exp-1-e 97.9%

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

       3. +-commutative 97.9%

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

   11. Simplified 97.9%

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

   if -1.69999999999999996 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 99.4%

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

      1. +-commutative 99.4%

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

      2. div-sub 99.4%

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

      3. mul-1-neg 99.4%

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

      4. sub-neg 99.4%

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

      5. *-inverses 99.4%

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

      6. *-rgt-identity 99.4%

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

      7. log1p-define 99.4%

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

      8. mul-1-neg 99.4%

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

   7. Simplified 99.4%

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

4. Final simplification 98.8%

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

Alternative 5: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5:\\ \;\;\;\;\log \left(e \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5)
   (log (* E (- y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (log (* y (/ E x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5) {
		tmp = log((((double) M_E) * -y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = log((y * (((double) M_E) / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5) {
		tmp = Math.log((Math.E * -y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = Math.log((y * (Math.E / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5:
		tmp = math.log((math.e * -y))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = math.log((y * (math.e / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5)
		tmp = log(Float64(exp(1) * Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = log(Float64(y * Float64(exp(1) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5], N[Log[N[(E * (-y)), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[(y * N[(E / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5

   1. Initial program 20.5%

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

      1. sub-neg 20.5%

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

      2. log1p-define 20.5%

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

      3. distribute-neg-frac2 20.5%

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

      4. neg-sub0 20.5%

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

      5. associate--r- 20.5%

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

      6. metadata-eval 20.5%

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

      7. +-commutative 20.5%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. unsub-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-diff 0.0%

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

      3. diff-log 97.7%

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

      4. add-exp-log 97.7%

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

      5. +-commutative 97.7%

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

   9. Applied egg-rr 97.7%

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

       1. associate-/r/ 97.7%

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

       2. exp-1-e 97.7%

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

       3. +-commutative 97.7%

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

   11. Simplified 97.7%

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

   12. Taylor expanded in x around 0 71.7%

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

       1. neg-mul-1 71.7%

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

   14. Simplified 71.7%

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

   if -9.5 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.9%

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

      1. +-commutative 98.9%

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

      2. div-sub 98.9%

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

      3. mul-1-neg 98.9%

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

      4. sub-neg 98.9%

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

      5. *-inverses 98.9%

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

      6. *-rgt-identity 98.9%

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

      7. log1p-define 98.9%

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

      8. mul-1-neg 98.9%

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

   7. Simplified 98.9%

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

   if 1 < y

   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 70.5%

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

      3. distribute-neg-frac2 70.5%

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

      4. neg-sub0 70.5%

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

      5. associate--r- 70.5%

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

      6. metadata-eval 70.5%

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

      7. +-commutative 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in y around inf 98.8%

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

      1. log-rec 98.8%

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

      2. unsub-neg 98.8%

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

      3. sub-neg 98.8%

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

      4. metadata-eval 98.8%

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

      5. +-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. add-log-exp 98.8%

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

      2. exp-diff 98.8%

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

      3. diff-log 99.9%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-/r/ 100.0%

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

       2. exp-1-e 100.0%

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

       3. +-commutative 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 98.3%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5:\\ \;\;\;\;\log \left(e \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8:\\ \;\;\;\;\log \left(e \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8)
   (log (* E (- y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (log (* y (/ E x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8) {
		tmp = log((((double) M_E) * -y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = log((y * (((double) M_E) / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.8) {
		tmp = Math.log((Math.E * -y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = Math.log((y * (Math.E / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8:
		tmp = math.log((math.e * -y))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = math.log((y * (math.e / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8)
		tmp = log(Float64(exp(1) * Float64(-y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = log(Float64(y * Float64(exp(1) / x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8], N[Log[N[(E * (-y)), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[Log[N[(y * N[(E / x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.79999999999999982

   1. Initial program 20.5%

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

      1. sub-neg 20.5%

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

      2. log1p-define 20.5%

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

      3. distribute-neg-frac2 20.5%

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

      4. neg-sub0 20.5%

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

      5. associate--r- 20.5%

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

      6. metadata-eval 20.5%

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

      7. +-commutative 20.5%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. unsub-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-diff 0.0%

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

      3. diff-log 97.7%

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

      4. add-exp-log 97.7%

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

      5. +-commutative 97.7%

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

   9. Applied egg-rr 97.7%

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

       1. associate-/r/ 97.7%

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

       2. exp-1-e 97.7%

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

       3. +-commutative 97.7%

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

   11. Simplified 97.7%

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

   12. Taylor expanded in x around 0 71.7%

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

       1. neg-mul-1 71.7%

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

   14. Simplified 71.7%

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

   if -7.79999999999999982 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.0%

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

      1. log1p-define 98.0%

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

      2. mul-1-neg 98.0%

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

   7. Simplified 98.0%

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

   if 1 < y

   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 70.5%

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

      3. distribute-neg-frac2 70.5%

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

      4. neg-sub0 70.5%

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

      5. associate--r- 70.5%

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

      6. metadata-eval 70.5%

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

      7. +-commutative 70.5%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in y around inf 98.8%

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

      1. log-rec 98.8%

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

      2. unsub-neg 98.8%

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

      3. sub-neg 98.8%

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

      4. metadata-eval 98.8%

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

      5. +-commutative 98.8%

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

   7. Simplified 98.8%

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

      1. add-log-exp 98.8%

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

      2. exp-diff 98.8%

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

      3. diff-log 99.9%

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

      4. add-exp-log 100.0%

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

      5. +-commutative 100.0%

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

   9. Applied egg-rr 100.0%

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

       1. associate-/r/ 100.0%

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

       2. exp-1-e 100.0%

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

       3. +-commutative 100.0%

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

   11. Simplified 100.0%

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

   12. Taylor expanded in x around inf 98.3%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8:\\ \;\;\;\;\log \left(e \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{e}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;\log \left(e \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-38}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -65.0)
   (log (* E (- y)))
   (if (<= y 1.7e-38) (- 1.0 (log1p (- x))) (- 1.0 (log1p x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -65.0) {
		tmp = log((((double) M_E) * -y));
	} else if (y <= 1.7e-38) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log1p(x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -65.0) {
		tmp = Math.log((Math.E * -y));
	} else if (y <= 1.7e-38) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log1p(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -65.0:
		tmp = math.log((math.e * -y))
	elif y <= 1.7e-38:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log1p(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -65.0)
		tmp = log(Float64(exp(1) * Float64(-y)));
	elseif (y <= 1.7e-38)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log1p(x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -65.0], N[Log[N[(E * (-y)), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.7e-38], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -65

   1. Initial program 20.5%

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

      1. sub-neg 20.5%

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

      2. log1p-define 20.5%

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

      3. distribute-neg-frac2 20.5%

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

      4. neg-sub0 20.5%

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

      5. associate--r- 20.5%

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

      6. metadata-eval 20.5%

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

      7. +-commutative 20.5%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. unsub-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-diff 0.0%

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

      3. diff-log 97.7%

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

      4. add-exp-log 97.7%

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

      5. +-commutative 97.7%

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

   9. Applied egg-rr 97.7%

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

       1. associate-/r/ 97.7%

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

       2. exp-1-e 97.7%

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

       3. +-commutative 97.7%

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

   11. Simplified 97.7%

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

   12. Taylor expanded in x around 0 71.7%

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

       1. neg-mul-1 71.7%

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

   14. Simplified 71.7%

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

   if -65 < y < 1.7000000000000001e-38

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.1%

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

      1. log1p-define 98.1%

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

      2. mul-1-neg 98.1%

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

   7. Simplified 98.1%

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

   if 1.7000000000000001e-38 < y

   1. Initial program 76.1%

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

      1. sub-neg 76.1%

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

      2. log1p-define 76.1%

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

      3. distribute-neg-frac2 76.1%

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

      4. neg-sub0 76.1%

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

      5. associate--r- 76.1%

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

      6. metadata-eval 76.1%

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

      7. +-commutative 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in y around 0 18.3%

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

      1. log1p-define 18.3%

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

      2. mul-1-neg 18.3%

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

   7. Simplified 18.3%

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

      1. sub-neg 18.3%

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

      2. add-sqr-sqrt 9.2%

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

      3. sqrt-unprod 22.5%

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

      4. sqr-neg 22.5%

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

      5. sqrt-unprod 19.8%

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

      6. add-sqr-sqrt 29.0%

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

   9. Applied egg-rr 29.0%

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

       1. sub-neg 29.0%

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

   11. Simplified 29.0%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -65:\\ \;\;\;\;\log \left(e \cdot \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-38}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\log \left(e \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{1 - y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.06) (log (* E (- y))) (+ 1.0 (/ x (- 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.06) {
		tmp = log((((double) M_E) * -y));
	} else {
		tmp = 1.0 + (x / (1.0 - y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.06) {
		tmp = Math.log((Math.E * -y));
	} else {
		tmp = 1.0 + (x / (1.0 - y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.06:
		tmp = math.log((math.e * -y))
	else:
		tmp = 1.0 + (x / (1.0 - y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.06)
		tmp = log(Float64(exp(1) * Float64(-y)));
	else
		tmp = Float64(1.0 + Float64(x / Float64(1.0 - y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.06)
		tmp = log((2.71828182845904523536 * -y));
	else
		tmp = 1.0 + (x / (1.0 - y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.06], N[Log[N[(E * (-y)), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.059999999999999998

   1. Initial program 22.7%

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

      1. sub-neg 22.7%

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

      2. log1p-define 22.7%

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

      3. distribute-neg-frac2 22.7%

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

      4. neg-sub0 22.7%

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

      5. associate--r- 22.7%

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

      6. metadata-eval 22.7%

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

      7. +-commutative 22.7%

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

   3. Simplified 22.7%

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

   5. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. unsub-neg 0.0%

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

      3. sub-neg 0.0%

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

      4. metadata-eval 0.0%

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

      5. +-commutative 0.0%

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

   7. Simplified 0.0%

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

      1. add-log-exp 0.0%

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

      2. exp-diff 0.0%

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

      3. diff-log 95.8%

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

      4. add-exp-log 95.8%

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

      5. +-commutative 95.8%

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

   9. Applied egg-rr 95.8%

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

       1. associate-/r/ 95.9%

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

       2. exp-1-e 95.9%

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

       3. +-commutative 95.9%

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

   11. Simplified 95.9%

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

   12. Taylor expanded in x around 0 69.9%

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

       1. neg-mul-1 69.9%

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

   14. Simplified 69.9%

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

   if -0.059999999999999998 < y

   1. Initial program 94.5%

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

      1. sub-neg 94.5%

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

      2. log1p-define 94.6%

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

      3. distribute-neg-frac2 94.6%

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

      4. neg-sub0 94.6%

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

      5. associate--r- 94.6%

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

      6. metadata-eval 94.6%

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

      7. +-commutative 94.6%

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

   3. Simplified 94.6%

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

   5. Taylor expanded in x around inf 93.1%

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

   6. Taylor expanded in x around 0 56.7%

      \[\leadsto 1 - \color{blue}{\frac{x}{y - 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.06:\\ \;\;\;\;\log \left(e \cdot \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{x}{1 - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.7% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. sub-neg 74.3%

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

   2. log1p-define 74.3%

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

   3. distribute-neg-frac2 74.3%

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

   4. neg-sub0 74.3%

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

   5. associate--r- 74.3%

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

   6. metadata-eval 74.3%

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

   7. +-commutative 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 74.6%

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

6. Taylor expanded in x around 0 44.4%

   \[\leadsto 1 - \color{blue}{\frac{x}{y - 1}} \]

7. Final simplification 44.4%

   \[\leadsto 1 + \frac{x}{1 - y} \]
8. Add Preprocessing

Alternative 10: 43.3% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. sub-neg 74.3%

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

   2. log1p-define 74.3%

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

   3. distribute-neg-frac2 74.3%

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

   4. neg-sub0 74.3%

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

   5. associate--r- 74.3%

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

   6. metadata-eval 74.3%

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

   7. +-commutative 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in y around 0 61.8%

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

   1. log1p-define 61.8%

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

   2. mul-1-neg 61.8%

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

7. Simplified 61.8%

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

8. Taylor expanded in x around 0 43.1%

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

9. Final simplification 43.1%

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

Alternative 11: 43.0% 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 74.3%

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

   1. sub-neg 74.3%

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

   2. log1p-define 74.3%

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

   3. distribute-neg-frac2 74.3%

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

   4. neg-sub0 74.3%

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

   5. associate--r- 74.3%

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

   6. metadata-eval 74.3%

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

   7. +-commutative 74.3%

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

3. Simplified 74.3%

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

5. Taylor expanded in x around inf 74.6%

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

6. Taylor expanded in x around 0 42.9%

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

Developer Target 1: 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 2024139 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8128475261947241/100000000) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 30094271212461764000000000) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))