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

Percentage Accurate: 71.2% → 99.1%

Time: 9.9s

Alternatives: 12

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: 71.2% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 30000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+19)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 30000000000000.0)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+19) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 30000000000000.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+19) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 30000000000000.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e+19:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 30000000000000.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+19)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 30000000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+19], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 30000000000000.0], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5e19

   1. Initial program 11.7%

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

      1. sub-neg 11.7%

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

      2. log1p-def 11.7%

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

      3. neg-sub0 11.7%

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

      4. div-sub 11.7%

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

      5. associate--r- 11.7%

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

      6. neg-sub0 11.7%

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

      7. +-commutative 11.7%

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

      8. sub-neg 11.7%

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

      9. div-sub 11.7%

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

   3. Simplified 11.7%

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

   4. Taylor expanded in y around -inf 99.5%

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

      1. sub-neg 99.5%

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

      2. metadata-eval 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. metadata-eval 99.5%

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

      5. +-commutative 99.5%

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

      6. log1p-def 99.5%

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

      7. mul-1-neg 99.5%

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

   6. Simplified 99.5%

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

   if -9.5e19 < y < 3e13

   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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   if 3e13 < y

   1. Initial program 62.9%

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

      1. sub-neg 62.9%

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

      2. log1p-def 62.9%

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

      3. neg-sub0 62.9%

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

      4. div-sub 62.9%

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

      5. associate--r- 62.9%

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

      6. neg-sub0 62.9%

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

      7. +-commutative 62.9%

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

      8. sub-neg 62.9%

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

      9. div-sub 62.9%

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

   3. Simplified 62.9%

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

   4. Taylor expanded in y around inf 98.0%

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

      1. +-commutative 98.0%

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

      2. log-rec 98.0%

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

      3. unsub-neg 98.0%

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

      4. sub-neg 98.0%

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

      5. metadata-eval 98.0%

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

      6. +-commutative 98.0%

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

   6. Simplified 98.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 30000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]

Alternative 2: 71.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 1

   1. Initial program 73.3%

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

      1. sub-neg 73.3%

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

      2. log1p-def 73.4%

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

      3. neg-sub0 73.4%

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

      4. div-sub 73.4%

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

      5. associate--r- 73.4%

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

      6. neg-sub0 73.4%

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

      7. +-commutative 73.4%

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

      8. sub-neg 73.4%

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

      9. div-sub 73.4%

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

   3. Simplified 73.4%

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

   if 1 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 73.3%

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

      1. sub-neg 73.3%

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

      2. log1p-def 73.4%

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

      3. neg-sub0 73.4%

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

      4. div-sub 73.4%

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

      5. associate--r- 73.4%

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

      6. neg-sub0 73.4%

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

      7. +-commutative 73.4%

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

      8. sub-neg 73.4%

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

      9. div-sub 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in x around 0 45.0%

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

      1. log1p-def 45.0%

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

   6. Simplified 45.0%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 21.7%

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

   9. Simplified 21.7%

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

4. Final simplification 73.4%

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

Alternative 3: 89.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.38 \cdot 10^{-5} \lor \neg \left(y \leq 10^{-5}\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+19)
   (- 1.0 (log (/ -1.0 y)))
   (if (or (<= y -1.38e-5) (not (<= y 1e-5)))
     (- 1.0 (log (/ x (+ y -1.0))))
     (- 1.0 (+ y (log1p (- x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+19) {
		tmp = 1.0 - log((-1.0 / y));
	} else if ((y <= -1.38e-5) || !(y <= 1e-5)) {
		tmp = 1.0 - log((x / (y + -1.0)));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+19) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if ((y <= -1.38e-5) || !(y <= 1e-5)) {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e+19:
		tmp = 1.0 - math.log((-1.0 / y))
	elif (y <= -1.38e-5) or not (y <= 1e-5):
		tmp = 1.0 - math.log((x / (y + -1.0)))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+19)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif ((y <= -1.38e-5) || !(y <= 1e-5))
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+19], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -1.38e-5], N[Not[LessEqual[y, 1e-5]], $MachinePrecision]], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5e19

   1. Initial program 11.7%

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

      1. sub-neg 11.7%

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

      2. log1p-def 11.7%

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

      3. neg-sub0 11.7%

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

      4. div-sub 11.7%

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

      5. associate--r- 11.7%

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

      6. neg-sub0 11.7%

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

      7. +-commutative 11.7%

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

      8. sub-neg 11.7%

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

      9. div-sub 11.7%

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

   3. Simplified 11.7%

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

   4. Taylor expanded in x around 0 2.9%

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

      1. log1p-def 2.9%

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

   6. Simplified 2.9%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 74.2%

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

   9. Simplified 74.2%

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

   if -9.5e19 < y < -1.38e-5 or 1.00000000000000008e-5 < y

   1. Initial program 71.8%

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

      1. sub-neg 71.8%

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

      2. log1p-def 71.8%

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

      3. neg-sub0 71.8%

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

      4. div-sub 71.8%

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

      5. associate--r- 71.8%

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

      6. neg-sub0 71.8%

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

      7. +-commutative 71.8%

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

      8. sub-neg 71.8%

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

      9. div-sub 71.8%

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

   3. Simplified 71.8%

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

   4. Taylor expanded in x around inf 69.7%

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

      1. neg-mul-1 69.7%

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

      2. distribute-neg-frac 69.7%

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

   6. Simplified 69.7%

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

      1. frac-2neg 69.7%

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

      2. div-inv 69.7%

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

      3. remove-double-neg 69.7%

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

   8. Applied egg-rr 69.7%

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

      1. associate-*r/ 69.7%

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

      2. *-rgt-identity 69.7%

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

      3. neg-sub0 69.7%

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

      4. associate--r- 69.7%

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

      5. metadata-eval 69.7%

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

   10. Simplified 69.7%

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

   11. Taylor expanded in x around inf 78.0%

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

       1. log-rec 78.0%

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

       2. unsub-neg 78.0%

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

       3. mul-1-neg 78.0%

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

       4. log-rec 78.0%

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

       5. remove-double-neg 78.0%

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

       6. sub-neg 78.0%

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

       7. metadata-eval 78.0%

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

       8. +-commutative 78.0%

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

       9. log-div 96.9%

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

       10. rem-exp-log 78.3%

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

       11. rem-exp-log 78.0%

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

       12. +-commutative 78.0%

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

       13. metadata-eval 78.0%

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

       14. sub-neg 78.0%

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

       15. rem-exp-log 78.7%

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

       16. sub-neg 78.7%

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

       17. metadata-eval 78.7%

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

       18. +-commutative 78.7%

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

       19. rem-exp-log 96.9%

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

       20. +-commutative 96.9%

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

   13. Simplified 96.9%

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

   if -1.38e-5 < y < 1.00000000000000008e-5

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

      1. flip-- 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 99.8%

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

      1. log1p-def 99.8%

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

      2. mul-1-neg 99.8%

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

   8. Simplified 99.8%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.38 \cdot 10^{-5} \lor \neg \left(y \leq 10^{-5}\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 4: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + -1}\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+20}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(t_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y -1.0))))
   (if (<= y -1.02e+20)
     (- 1.0 (log (/ -1.0 y)))
     (if (<= y 1.0) (- 1.0 (log1p t_0)) (- 1.0 (log t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (y + -1.0);
	double tmp;
	if (y <= -1.02e+20) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(t_0);
	} else {
		tmp = 1.0 - log(t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = x / (y + -1.0);
	double tmp;
	if (y <= -1.02e+20) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(t_0);
	} else {
		tmp = 1.0 - Math.log(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + -1.0)
	tmp = 0
	if y <= -1.02e+20:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(t_0)
	else:
		tmp = 1.0 - math.log(t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + -1.0))
	tmp = 0.0
	if (y <= -1.02e+20)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(t_0));
	else
		tmp = Float64(1.0 - log(t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.02e+20], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02e20

   1. Initial program 11.7%

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

      1. sub-neg 11.7%

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

      2. log1p-def 11.7%

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

      3. neg-sub0 11.7%

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

      4. div-sub 11.7%

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

      5. associate--r- 11.7%

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

      6. neg-sub0 11.7%

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

      7. +-commutative 11.7%

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

      8. sub-neg 11.7%

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

      9. div-sub 11.7%

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

   3. Simplified 11.7%

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

   4. Taylor expanded in x around 0 2.9%

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

      1. log1p-def 2.9%

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

   6. Simplified 2.9%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 74.2%

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

   9. Simplified 74.2%

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

   if -1.02e20 < 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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.4%

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

      1. neg-mul-1 99.4%

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

      2. distribute-neg-frac 99.4%

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

   6. Simplified 99.4%

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

      1. frac-2neg 99.4%

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

      2. div-inv 99.4%

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

      3. remove-double-neg 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate-*r/ 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. neg-sub0 99.4%

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

      4. associate--r- 99.4%

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

      5. metadata-eval 99.4%

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

   10. Simplified 99.4%

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

   if 1 < y

   1. Initial program 66.0%

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

      1. sub-neg 66.0%

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

      2. log1p-def 66.0%

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

      3. neg-sub0 66.0%

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

      4. div-sub 66.0%

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

      5. associate--r- 66.0%

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

      6. neg-sub0 66.0%

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

      7. +-commutative 66.0%

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

      8. sub-neg 66.0%

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

      9. div-sub 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in x around inf 63.4%

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

      1. neg-mul-1 63.4%

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

      2. distribute-neg-frac 63.4%

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

   6. Simplified 63.4%

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

      1. frac-2neg 63.4%

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

      2. div-inv 63.4%

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

      3. remove-double-neg 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. associate-*r/ 63.4%

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

      2. *-rgt-identity 63.4%

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

      3. neg-sub0 63.4%

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

      4. associate--r- 63.4%

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

      5. metadata-eval 63.4%

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

   10. Simplified 63.4%

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

   11. Taylor expanded in x around inf 94.3%

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

       1. log-rec 94.3%

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

       2. unsub-neg 94.3%

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

       3. mul-1-neg 94.3%

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

       4. log-rec 94.3%

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

       5. remove-double-neg 94.3%

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

       6. sub-neg 94.3%

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

       7. metadata-eval 94.3%

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

       8. +-commutative 94.3%

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

       9. log-div 96.2%

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

       10. rem-exp-log 94.6%

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

       11. rem-exp-log 94.3%

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

       12. +-commutative 94.3%

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

       13. metadata-eval 94.3%

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

       14. sub-neg 94.3%

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

       15. rem-exp-log 95.1%

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

       16. sub-neg 95.1%

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

       17. metadata-eval 95.1%

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

       18. +-commutative 95.1%

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

       19. rem-exp-log 96.2%

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

       20. +-commutative 96.2%

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

   13. Simplified 96.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+20}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 5: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.2)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.2) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.2) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.2:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.2)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.2], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.1999999999999993

   1. Initial program 16.6%

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

      1. sub-neg 16.6%

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

      2. log1p-def 16.6%

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

      3. neg-sub0 16.6%

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

      4. div-sub 16.6%

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

      5. associate--r- 16.6%

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

      6. neg-sub0 16.6%

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

      7. +-commutative 16.6%

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

      8. sub-neg 16.6%

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

      9. div-sub 16.6%

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

   3. Simplified 16.6%

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

   4. Taylor expanded in x around 0 2.8%

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

      1. log1p-def 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 70.2%

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

   9. Simplified 70.2%

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

   if -8.1999999999999993 < 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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

      1. flip-- 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 99.7%

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

      1. log1p-def 99.7%

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

      2. mul-1-neg 99.7%

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

   8. Simplified 99.7%

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

   if 1 < y

   1. Initial program 66.0%

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

      1. sub-neg 66.0%

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

      2. log1p-def 66.0%

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

      3. neg-sub0 66.0%

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

      4. div-sub 66.0%

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

      5. associate--r- 66.0%

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

      6. neg-sub0 66.0%

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

      7. +-commutative 66.0%

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

      8. sub-neg 66.0%

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

      9. div-sub 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in x around inf 63.4%

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

      1. neg-mul-1 63.4%

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

      2. distribute-neg-frac 63.4%

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

   6. Simplified 63.4%

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

      1. frac-2neg 63.4%

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

      2. div-inv 63.4%

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

      3. remove-double-neg 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. associate-*r/ 63.4%

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

      2. *-rgt-identity 63.4%

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

      3. neg-sub0 63.4%

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

      4. associate--r- 63.4%

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

      5. metadata-eval 63.4%

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

   10. Simplified 63.4%

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

   11. Taylor expanded in x around inf 94.3%

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

       1. log-rec 94.3%

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

       2. unsub-neg 94.3%

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

       3. mul-1-neg 94.3%

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

       4. log-rec 94.3%

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

       5. remove-double-neg 94.3%

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

       6. sub-neg 94.3%

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

       7. metadata-eval 94.3%

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

       8. +-commutative 94.3%

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

       9. log-div 96.2%

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

       10. rem-exp-log 94.6%

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

       11. rem-exp-log 94.3%

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

       12. +-commutative 94.3%

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

       13. metadata-eval 94.3%

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

       14. sub-neg 94.3%

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

       15. rem-exp-log 95.1%

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

       16. sub-neg 95.1%

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

       17. metadata-eval 95.1%

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

       18. +-commutative 95.1%

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

       19. rem-exp-log 96.2%

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

       20. +-commutative 96.2%

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

   13. Simplified 96.2%

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

   14. Taylor expanded in y around inf 94.8%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 6: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+19)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+19) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+19) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.5e+19:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.5e+19)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.5e+19], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5e19

   1. Initial program 11.7%

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

      1. sub-neg 11.7%

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

      2. log1p-def 11.7%

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

      3. neg-sub0 11.7%

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

      4. div-sub 11.7%

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

      5. associate--r- 11.7%

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

      6. neg-sub0 11.7%

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

      7. +-commutative 11.7%

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

      8. sub-neg 11.7%

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

      9. div-sub 11.7%

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

   3. Simplified 11.7%

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

   4. Taylor expanded in x around 0 2.9%

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

      1. log1p-def 2.9%

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

   6. Simplified 2.9%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 74.2%

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

   9. Simplified 74.2%

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

   if -9.5e19 < 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-def 100.0%

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

      3. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.0%

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

      1. log1p-def 97.0%

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

      2. mul-1-neg 97.0%

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

   6. Simplified 97.0%

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

   if 1 < y

   1. Initial program 66.0%

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

      1. sub-neg 66.0%

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

      2. log1p-def 66.0%

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

      3. neg-sub0 66.0%

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

      4. div-sub 66.0%

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

      5. associate--r- 66.0%

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

      6. neg-sub0 66.0%

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

      7. +-commutative 66.0%

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

      8. sub-neg 66.0%

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

      9. div-sub 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in x around inf 63.4%

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

      1. neg-mul-1 63.4%

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

      2. distribute-neg-frac 63.4%

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

   6. Simplified 63.4%

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

      1. frac-2neg 63.4%

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

      2. div-inv 63.4%

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

      3. remove-double-neg 63.4%

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

   8. Applied egg-rr 63.4%

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

      1. associate-*r/ 63.4%

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

      2. *-rgt-identity 63.4%

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

      3. neg-sub0 63.4%

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

      4. associate--r- 63.4%

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

      5. metadata-eval 63.4%

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

   10. Simplified 63.4%

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

   11. Taylor expanded in x around inf 94.3%

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

       1. log-rec 94.3%

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

       2. unsub-neg 94.3%

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

       3. mul-1-neg 94.3%

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

       4. log-rec 94.3%

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

       5. remove-double-neg 94.3%

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

       6. sub-neg 94.3%

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

       7. metadata-eval 94.3%

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

       8. +-commutative 94.3%

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

       9. log-div 96.2%

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

       10. rem-exp-log 94.6%

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

       11. rem-exp-log 94.3%

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

       12. +-commutative 94.3%

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

       13. metadata-eval 94.3%

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

       14. sub-neg 94.3%

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

       15. rem-exp-log 95.1%

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

       16. sub-neg 95.1%

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

       17. metadata-eval 95.1%

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

       18. +-commutative 95.1%

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

       19. rem-exp-log 96.2%

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

       20. +-commutative 96.2%

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

   13. Simplified 96.2%

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

   14. Taylor expanded in y around inf 94.8%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 7: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.5e+19) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.5e+19) {
		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 <= -9.5e+19) {
		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 <= -9.5e+19:
		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 <= -9.5e+19)
		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, -9.5e+19], 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 < -9.5e19

   1. Initial program 11.7%

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

      1. sub-neg 11.7%

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

      2. log1p-def 11.7%

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

      3. neg-sub0 11.7%

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

      4. div-sub 11.7%

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

      5. associate--r- 11.7%

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

      6. neg-sub0 11.7%

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

      7. +-commutative 11.7%

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

      8. sub-neg 11.7%

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

      9. div-sub 11.7%

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

   3. Simplified 11.7%

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

   4. Taylor expanded in x around 0 2.9%

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

      1. log1p-def 2.9%

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

   6. Simplified 2.9%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 74.2%

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

   9. Simplified 74.2%

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

   if -9.5e19 < y

   1. Initial program 95.6%

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

      1. sub-neg 95.6%

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

      2. log1p-def 95.7%

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

      3. neg-sub0 95.7%

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

      4. div-sub 95.7%

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

      5. associate--r- 95.7%

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

      6. neg-sub0 95.7%

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

      7. +-commutative 95.7%

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

      8. sub-neg 95.7%

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

      9. div-sub 95.7%

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

   3. Simplified 95.7%

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

   4. Taylor expanded in y around 0 84.6%

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

      1. log1p-def 84.6%

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

      2. mul-1-neg 84.6%

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

   6. Simplified 84.6%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+19}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 8: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y + -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.88) (- 1.0 (log (- x))) (- 1.0 (/ x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.88) {
		tmp = 1.0 - log(-x);
	} else {
		tmp = 1.0 - (x / (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.88d0)) then
        tmp = 1.0d0 - log(-x)
    else
        tmp = 1.0d0 - (x / (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.88) {
		tmp = 1.0 - Math.log(-x);
	} else {
		tmp = 1.0 - (x / (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.88:
		tmp = 1.0 - math.log(-x)
	else:
		tmp = 1.0 - (x / (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.88)
		tmp = Float64(1.0 - log(Float64(-x)));
	else
		tmp = Float64(1.0 - Float64(x / Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.88)
		tmp = 1.0 - log(-x);
	else
		tmp = 1.0 - (x / (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.88], N[(1.0 - N[Log[(-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.880000000000000004

   1. Initial program 80.3%

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

      1. sub-neg 80.3%

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

      2. log1p-def 80.3%

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

      3. neg-sub0 80.3%

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

      4. div-sub 80.3%

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

      5. associate--r- 80.3%

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

      6. neg-sub0 80.3%

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

      7. +-commutative 80.3%

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

      8. sub-neg 80.3%

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

      9. div-sub 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in x around inf 82.6%

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

      1. neg-mul-1 82.6%

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

      2. distribute-neg-frac 82.6%

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

   6. Simplified 82.6%

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

      1. frac-2neg 82.6%

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

      2. div-inv 82.6%

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

      3. remove-double-neg 82.6%

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

   8. Applied egg-rr 82.6%

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

      1. associate-*r/ 82.6%

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

      2. *-rgt-identity 82.6%

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

      3. neg-sub0 82.6%

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

      4. associate--r- 82.6%

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

      5. metadata-eval 82.6%

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

   10. Simplified 82.6%

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

   11. Taylor expanded in x around inf 0.0%

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

       1. log-rec 0.0%

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

       2. unsub-neg 0.0%

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

       3. mul-1-neg 0.0%

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

       4. log-rec 0.0%

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

       5. remove-double-neg 0.0%

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

       6. sub-neg 0.0%

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

       7. metadata-eval 0.0%

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

       8. +-commutative 0.0%

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

       9. log-div 99.0%

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

       10. rem-exp-log 0.0%

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

       11. rem-exp-log 0.0%

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

       12. +-commutative 0.0%

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

       13. metadata-eval 0.0%

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

       14. sub-neg 0.0%

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

       15. rem-exp-log 0.0%

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

       16. sub-neg 0.0%

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

       17. metadata-eval 0.0%

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

       18. +-commutative 0.0%

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

       19. rem-exp-log 99.0%

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

       20. +-commutative 99.0%

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

   13. Simplified 99.0%

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

   14. Taylor expanded in y around 0 66.3%

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

       1. neg-mul-1 66.3%

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

   16. Simplified 66.3%

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

   if -0.880000000000000004 < x

   1. Initial program 70.7%

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

      1. sub-neg 70.7%

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

      2. log1p-def 70.8%

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

      3. neg-sub0 70.8%

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

      4. div-sub 70.8%

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

      5. associate--r- 70.8%

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

      6. neg-sub0 70.8%

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

      7. +-commutative 70.8%

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

      8. sub-neg 70.8%

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

      9. div-sub 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in x around inf 72.7%

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

      1. neg-mul-1 72.7%

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

      2. distribute-neg-frac 72.7%

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

   6. Simplified 72.7%

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

      1. frac-2neg 72.7%

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

      2. div-inv 72.7%

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

      3. remove-double-neg 72.7%

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

   8. Applied egg-rr 72.7%

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

      1. associate-*r/ 72.7%

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

      2. *-rgt-identity 72.7%

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

      3. neg-sub0 72.7%

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

      4. associate--r- 72.7%

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

      5. metadata-eval 72.7%

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

   10. Simplified 72.7%

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

   11. Taylor expanded in x around 0 65.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.88:\\ \;\;\;\;1 - \log \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y + -1}\\ \end{array} \]

Alternative 9: 61.9% 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 73.3%

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

   1. sub-neg 73.3%

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

   2. log1p-def 73.4%

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

   3. neg-sub0 73.4%

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

   4. div-sub 73.4%

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

   5. associate--r- 73.4%

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

   6. neg-sub0 73.4%

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

   7. +-commutative 73.4%

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

   8. sub-neg 73.4%

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

   9. div-sub 73.4%

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

3. Simplified 73.4%

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

4. Taylor expanded in y around 0 65.3%

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

   1. log1p-def 65.3%

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

   2. mul-1-neg 65.3%

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

6. Simplified 65.3%

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

7. Final simplification 65.3%

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

Alternative 10: 42.6% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.38 \cdot 10^{-5}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.38e-5) (- 1.0 (/ x y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.38e-5) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.38d-5)) then
        tmp = 1.0d0 - (x / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.38e-5) {
		tmp = 1.0 - (x / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.38e-5:
		tmp = 1.0 - (x / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.38e-5)
		tmp = Float64(1.0 - Float64(x / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.38e-5)
		tmp = 1.0 - (x / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.38e-5], N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.38e-5

   1. Initial program 17.7%

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

      1. sub-neg 17.7%

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

      2. log1p-def 17.7%

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

      3. neg-sub0 17.7%

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

      4. div-sub 17.7%

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

      5. associate--r- 17.7%

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

      6. neg-sub0 17.7%

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

      7. +-commutative 17.7%

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

      8. sub-neg 17.7%

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

      9. div-sub 17.7%

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

   3. Simplified 17.7%

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

   4. Taylor expanded in x around inf 27.0%

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

      1. neg-mul-1 27.0%

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

      2. distribute-neg-frac 27.0%

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

   6. Simplified 27.0%

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

      1. frac-2neg 27.0%

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

      2. div-inv 27.0%

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

      3. remove-double-neg 27.0%

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

   8. Applied egg-rr 27.0%

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

      1. associate-*r/ 27.0%

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

      2. *-rgt-identity 27.0%

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

      3. neg-sub0 27.0%

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

      4. associate--r- 27.0%

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

      5. metadata-eval 27.0%

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

   10. Simplified 27.0%

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

   11. Taylor expanded in y around inf 12.9%

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

   if -1.38e-5 < y

   1. Initial program 95.5%

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

      1. sub-neg 95.5%

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

      2. log1p-def 95.5%

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

      3. neg-sub0 95.5%

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

      4. div-sub 95.5%

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

      5. associate--r- 95.5%

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

      6. neg-sub0 95.5%

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

      7. +-commutative 95.5%

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

      8. sub-neg 95.5%

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

      9. div-sub 95.5%

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

   3. Simplified 95.5%

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

   4. Taylor expanded in x around 0 61.8%

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

      1. log1p-def 61.8%

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

   6. Simplified 61.8%

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

   7. Taylor expanded in y around 0 62.1%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.38 \cdot 10^{-5}:\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]

Alternative 11: 44.1% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sub-neg 73.3%

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

   2. log1p-def 73.4%

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

   3. neg-sub0 73.4%

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

   4. div-sub 73.4%

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

   5. associate--r- 73.4%

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

   6. neg-sub0 73.4%

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

   7. +-commutative 73.4%

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

   8. sub-neg 73.4%

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

   9. div-sub 73.4%

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

3. Simplified 73.4%

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

4. Taylor expanded in x around inf 75.4%

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

   1. neg-mul-1 75.4%

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

   2. distribute-neg-frac 75.4%

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

6. Simplified 75.4%

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

   1. frac-2neg 75.4%

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

   2. div-inv 75.4%

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

   3. remove-double-neg 75.4%

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

8. Applied egg-rr 75.4%

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

   1. associate-*r/ 75.4%

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

   2. *-rgt-identity 75.4%

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

   3. neg-sub0 75.4%

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

   4. associate--r- 75.4%

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

   5. metadata-eval 75.4%

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

10. Simplified 75.4%

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

11. Taylor expanded in x around 0 49.4%

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

12. Final simplification 49.4%

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

Alternative 12: 40.1% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.3%

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

   1. sub-neg 73.3%

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

   2. log1p-def 73.4%

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

   3. neg-sub0 73.4%

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

   4. div-sub 73.4%

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

   5. associate--r- 73.4%

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

   6. neg-sub0 73.4%

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

   7. +-commutative 73.4%

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

   8. sub-neg 73.4%

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

   9. div-sub 73.4%

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

3. Simplified 73.4%

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

4. Taylor expanded in x around 0 45.0%

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

   1. log1p-def 45.0%

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

6. Simplified 45.0%

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

7. Taylor expanded in y around 0 45.7%

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

8. Final simplification 45.7%

   \[\leadsto 1 - y \]

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 2023224 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

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