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

Percentage Accurate: 72.7% → 99.7%

Time: 16.1s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -445000:\\ \;\;\;\;1 + \left(\frac{\frac{1 - x}{y}}{x + -1} - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -445000.0)
   (+
    1.0
    (- (/ (/ (- 1.0 x) y) (+ x -1.0)) (+ (log1p (- x)) (log (/ -1.0 y)))))
   (if (<= y 1e+15)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (- (log (+ x -1.0)) (log y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -445000.0) {
		tmp = 1.0 + ((((1.0 - x) / y) / (x + -1.0)) - (log1p(-x) + log((-1.0 / y))));
	} else if (y <= 1e+15) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (log((x + -1.0)) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -445000.0) {
		tmp = 1.0 + ((((1.0 - x) / y) / (x + -1.0)) - (Math.log1p(-x) + Math.log((-1.0 / y))));
	} else if (y <= 1e+15) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -445000.0:
		tmp = 1.0 + ((((1.0 - x) / y) / (x + -1.0)) - (math.log1p(-x) + math.log((-1.0 / y))))
	elif y <= 1e+15:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) - math.log(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -445000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(1.0 - x) / y) / Float64(x + -1.0)) - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y)))));
	elseif (y <= 1e+15)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - Float64(log(Float64(x + -1.0)) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -445000.0], N[(1.0 + N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+15], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -445000

   1. Initial program 24.1%

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

      1. sub-neg 24.1%

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

      2. log1p-define 24.1%

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

      3. distribute-neg-frac2 24.1%

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

      4. neg-sub0 24.1%

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

      5. associate--r- 24.1%

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

      6. metadata-eval 24.1%

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

      7. +-commutative 24.1%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in y around -inf 99.3%

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

      1. associate-+r+ 99.3%

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

      2. div-sub 99.3%

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

      3. associate-/l/ 99.3%

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

      4. sub-neg 99.3%

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

      5. mul-1-neg 99.3%

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

      6. +-commutative 99.3%

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

   7. Simplified 99.3%

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

   if -445000 < y < 1e15

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 1e15 < y

   1. Initial program 46.3%

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

      1. sub-neg 46.3%

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

      2. log1p-define 46.3%

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

      3. distribute-neg-frac2 46.3%

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

      4. neg-sub0 46.3%

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

      5. associate--r- 46.3%

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

      6. metadata-eval 46.3%

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

      7. +-commutative 46.3%

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

   3. Simplified 46.3%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. log-rec 99.1%

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

      2. unsub-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -445000:\\ \;\;\;\;1 + \left(\frac{\frac{1 - x}{y}}{x + -1} - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1000000000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1000000000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 7.5e+14)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (- (log (+ x -1.0)) (log y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1000000000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 7.5e+14) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (log((x + -1.0)) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1000000000000.0) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 7.5e+14) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1000000000000.0:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 7.5e+14:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) - math.log(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1000000000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 7.5e+14)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - Float64(log(Float64(x + -1.0)) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1000000000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.5e+14], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e12

   1. Initial program 20.7%

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

      1. sub-neg 20.7%

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

      2. log1p-define 20.7%

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

      3. distribute-neg-frac2 20.7%

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

      4. neg-sub0 20.7%

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

      5. associate--r- 20.7%

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

      6. metadata-eval 20.7%

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

      7. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in y around -inf 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. distribute-lft-in 99.3%

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

      4. metadata-eval 99.3%

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

      5. +-commutative 99.3%

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

      6. log1p-define 99.3%

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

      7. mul-1-neg 99.3%

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

   7. Simplified 99.3%

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

   if -1e12 < y < 7.5e14

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. log1p-define 99.6%

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

      3. distribute-neg-frac2 99.6%

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

      4. neg-sub0 99.6%

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

      5. associate--r- 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   if 7.5e14 < y

   1. Initial program 46.3%

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

      1. sub-neg 46.3%

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

      2. log1p-define 46.3%

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

      3. distribute-neg-frac2 46.3%

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

      4. neg-sub0 46.3%

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

      5. associate--r- 46.3%

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

      6. metadata-eval 46.3%

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

      7. +-commutative 46.3%

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

   3. Simplified 46.3%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. log-rec 99.1%

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

      2. unsub-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1000000000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1000000000000:\\ \;\;\;\;1 + \left(\log \left(-y\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1000000000000.0)
   (+ 1.0 (- (log (- y)) (log1p (- x))))
   (if (<= y 2.7e+15)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (- (log (+ x -1.0)) (log y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1000000000000.0) {
		tmp = 1.0 + (log(-y) - log1p(-x));
	} else if (y <= 2.7e+15) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (log((x + -1.0)) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1000000000000.0) {
		tmp = 1.0 + (Math.log(-y) - Math.log1p(-x));
	} else if (y <= 2.7e+15) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1000000000000.0:
		tmp = 1.0 + (math.log(-y) - math.log1p(-x))
	elif y <= 2.7e+15:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) - math.log(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1000000000000.0)
		tmp = Float64(1.0 + Float64(log(Float64(-y)) - log1p(Float64(-x))));
	elseif (y <= 2.7e+15)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - Float64(log(Float64(x + -1.0)) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1000000000000.0], N[(1.0 + N[(N[Log[(-y)], $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+15], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e12

   1. Initial program 20.7%

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

      1. sub-neg 20.7%

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

      2. log1p-define 20.7%

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

      3. distribute-neg-frac2 20.7%

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

      4. neg-sub0 20.7%

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

      5. associate--r- 20.7%

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

      6. metadata-eval 20.7%

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

      7. +-commutative 20.7%

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

   3. Simplified 20.7%

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

   5. Taylor expanded in y around -inf 99.3%

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

      1. sub-neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. distribute-lft-in 99.3%

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

      4. metadata-eval 99.3%

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

      5. +-commutative 99.3%

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

      6. log1p-define 99.3%

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

      7. mul-1-neg 99.3%

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

   7. Simplified 99.3%

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

      1. frac-2neg 99.3%

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

      2. metadata-eval 99.3%

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

      3. log-rec 99.3%

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

   9. Applied egg-rr 99.3%

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

   if -1e12 < y < 2.7e15

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. log1p-define 99.6%

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

      3. distribute-neg-frac2 99.6%

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

      4. neg-sub0 99.6%

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

      5. associate--r- 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   if 2.7e15 < y

   1. Initial program 46.3%

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

      1. sub-neg 46.3%

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

      2. log1p-define 46.3%

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

      3. distribute-neg-frac2 46.3%

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

      4. neg-sub0 46.3%

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

      5. associate--r- 46.3%

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

      6. metadata-eval 46.3%

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

      7. +-commutative 46.3%

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

   3. Simplified 46.3%

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

   5. Taylor expanded in y around inf 99.1%

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

      1. log-rec 99.1%

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

      2. unsub-neg 99.1%

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

      3. sub-neg 99.1%

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

      4. metadata-eval 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1000000000000:\\ \;\;\;\;1 + \left(\log \left(-y\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{+281}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+185}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -34000000000000:\\ \;\;\;\;\left(1 + x\right) - t\_0\\ \mathbf{elif}\;y \leq 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log (/ -1.0 y))) (t_1 (- 1.0 (log (/ (+ 1.0 x) y)))))
   (if (<= y -5.7e+281)
     t_1
     (if (<= y -2.7e+185)
       (- 1.0 t_0)
       (if (<= y -5.5e+113)
         t_1
         (if (<= y -34000000000000.0)
           (- (+ 1.0 x) t_0)
           (if (<= y 1e+15) (- 1.0 (log1p (/ (- x y) (+ y -1.0)))) t_1)))))))

C

double code(double x, double y) {
	double t_0 = log((-1.0 / y));
	double t_1 = 1.0 - log(((1.0 + x) / y));
	double tmp;
	if (y <= -5.7e+281) {
		tmp = t_1;
	} else if (y <= -2.7e+185) {
		tmp = 1.0 - t_0;
	} else if (y <= -5.5e+113) {
		tmp = t_1;
	} else if (y <= -34000000000000.0) {
		tmp = (1.0 + x) - t_0;
	} else if (y <= 1e+15) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.log((-1.0 / y));
	double t_1 = 1.0 - Math.log(((1.0 + x) / y));
	double tmp;
	if (y <= -5.7e+281) {
		tmp = t_1;
	} else if (y <= -2.7e+185) {
		tmp = 1.0 - t_0;
	} else if (y <= -5.5e+113) {
		tmp = t_1;
	} else if (y <= -34000000000000.0) {
		tmp = (1.0 + x) - t_0;
	} else if (y <= 1e+15) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((-1.0 / y))
	t_1 = 1.0 - math.log(((1.0 + x) / y))
	tmp = 0
	if y <= -5.7e+281:
		tmp = t_1
	elif y <= -2.7e+185:
		tmp = 1.0 - t_0
	elif y <= -5.5e+113:
		tmp = t_1
	elif y <= -34000000000000.0:
		tmp = (1.0 + x) - t_0
	elif y <= 1e+15:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = log(Float64(-1.0 / y))
	t_1 = Float64(1.0 - log(Float64(Float64(1.0 + x) / y)))
	tmp = 0.0
	if (y <= -5.7e+281)
		tmp = t_1;
	elseif (y <= -2.7e+185)
		tmp = Float64(1.0 - t_0);
	elseif (y <= -5.5e+113)
		tmp = t_1;
	elseif (y <= -34000000000000.0)
		tmp = Float64(Float64(1.0 + x) - t_0);
	elseif (y <= 1e+15)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.7e+281], t$95$1, If[LessEqual[y, -2.7e+185], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[y, -5.5e+113], t$95$1, If[LessEqual[y, -34000000000000.0], N[(N[(1.0 + x), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[y, 1e+15], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.69999999999999986e281 or -2.70000000000000007e185 < y < -5.5000000000000001e113 or 1e15 < y

   1. Initial program 38.8%

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

      1. sub-neg 38.8%

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

      2. log1p-define 38.8%

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

      3. distribute-neg-frac2 38.8%

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

      4. neg-sub0 38.8%

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

      5. associate--r- 38.8%

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

      6. metadata-eval 38.8%

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

      7. +-commutative 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in y around -inf 43.1%

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

      1. sub-neg 43.1%

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

      2. metadata-eval 43.1%

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

      3. distribute-lft-in 43.1%

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

      4. metadata-eval 43.1%

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

      5. +-commutative 43.1%

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

      6. log1p-define 43.1%

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

      7. mul-1-neg 43.1%

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

   7. Simplified 43.1%

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

      1. log1p-undefine 43.1%

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

      2. sub-neg 43.1%

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

      3. sum-log 100.0%

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

      4. add-sqr-sqrt 43.6%

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

      5. sqrt-unprod 22.4%

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

      6. frac-times 21.5%

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

      7. metadata-eval 21.5%

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

      8. metadata-eval 21.5%

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

      9. frac-times 22.4%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. div-inv 0.0%

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

      13. sub-neg 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 22.3%

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

      16. sqr-neg 22.3%

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

      17. sqrt-unprod 54.2%

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

      18. add-sqr-sqrt 86.9%

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

   9. Applied egg-rr 86.9%

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

   if -5.69999999999999986e281 < y < -2.70000000000000007e185

   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-define 3.1%

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

      3. distribute-neg-frac2 3.1%

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

      4. neg-sub0 3.1%

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

      5. associate--r- 3.1%

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

      6. metadata-eval 3.1%

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

      7. +-commutative 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 3.1%

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

      1. sub-neg 3.1%

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

      2. metadata-eval 3.1%

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

      3. neg-mul-1 3.1%

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

      4. distribute-neg-frac2 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in y around inf 0.0%

      \[\leadsto 1 - \color{blue}{\left(\log -1 + \log \left(\frac{1}{y}\right)\right)} \]
   9. 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 70.5%

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

   10. Simplified 70.5%

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

   if -5.5000000000000001e113 < y < -3.4e13

   1. Initial program 22.1%

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

      1. sub-neg 22.1%

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

      2. log1p-define 22.1%

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

      3. distribute-neg-frac2 22.1%

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

      4. neg-sub0 22.1%

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

      5. associate--r- 22.1%

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

      6. metadata-eval 22.1%

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

      7. +-commutative 22.1%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in y around -inf 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

      6. log1p-define 99.4%

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

      7. mul-1-neg 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0 77.8%

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

   if -3.4e13 < y < 1e15

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. log1p-define 99.6%

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

      3. distribute-neg-frac2 99.6%

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

      4. neg-sub0 99.6%

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

      5. associate--r- 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   3. Simplified 99.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.7 \cdot 10^{+281}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+185}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+113}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{elif}\;y \leq -34000000000000:\\ \;\;\;\;\left(1 + x\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{1 + x}{y}\right)\\ t_1 := 1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{if}\;y \leq -9.2 \cdot 10^{+280}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -22.5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ (+ 1.0 x) y)))) (t_1 (- 1.0 (log (/ -1.0 y)))))
   (if (<= y -9.2e+280)
     t_0
     (if (<= y -2.6e+185)
       t_1
       (if (<= y -5.5e+113)
         t_0
         (if (<= y -22.5)
           t_1
           (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((1.0 + x) / y));
	double t_1 = 1.0 - log((-1.0 / y));
	double tmp;
	if (y <= -9.2e+280) {
		tmp = t_0;
	} else if (y <= -2.6e+185) {
		tmp = t_1;
	} else if (y <= -5.5e+113) {
		tmp = t_0;
	} else if (y <= -22.5) {
		tmp = t_1;
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((1.0 + x) / y));
	double t_1 = 1.0 - Math.log((-1.0 / y));
	double tmp;
	if (y <= -9.2e+280) {
		tmp = t_0;
	} else if (y <= -2.6e+185) {
		tmp = t_1;
	} else if (y <= -5.5e+113) {
		tmp = t_0;
	} else if (y <= -22.5) {
		tmp = t_1;
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((1.0 + x) / y))
	t_1 = 1.0 - math.log((-1.0 / y))
	tmp = 0
	if y <= -9.2e+280:
		tmp = t_0
	elif y <= -2.6e+185:
		tmp = t_1
	elif y <= -5.5e+113:
		tmp = t_0
	elif y <= -22.5:
		tmp = t_1
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(1.0 + x) / y)))
	t_1 = Float64(1.0 - log(Float64(-1.0 / y)))
	tmp = 0.0
	if (y <= -9.2e+280)
		tmp = t_0;
	elseif (y <= -2.6e+185)
		tmp = t_1;
	elseif (y <= -5.5e+113)
		tmp = t_0;
	elseif (y <= -22.5)
		tmp = t_1;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.2e+280], t$95$0, If[LessEqual[y, -2.6e+185], t$95$1, If[LessEqual[y, -5.5e+113], t$95$0, If[LessEqual[y, -22.5], t$95$1, If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.19999999999999998e280 or -2.60000000000000001e185 < y < -5.5000000000000001e113 or 1 < y

   1. Initial program 40.9%

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

      1. sub-neg 40.9%

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

      2. log1p-define 40.9%

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

      3. distribute-neg-frac2 40.9%

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

      4. neg-sub0 40.9%

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

      5. associate--r- 40.9%

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

      6. metadata-eval 40.9%

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

      7. +-commutative 40.9%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in y around -inf 41.6%

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

      1. sub-neg 41.6%

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

      2. metadata-eval 41.6%

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

      3. distribute-lft-in 41.6%

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

      4. metadata-eval 41.6%

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

      5. +-commutative 41.6%

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

      6. log1p-define 41.6%

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

      7. mul-1-neg 41.6%

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

   7. Simplified 41.6%

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

      1. log1p-undefine 41.6%

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

      2. sub-neg 41.6%

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

      3. sum-log 98.8%

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

      4. add-sqr-sqrt 42.1%

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

      5. sqrt-unprod 21.6%

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

      6. frac-times 20.7%

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

      7. metadata-eval 20.7%

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

      8. metadata-eval 20.7%

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

      9. frac-times 21.6%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. div-inv 0.0%

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

      13. sub-neg 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 21.6%

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

      16. sqr-neg 21.6%

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

      17. sqrt-unprod 54.6%

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

      18. add-sqr-sqrt 86.1%

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

   9. Applied egg-rr 86.1%

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

   if -9.19999999999999998e280 < y < -2.60000000000000001e185 or -5.5000000000000001e113 < y < -22.5

   1. Initial program 23.3%

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

      1. sub-neg 23.3%

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

      2. log1p-define 23.3%

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

      3. distribute-neg-frac2 23.3%

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

      4. neg-sub0 23.3%

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

      5. associate--r- 23.3%

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

      6. metadata-eval 23.3%

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

      7. +-commutative 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in x around 0 6.8%

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

      1. sub-neg 6.8%

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

      2. metadata-eval 6.8%

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

      3. neg-mul-1 6.8%

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

      4. distribute-neg-frac2 6.8%

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

   7. Simplified 6.8%

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

   8. Taylor expanded in y around inf 0.0%

      \[\leadsto 1 - \color{blue}{\left(\log -1 + \log \left(\frac{1}{y}\right)\right)} \]
   9. 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 70.0%

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

   10. Simplified 70.0%

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

   if -22.5 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. div-sub 98.6%

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

      3. fma-define 98.6%

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

      4. mul-1-neg 98.6%

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

      5. sub-neg 98.6%

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

      6. *-inverses 98.6%

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

      7. +-commutative 98.6%

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

      8. metadata-eval 98.6%

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

      9. distribute-lft-in 98.6%

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

      10. metadata-eval 98.6%

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

      11. sub-neg 98.6%

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

      12. fma-undefine 98.6%

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

      13. *-rgt-identity 98.6%

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

      14. sub-neg 98.6%

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

      15. metadata-eval 98.6%

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

      16. distribute-lft-in 98.6%

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

      17. metadata-eval 98.6%

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

      18. +-commutative 98.6%

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

   7. Simplified 98.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+280}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+185}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+113}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{elif}\;y \leq -22.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{1 + x}{y}\right)\\ t_1 := 1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+281}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+113}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -2000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ (+ 1.0 x) y)))) (t_1 (- 1.0 (log (/ -1.0 y)))))
   (if (<= y -4.8e+281)
     t_0
     (if (<= y -8.6e+185)
       t_1
       (if (<= y -5.5e+113)
         t_0
         (if (<= y -2000000000000.0)
           t_1
           (if (<= y 7.5e+14) (- 1.0 (log1p (/ x (+ y -1.0)))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((1.0 + x) / y));
	double t_1 = 1.0 - log((-1.0 / y));
	double tmp;
	if (y <= -4.8e+281) {
		tmp = t_0;
	} else if (y <= -8.6e+185) {
		tmp = t_1;
	} else if (y <= -5.5e+113) {
		tmp = t_0;
	} else if (y <= -2000000000000.0) {
		tmp = t_1;
	} else if (y <= 7.5e+14) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((1.0 + x) / y));
	double t_1 = 1.0 - Math.log((-1.0 / y));
	double tmp;
	if (y <= -4.8e+281) {
		tmp = t_0;
	} else if (y <= -8.6e+185) {
		tmp = t_1;
	} else if (y <= -5.5e+113) {
		tmp = t_0;
	} else if (y <= -2000000000000.0) {
		tmp = t_1;
	} else if (y <= 7.5e+14) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((1.0 + x) / y))
	t_1 = 1.0 - math.log((-1.0 / y))
	tmp = 0
	if y <= -4.8e+281:
		tmp = t_0
	elif y <= -8.6e+185:
		tmp = t_1
	elif y <= -5.5e+113:
		tmp = t_0
	elif y <= -2000000000000.0:
		tmp = t_1
	elif y <= 7.5e+14:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(1.0 + x) / y)))
	t_1 = Float64(1.0 - log(Float64(-1.0 / y)))
	tmp = 0.0
	if (y <= -4.8e+281)
		tmp = t_0;
	elseif (y <= -8.6e+185)
		tmp = t_1;
	elseif (y <= -5.5e+113)
		tmp = t_0;
	elseif (y <= -2000000000000.0)
		tmp = t_1;
	elseif (y <= 7.5e+14)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+281], t$95$0, If[LessEqual[y, -8.6e+185], t$95$1, If[LessEqual[y, -5.5e+113], t$95$0, If[LessEqual[y, -2000000000000.0], t$95$1, If[LessEqual[y, 7.5e+14], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8000000000000002e281 or -8.6000000000000002e185 < y < -5.5000000000000001e113 or 7.5e14 < y

   1. Initial program 38.8%

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

      1. sub-neg 38.8%

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

      2. log1p-define 38.8%

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

      3. distribute-neg-frac2 38.8%

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

      4. neg-sub0 38.8%

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

      5. associate--r- 38.8%

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

      6. metadata-eval 38.8%

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

      7. +-commutative 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in y around -inf 43.1%

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

      1. sub-neg 43.1%

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

      2. metadata-eval 43.1%

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

      3. distribute-lft-in 43.1%

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

      4. metadata-eval 43.1%

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

      5. +-commutative 43.1%

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

      6. log1p-define 43.1%

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

      7. mul-1-neg 43.1%

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

   7. Simplified 43.1%

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

      1. log1p-undefine 43.1%

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

      2. sub-neg 43.1%

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

      3. sum-log 100.0%

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

      4. add-sqr-sqrt 43.6%

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

      5. sqrt-unprod 22.4%

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

      6. frac-times 21.5%

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

      7. metadata-eval 21.5%

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

      8. metadata-eval 21.5%

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

      9. frac-times 22.4%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. div-inv 0.0%

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

      13. sub-neg 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 22.3%

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

      16. sqr-neg 22.3%

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

      17. sqrt-unprod 54.2%

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

      18. add-sqr-sqrt 86.9%

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

   9. Applied egg-rr 86.9%

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

   if -4.8000000000000002e281 < y < -8.6000000000000002e185 or -5.5000000000000001e113 < y < -2e12

   1. Initial program 16.5%

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

      1. sub-neg 16.5%

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

      2. log1p-define 16.5%

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

      3. distribute-neg-frac2 16.5%

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

      4. neg-sub0 16.5%

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

      5. associate--r- 16.5%

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

      6. metadata-eval 16.5%

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

      7. +-commutative 16.5%

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

   3. Simplified 16.5%

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

   5. Taylor expanded in x around 0 4.4%

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

      1. sub-neg 4.4%

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

      2. metadata-eval 4.4%

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

      3. neg-mul-1 4.4%

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

      4. distribute-neg-frac2 4.4%

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

   7. Simplified 4.4%

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

   8. Taylor expanded in y around inf 0.0%

      \[\leadsto 1 - \color{blue}{\left(\log -1 + \log \left(\frac{1}{y}\right)\right)} \]
   9. 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 74.6%

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

   10. Simplified 74.6%

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

   if -2e12 < y < 7.5e14

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. log1p-define 99.6%

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

      3. distribute-neg-frac2 99.6%

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

      4. neg-sub0 99.6%

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

      5. associate--r- 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 97.3%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+281}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{elif}\;y \leq -8.6 \cdot 10^{+185}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+113}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{elif}\;y \leq -2000000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{+281}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+185}:\\ \;\;\;\;1 - t\_0\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3200000000000:\\ \;\;\;\;\left(1 + x\right) - t\_0\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log (/ -1.0 y))) (t_1 (- 1.0 (log (/ (+ 1.0 x) y)))))
   (if (<= y -7e+281)
     t_1
     (if (<= y -2.7e+185)
       (- 1.0 t_0)
       (if (<= y -5.5e+113)
         t_1
         (if (<= y -3200000000000.0)
           (- (+ 1.0 x) t_0)
           (if (<= y 7.8e+14) (- 1.0 (log1p (/ x (+ y -1.0)))) t_1)))))))

C

double code(double x, double y) {
	double t_0 = log((-1.0 / y));
	double t_1 = 1.0 - log(((1.0 + x) / y));
	double tmp;
	if (y <= -7e+281) {
		tmp = t_1;
	} else if (y <= -2.7e+185) {
		tmp = 1.0 - t_0;
	} else if (y <= -5.5e+113) {
		tmp = t_1;
	} else if (y <= -3200000000000.0) {
		tmp = (1.0 + x) - t_0;
	} else if (y <= 7.8e+14) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.log((-1.0 / y));
	double t_1 = 1.0 - Math.log(((1.0 + x) / y));
	double tmp;
	if (y <= -7e+281) {
		tmp = t_1;
	} else if (y <= -2.7e+185) {
		tmp = 1.0 - t_0;
	} else if (y <= -5.5e+113) {
		tmp = t_1;
	} else if (y <= -3200000000000.0) {
		tmp = (1.0 + x) - t_0;
	} else if (y <= 7.8e+14) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((-1.0 / y))
	t_1 = 1.0 - math.log(((1.0 + x) / y))
	tmp = 0
	if y <= -7e+281:
		tmp = t_1
	elif y <= -2.7e+185:
		tmp = 1.0 - t_0
	elif y <= -5.5e+113:
		tmp = t_1
	elif y <= -3200000000000.0:
		tmp = (1.0 + x) - t_0
	elif y <= 7.8e+14:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = log(Float64(-1.0 / y))
	t_1 = Float64(1.0 - log(Float64(Float64(1.0 + x) / y)))
	tmp = 0.0
	if (y <= -7e+281)
		tmp = t_1;
	elseif (y <= -2.7e+185)
		tmp = Float64(1.0 - t_0);
	elseif (y <= -5.5e+113)
		tmp = t_1;
	elseif (y <= -3200000000000.0)
		tmp = Float64(Float64(1.0 + x) - t_0);
	elseif (y <= 7.8e+14)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7e+281], t$95$1, If[LessEqual[y, -2.7e+185], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[y, -5.5e+113], t$95$1, If[LessEqual[y, -3200000000000.0], N[(N[(1.0 + x), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[y, 7.8e+14], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.9999999999999996e281 or -2.70000000000000007e185 < y < -5.5000000000000001e113 or 7.8e14 < y

   1. Initial program 38.8%

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

      1. sub-neg 38.8%

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

      2. log1p-define 38.8%

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

      3. distribute-neg-frac2 38.8%

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

      4. neg-sub0 38.8%

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

      5. associate--r- 38.8%

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

      6. metadata-eval 38.8%

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

      7. +-commutative 38.8%

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

   3. Simplified 38.8%

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

   5. Taylor expanded in y around -inf 43.1%

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

      1. sub-neg 43.1%

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

      2. metadata-eval 43.1%

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

      3. distribute-lft-in 43.1%

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

      4. metadata-eval 43.1%

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

      5. +-commutative 43.1%

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

      6. log1p-define 43.1%

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

      7. mul-1-neg 43.1%

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

   7. Simplified 43.1%

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

      1. log1p-undefine 43.1%

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

      2. sub-neg 43.1%

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

      3. sum-log 100.0%

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

      4. add-sqr-sqrt 43.6%

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

      5. sqrt-unprod 22.4%

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

      6. frac-times 21.5%

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

      7. metadata-eval 21.5%

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

      8. metadata-eval 21.5%

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

      9. frac-times 22.4%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 0.0%

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

      12. div-inv 0.0%

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

      13. sub-neg 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 22.3%

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

      16. sqr-neg 22.3%

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

      17. sqrt-unprod 54.2%

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

      18. add-sqr-sqrt 86.9%

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

   9. Applied egg-rr 86.9%

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

   if -6.9999999999999996e281 < y < -2.70000000000000007e185

   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-define 3.1%

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

      3. distribute-neg-frac2 3.1%

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

      4. neg-sub0 3.1%

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

      5. associate--r- 3.1%

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

      6. metadata-eval 3.1%

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

      7. +-commutative 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in x around 0 3.1%

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

      1. sub-neg 3.1%

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

      2. metadata-eval 3.1%

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

      3. neg-mul-1 3.1%

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

      4. distribute-neg-frac2 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in y around inf 0.0%

      \[\leadsto 1 - \color{blue}{\left(\log -1 + \log \left(\frac{1}{y}\right)\right)} \]
   9. 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 70.5%

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

   10. Simplified 70.5%

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

   if -5.5000000000000001e113 < y < -3.2e12

   1. Initial program 22.1%

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

      1. sub-neg 22.1%

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

      2. log1p-define 22.1%

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

      3. distribute-neg-frac2 22.1%

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

      4. neg-sub0 22.1%

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

      5. associate--r- 22.1%

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

      6. metadata-eval 22.1%

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

      7. +-commutative 22.1%

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

   3. Simplified 22.1%

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

   5. Taylor expanded in y around -inf 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

      6. log1p-define 99.4%

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

      7. mul-1-neg 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0 77.8%

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

   if -3.2e12 < y < 7.8e14

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. log1p-define 99.6%

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

      3. distribute-neg-frac2 99.6%

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

      4. neg-sub0 99.6%

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

      5. associate--r- 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 97.3%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{\frac{x}{y - 1}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+281}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+185}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{+113}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \mathbf{elif}\;y \leq -3200000000000:\\ \;\;\;\;\left(1 + x\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1 + x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999999995)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (+ 1.0 (- (/ -1.0 y) (log (/ -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999999995) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - log((-1.0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999999995) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - Math.log((-1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999999995:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + ((-1.0 / y) - math.log((-1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999999995)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - 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], 0.999999999995], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / y), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999999999995

   1. Initial program 98.9%

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

      1. sub-neg 98.9%

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

      2. log1p-define 98.9%

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

      3. distribute-neg-frac2 98.9%

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

      4. neg-sub0 98.9%

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

      5. associate--r- 98.9%

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

      6. metadata-eval 98.9%

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

      7. +-commutative 98.9%

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

   3. Simplified 98.9%

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

   if 0.999999999995 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 4.5%

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

      1. sub-neg 4.5%

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

      2. log1p-define 4.5%

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

      3. distribute-neg-frac2 4.5%

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

      4. neg-sub0 4.5%

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

      5. associate--r- 4.5%

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

      6. metadata-eval 4.5%

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

      7. +-commutative 4.5%

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

   3. Simplified 4.5%

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

   5. Taylor expanded in x around 0 4.1%

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

      1. sub-neg 4.1%

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

      2. metadata-eval 4.1%

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

      3. neg-mul-1 4.1%

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

      4. distribute-neg-frac2 4.1%

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

   7. Simplified 4.1%

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

   8. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. associate-+r+ 0.0%

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

      3. sub-neg 0.0%

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

      4. log-div 57.2%

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

      5. +-commutative 57.2%

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

   10. Simplified 57.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -66:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -66.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -66.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -66.0) {
		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.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -66.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -66.0)
		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 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -66.0], 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[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -66

   1. Initial program 25.1%

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

      1. sub-neg 25.1%

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

      2. log1p-define 25.1%

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

      3. distribute-neg-frac2 25.1%

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

      4. neg-sub0 25.1%

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

      5. associate--r- 25.1%

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

      6. metadata-eval 25.1%

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

      7. +-commutative 25.1%

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

   3. Simplified 25.1%

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

   5. Taylor expanded in x around 0 5.5%

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

      1. sub-neg 5.5%

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

      2. metadata-eval 5.5%

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

      3. neg-mul-1 5.5%

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

      4. distribute-neg-frac2 5.5%

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

   7. Simplified 5.5%

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

   8. Taylor expanded in y around inf 0.0%

      \[\leadsto 1 - \color{blue}{\left(\log -1 + \log \left(\frac{1}{y}\right)\right)} \]
   9. 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 59.1%

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

   10. Simplified 59.1%

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

   if -66 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 98.6%

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

      1. +-commutative 98.6%

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

      2. div-sub 98.6%

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

      3. fma-define 98.6%

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

      4. mul-1-neg 98.6%

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

      5. sub-neg 98.6%

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

      6. *-inverses 98.6%

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

      7. +-commutative 98.6%

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

      8. metadata-eval 98.6%

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

      9. distribute-lft-in 98.6%

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

      10. metadata-eval 98.6%

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

      11. sub-neg 98.6%

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

      12. fma-undefine 98.6%

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

      13. *-rgt-identity 98.6%

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

      14. sub-neg 98.6%

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

      15. metadata-eval 98.6%

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

      16. distribute-lft-in 98.6%

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

      17. metadata-eval 98.6%

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

      18. +-commutative 98.6%

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

   7. Simplified 98.7%

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

   if 1 < y

   1. Initial program 49.6%

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

      1. sub-neg 49.6%

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

      2. log1p-define 49.6%

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

      3. distribute-neg-frac2 49.6%

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

      4. neg-sub0 49.6%

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

      5. associate--r- 49.6%

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

      6. metadata-eval 49.6%

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

      7. +-commutative 49.6%

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

   3. Simplified 49.6%

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

   5. Taylor expanded in y around inf 47.5%

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

   6. Taylor expanded in x around inf 51.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -66:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -220000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-19}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -220000.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 6.7e-19) (- 1.0 (log1p (- x))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -220000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 6.7e-19) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -220000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 6.7e-19) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -220000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 6.7e-19:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -220000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 6.7e-19)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -220000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.7e-19], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e5

   1. Initial program 24.1%

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

      1. sub-neg 24.1%

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

      2. log1p-define 24.1%

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

      3. distribute-neg-frac2 24.1%

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

      4. neg-sub0 24.1%

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

      5. associate--r- 24.1%

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

      6. metadata-eval 24.1%

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

      7. +-commutative 24.1%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in x around 0 5.5%

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

      1. sub-neg 5.5%

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

      2. metadata-eval 5.5%

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

      3. neg-mul-1 5.5%

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

      4. distribute-neg-frac2 5.5%

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

   7. Simplified 5.5%

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

   8. Taylor expanded in y around inf 0.0%

      \[\leadsto 1 - \color{blue}{\left(\log -1 + \log \left(\frac{1}{y}\right)\right)} \]
   9. 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 59.8%

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

   10. Simplified 59.8%

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

   if -2.2e5 < y < 6.69999999999999998e-19

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 97.4%

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

      1. log1p-define 97.4%

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

      2. mul-1-neg 97.4%

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

   7. Simplified 97.4%

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

   if 6.69999999999999998e-19 < y

   1. Initial program 53.7%

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

      1. sub-neg 53.7%

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

      2. log1p-define 53.8%

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

      3. distribute-neg-frac2 53.8%

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

      4. neg-sub0 53.8%

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

      5. associate--r- 53.8%

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

      6. metadata-eval 53.8%

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

      7. +-commutative 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in y around inf 43.5%

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

   6. Taylor expanded in x around inf 53.7%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -220000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-19}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -220000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -220000.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -220000.0) {
		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 <= -220000.0) {
		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 <= -220000.0:
		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 <= -220000.0)
		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, -220000.0], 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 < -2.2e5

   1. Initial program 24.1%

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

      1. sub-neg 24.1%

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

      2. log1p-define 24.1%

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

      3. distribute-neg-frac2 24.1%

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

      4. neg-sub0 24.1%

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

      5. associate--r- 24.1%

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

      6. metadata-eval 24.1%

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

      7. +-commutative 24.1%

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

   3. Simplified 24.1%

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

   5. Taylor expanded in x around 0 5.5%

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

      1. sub-neg 5.5%

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

      2. metadata-eval 5.5%

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

      3. neg-mul-1 5.5%

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

      4. distribute-neg-frac2 5.5%

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

   7. Simplified 5.5%

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

   8. Taylor expanded in y around inf 0.0%

      \[\leadsto 1 - \color{blue}{\left(\log -1 + \log \left(\frac{1}{y}\right)\right)} \]
   9. 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 59.8%

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

   10. Simplified 59.8%

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

   if -2.2e5 < y

   1. Initial program 90.8%

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

      1. sub-neg 90.8%

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

      2. log1p-define 90.8%

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

      3. distribute-neg-frac2 90.8%

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

      4. neg-sub0 90.8%

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

      5. associate--r- 90.8%

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

      6. metadata-eval 90.8%

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

      7. +-commutative 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in y around 0 79.4%

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

      1. log1p-define 79.4%

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

      2. mul-1-neg 79.4%

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

   7. Simplified 79.4%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -220000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.1% 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 71.2%

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

   1. sub-neg 71.2%

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

   2. log1p-define 71.3%

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

   3. distribute-neg-frac2 71.3%

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

   4. neg-sub0 71.3%

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

   5. associate--r- 71.3%

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

   6. metadata-eval 71.3%

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

   7. +-commutative 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in y around 0 59.7%

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

   1. log1p-define 59.7%

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

   2. mul-1-neg 59.7%

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

7. Simplified 59.7%

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

8. Final simplification 59.7%

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

Alternative 13: 41.9% 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 71.2%

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

   1. sub-neg 71.2%

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

   2. log1p-define 71.3%

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

   3. distribute-neg-frac2 71.3%

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

   4. neg-sub0 71.3%

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

   5. associate--r- 71.3%

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

   6. metadata-eval 71.3%

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

   7. +-commutative 71.3%

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

3. Simplified 71.3%

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

5. Taylor expanded in x around 0 39.8%

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

   1. sub-neg 39.8%

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

   2. metadata-eval 39.8%

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

   3. neg-mul-1 39.8%

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

   4. distribute-neg-frac2 39.8%

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

7. Simplified 39.8%

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

8. Taylor expanded in y around 0 39.5%

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

9. Final simplification 39.5%

   \[\leadsto 1 - y \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{if}\;y < -81284752.61947241:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))
   (if (< y -81284752.61947241)
     t_0
     (if (< y 3.0094271212461764e+25)
       (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y)))))
       t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - log(((x / (y * y)) - ((1.0d0 / y) - (x / y))))
    if (y < (-81284752.61947241d0)) then
        tmp = t_0
    else if (y < 3.0094271212461764d+25) then
        tmp = log((exp(1.0d0) / (1.0d0 - ((x - y) / (1.0d0 - y)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	double tmp;
	if (y < -81284752.61947241) {
		tmp = t_0;
	} else if (y < 3.0094271212461764e+25) {
		tmp = Math.log((Math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log(((x / (y * y)) - ((1.0 / y) - (x / y))))
	tmp = 0
	if y < -81284752.61947241:
		tmp = t_0
	elif y < 3.0094271212461764e+25:
		tmp = math.log((math.exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(Float64(x / Float64(y * y)) - Float64(Float64(1.0 / y) - Float64(x / y)))))
	tmp = 0.0
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log(Float64(exp(1.0) / Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 - log(((x / (y * y)) - ((1.0 / y) - (x / y))));
	tmp = 0.0;
	if (y < -81284752.61947241)
		tmp = t_0;
	elseif (y < 3.0094271212461764e+25)
		tmp = log((exp(1.0) / (1.0 - ((x - y) / (1.0 - y)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / y), $MachinePrecision] - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -81284752.61947241], t$95$0, If[Less[y, 3.0094271212461764e+25], N[Log[N[(N[Exp[1.0], $MachinePrecision] / N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024053 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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