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

Percentage Accurate: 72.2% → 99.4%

Time: 12.0s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -420000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+28}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -420000.0)
   (+ (- 1.0 (log1p (- x))) (- (/ -1.0 y) (log (/ -1.0 y))))
   (if (<= y 1.45e+28)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (- 1.0 (- (log x) (log y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -420000.0) {
		tmp = (1.0 - log1p(-x)) + ((-1.0 / y) - log((-1.0 / y)));
	} else if (y <= 1.45e+28) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -420000.0) {
		tmp = (1.0 - Math.log1p(-x)) + ((-1.0 / y) - Math.log((-1.0 / y)));
	} else if (y <= 1.45e+28) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -420000.0:
		tmp = (1.0 - math.log1p(-x)) + ((-1.0 / y) - math.log((-1.0 / y)))
	elif y <= 1.45e+28:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -420000.0)
		tmp = Float64(Float64(1.0 - log1p(Float64(-x))) + Float64(Float64(-1.0 / y) - log(Float64(-1.0 / y))));
	elseif (y <= 1.45e+28)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - Float64(log(x) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -420000.0], N[(N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 / y), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+28], 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[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.2e5

   1. Initial program 19.7%

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

      1. sub-neg 19.7%

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

      2. log1p-def 19.7%

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

      3. distribute-neg-frac 19.7%

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

      4. sub-neg 19.7%

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

      5. distribute-neg-in 19.7%

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

      6. remove-double-neg 19.7%

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

      7. +-commutative 19.7%

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

      8. sub-neg 19.7%

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

   3. Simplified 19.7%

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

      1. flip-- 18.7%

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

      2. associate-/r/ 19.8%

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

      3. metadata-eval 19.8%

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

      4. +-commutative 19.8%

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

   5. Applied egg-rr 19.8%

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

   6. Taylor expanded in y around -inf 99.4%

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

      1. associate--r+ 99.4%

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

      2. log1p-def 99.4%

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

      3. mul-1-neg 99.4%

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

      4. +-commutative 99.4%

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

   8. Simplified 99.4%

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

   if -4.2e5 < y < 1.4500000000000001e28

   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. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 1.4500000000000001e28 < y

   1. Initial program 57.9%

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

      1. sub-neg 57.9%

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

      2. log1p-def 57.9%

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

      3. distribute-neg-frac 57.9%

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

      4. sub-neg 57.9%

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

      5. distribute-neg-in 57.9%

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

      6. remove-double-neg 57.9%

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

      7. +-commutative 57.9%

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

      8. sub-neg 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in y around inf 58.0%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. log-rec 98.6%

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

      3. unsub-neg 98.6%

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

      4. mul-1-neg 98.6%

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

      5. log-rec 98.6%

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

      6. remove-double-neg 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -420000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+28}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log x - \log y\right)\\ \end{array} \]

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1600000000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+21}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log x - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1600000000.0)
   (- (- 1.0 (log1p (- x))) (log (/ -1.0 y)))
   (if (<= y 2.3e+21)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (- 1.0 (- (log x) (log y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1600000000.0) {
		tmp = (1.0 - log1p(-x)) - log((-1.0 / y));
	} else if (y <= 2.3e+21) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - (log(x) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1600000000.0) {
		tmp = (1.0 - Math.log1p(-x)) - Math.log((-1.0 / y));
	} else if (y <= 2.3e+21) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - (Math.log(x) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1600000000.0:
		tmp = (1.0 - math.log1p(-x)) - math.log((-1.0 / y))
	elif y <= 2.3e+21:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - (math.log(x) - math.log(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1600000000.0)
		tmp = Float64(Float64(1.0 - log1p(Float64(-x))) - log(Float64(-1.0 / y)));
	elseif (y <= 2.3e+21)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - Float64(log(x) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1600000000.0], N[(N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+21], 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[x], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6e9

   1. Initial program 19.7%

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

      1. sub-neg 19.7%

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

      2. log1p-def 19.7%

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

      3. distribute-neg-frac 19.7%

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

      4. sub-neg 19.7%

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

      5. distribute-neg-in 19.7%

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

      6. remove-double-neg 19.7%

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

      7. +-commutative 19.7%

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

      8. sub-neg 19.7%

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

   3. Simplified 19.7%

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

   4. Taylor expanded in y around -inf 99.2%

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

      1. associate--r+ 99.2%

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

      2. sub-neg 99.2%

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

      3. metadata-eval 99.2%

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

      4. distribute-lft-in 99.2%

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

      5. metadata-eval 99.2%

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

      6. +-commutative 99.2%

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

      7. log1p-def 99.2%

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

      8. mul-1-neg 99.2%

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

   6. Simplified 99.2%

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

   if -1.6e9 < y < 2.3e21

   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. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 2.3e21 < y

   1. Initial program 57.9%

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

      1. sub-neg 57.9%

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

      2. log1p-def 57.9%

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

      3. distribute-neg-frac 57.9%

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

      4. sub-neg 57.9%

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

      5. distribute-neg-in 57.9%

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

      6. remove-double-neg 57.9%

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

      7. +-commutative 57.9%

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

      8. sub-neg 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in y around inf 58.0%

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

   5. Taylor expanded in x around inf 98.6%

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

      1. +-commutative 98.6%

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

      2. log-rec 98.6%

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

      3. unsub-neg 98.6%

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

      4. mul-1-neg 98.6%

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

      5. log-rec 98.6%

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

      6. remove-double-neg 98.6%

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

   7. Simplified 98.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1600000000:\\ \;\;\;\;\left(1 - \mathsf{log1p}\left(-x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+21}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log x - \log y\right)\\ \end{array} \]

Alternative 3: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + \frac{y - x}{1 - y} \leq 0:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(y - x\right) \cdot \frac{1}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (/ (- y x) (- 1.0 y))) 0.0)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (* (- y x) (/ 1.0 (- 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + ((y - x) / (1.0 - y))) <= 0.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(((y - x) * (1.0 / (1.0 - y))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 + ((y - x) / (1.0 - y))) <= 0.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(((y - x) * (1.0 / (1.0 - y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 + ((y - x) / (1.0 - y))) <= 0.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(((y - x) * (1.0 / (1.0 - y))))
	return tmp

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] * N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y))) < 0.0

   1. Initial program 3.1%

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

      1. sub-neg 3.1%

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

      2. log1p-def 3.1%

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

      3. distribute-neg-frac 3.1%

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

      4. sub-neg 3.1%

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

      5. distribute-neg-in 3.1%

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

      6. remove-double-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around inf 3.1%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. distribute-neg-frac 73.5%

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

      2. metadata-eval 73.5%

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

   7. Simplified 73.5%

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

   if 0.0 < (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))

   1. Initial program 98.7%

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

      1. sub-neg 98.7%

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

      2. log1p-def 98.7%

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

      3. distribute-neg-frac 98.7%

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

      4. sub-neg 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. remove-double-neg 98.7%

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

      7. +-commutative 98.7%

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

      8. sub-neg 98.7%

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

   3. Simplified 98.7%

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

      1. clear-num 98.8%

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

      2. associate-/r/ 98.8%

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

   5. Applied egg-rr 98.8%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y - x}{1 - y} \leq 0:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(y - x\right) \cdot \frac{1}{1 - y}\right)\\ \end{array} \]

Alternative 4: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{1 - y}\\ \mathbf{if}\;1 + t_0 \leq 0:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- 1.0 y))))
   (if (<= (+ 1.0 t_0) 0.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p t_0)))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (1.0 - y);
	double tmp;
	if ((1.0 + t_0) <= 0.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (1.0 - y);
	double tmp;
	if ((1.0 + t_0) <= 0.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (1.0 - y)
	tmp = 0
	if (1.0 + t_0) <= 0.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(t_0)
	return tmp

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 + t$95$0), $MachinePrecision], 0.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y))) < 0.0

   1. Initial program 3.1%

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

      1. sub-neg 3.1%

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

      2. log1p-def 3.1%

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

      3. distribute-neg-frac 3.1%

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

      4. sub-neg 3.1%

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

      5. distribute-neg-in 3.1%

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

      6. remove-double-neg 3.1%

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

      7. +-commutative 3.1%

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

      8. sub-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around inf 3.1%

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

   5. Taylor expanded in x around 0 73.5%

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

      1. distribute-neg-frac 73.5%

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

      2. metadata-eval 73.5%

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

   7. Simplified 73.5%

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

   if 0.0 < (-.f64 1 (/.f64 (-.f64 x y) (-.f64 1 y)))

   1. Initial program 98.7%

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

      1. sub-neg 98.7%

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

      2. log1p-def 98.7%

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

      3. distribute-neg-frac 98.7%

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

      4. sub-neg 98.7%

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

      5. distribute-neg-in 98.7%

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

      6. remove-double-neg 98.7%

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

      7. +-commutative 98.7%

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

      8. sub-neg 98.7%

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

   3. Simplified 98.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{y - x}{1 - y} \leq 0:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \end{array} \]

Alternative 5: 85.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -30000.0)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ (- x) (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -30000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -30000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -30000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p((-x / (1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -30000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(-x) / Float64(1.0 - y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -30000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3e4

   1. Initial program 19.7%

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

      1. sub-neg 19.7%

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

      2. log1p-def 19.7%

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

      3. distribute-neg-frac 19.7%

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

      4. sub-neg 19.7%

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

      5. distribute-neg-in 19.7%

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

      6. remove-double-neg 19.7%

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

      7. +-commutative 19.7%

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

      8. sub-neg 19.7%

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

   3. Simplified 19.7%

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

   4. Taylor expanded in y around inf 19.7%

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

   5. Taylor expanded in x around 0 72.2%

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

      1. distribute-neg-frac 72.2%

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

      2. metadata-eval 72.2%

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

   7. Simplified 72.2%

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

   if -3e4 < y

   1. Initial program 93.5%

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

      1. sub-neg 93.5%

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

      2. log1p-def 93.5%

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

      3. distribute-neg-frac 93.5%

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

      4. sub-neg 93.5%

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

      5. distribute-neg-in 93.5%

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

      6. remove-double-neg 93.5%

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

      7. +-commutative 93.5%

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

      8. sub-neg 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in x around inf 92.0%

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

      1. neg-mul-1 92.0%

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

      2. distribute-neg-frac 92.0%

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

   6. Simplified 92.0%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -30000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]

Alternative 6: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.5) (- 1.0 (log (/ -1.0 y))) (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.5) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.5) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.5:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.5)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.5], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.5

   1. Initial program 19.7%

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

      1. sub-neg 19.7%

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

      2. log1p-def 19.7%

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

      3. distribute-neg-frac 19.7%

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

      4. sub-neg 19.7%

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

      5. distribute-neg-in 19.7%

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

      6. remove-double-neg 19.7%

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

      7. +-commutative 19.7%

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

      8. sub-neg 19.7%

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

   3. Simplified 19.7%

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

   4. Taylor expanded in y around inf 19.7%

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

   5. Taylor expanded in x around 0 72.2%

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

      1. distribute-neg-frac 72.2%

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

      2. metadata-eval 72.2%

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

   7. Simplified 72.2%

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

   if -6.5 < y

   1. Initial program 93.5%

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

      1. sub-neg 93.5%

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

      2. log1p-def 93.5%

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

      3. distribute-neg-frac 93.5%

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

      4. sub-neg 93.5%

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

      5. distribute-neg-in 93.5%

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

      6. remove-double-neg 93.5%

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

      7. +-commutative 93.5%

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

      8. sub-neg 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around 0 82.6%

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

      1. +-commutative 82.6%

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

      2. div-sub 82.6%

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

      3. mul-1-neg 82.6%

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

      4. sub-neg 82.6%

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

      5. *-inverses 82.6%

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

      6. *-rgt-identity 82.6%

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

      7. log1p-def 82.6%

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

      8. mul-1-neg 82.6%

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

   6. Simplified 82.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 7: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -270:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -270.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -270.0) {
		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 <= -270.0) {
		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 <= -270.0:
		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 <= -270.0)
		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, -270.0], 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 < -270

   1. Initial program 19.7%

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

      1. sub-neg 19.7%

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

      2. log1p-def 19.7%

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

      3. distribute-neg-frac 19.7%

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

      4. sub-neg 19.7%

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

      5. distribute-neg-in 19.7%

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

      6. remove-double-neg 19.7%

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

      7. +-commutative 19.7%

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

      8. sub-neg 19.7%

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

   3. Simplified 19.7%

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

   4. Taylor expanded in y around inf 19.7%

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

   5. Taylor expanded in x around 0 72.2%

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

      1. distribute-neg-frac 72.2%

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

      2. metadata-eval 72.2%

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

   7. Simplified 72.2%

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

   if -270 < y

   1. Initial program 93.5%

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

      1. sub-neg 93.5%

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

      2. log1p-def 93.5%

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

      3. distribute-neg-frac 93.5%

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

      4. sub-neg 93.5%

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

      5. distribute-neg-in 93.5%

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

      6. remove-double-neg 93.5%

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

      7. +-commutative 93.5%

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

      8. sub-neg 93.5%

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

   3. Simplified 93.5%

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

   4. Taylor expanded in y around 0 81.9%

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

      1. log1p-def 81.9%

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

      2. mul-1-neg 81.9%

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

   6. Simplified 81.9%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 8: 62.7% 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 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 71.9%

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

   3. distribute-neg-frac 71.9%

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

   4. sub-neg 71.9%

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

   5. distribute-neg-in 71.9%

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

   6. remove-double-neg 71.9%

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

   7. +-commutative 71.9%

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

   8. sub-neg 71.9%

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

3. Simplified 71.9%

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

4. Taylor expanded in y around 0 61.7%

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

   1. log1p-def 61.7%

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

   2. mul-1-neg 61.7%

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

6. Simplified 61.7%

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

7. Final simplification 61.7%

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

Alternative 9: 43.4% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 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 71.9%

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

   3. distribute-neg-frac 71.9%

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

   4. sub-neg 71.9%

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

   5. distribute-neg-in 71.9%

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

   6. remove-double-neg 71.9%

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

   7. +-commutative 71.9%

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

   8. sub-neg 71.9%

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

3. Simplified 71.9%

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

4. Taylor expanded in x around inf 72.6%

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

   1. neg-mul-1 72.6%

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

   2. distribute-neg-frac 72.6%

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

6. Simplified 72.6%

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

7. Taylor expanded in x around 0 42.5%

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

8. Final simplification 42.5%

   \[\leadsto 1 \]

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 2023275 
(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))))))