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

Percentage Accurate: 73.2% → 99.7%

Time: 12.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1550:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5 + \frac{-0.3333333333333333}{y}}{y} + -1}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1550.0)
   (-
    (+ 1.0 (/ (+ (/ (+ -0.5 (/ -0.3333333333333333 y)) y) -1.0) y))
    (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 4.2e+18)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (+ (log (+ x -1.0)) (log (/ 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1550.0) {
		tmp = (1.0 + ((((-0.5 + (-0.3333333333333333 / y)) / y) + -1.0) / y)) - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 4.2e+18) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (log((x + -1.0)) + log((1.0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1550.0) {
		tmp = (1.0 + ((((-0.5 + (-0.3333333333333333 / y)) / y) + -1.0) / y)) - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 4.2e+18) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) + Math.log((1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1550.0:
		tmp = (1.0 + ((((-0.5 + (-0.3333333333333333 / y)) / y) + -1.0) / y)) - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 4.2e+18:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) + math.log((1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1550.0)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(-0.5 + Float64(-0.3333333333333333 / y)) / y) + -1.0) / y)) - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 4.2e+18)
		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(Float64(1.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1550.0], N[(N[(1.0 + N[(N[(N[(N[(-0.5 + N[(-0.3333333333333333 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+18], 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[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1550

   1. Initial program 23.6%

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

      1. sub-neg 23.6%

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

      2. log1p-define 23.6%

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

      3. distribute-neg-frac2 23.6%

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

      4. neg-sub0 23.6%

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

      5. associate--r- 23.6%

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

      6. metadata-eval 23.6%

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

      7. +-commutative 23.6%

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

   3. Simplified 23.6%

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

   5. Taylor expanded in y around -inf 76.7%

      \[\leadsto \color{blue}{\left(1 + -1 \cdot \frac{\left(-1 \cdot \frac{0.16666666666666666 \cdot \frac{-6 \cdot \frac{1 - x}{x - 1} + \left(2 \cdot \frac{{\left(1 - x\right)}^{3}}{{\left(x - 1\right)}^{3}} + 6 \cdot \frac{1 - x}{x - 1}\right)}{y} - 0.5 \cdot \left(2 + -1 \cdot \frac{{\left(1 - x\right)}^{2}}{{\left(x - 1\right)}^{2}}\right)}{y} + \frac{x}{x - 1}\right) - \frac{1}{x - 1}}{y}\right) - \left(\log \left(-1 \cdot \left(x - 1\right)\right) + \log \left(\frac{-1}{y}\right)\right)} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\left(1 - \frac{1 - \frac{-0.5 + \frac{-0.3333333333333333}{y}}{y}}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)} \]

   if -1550 < y < 4.2e18

   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

   if 4.2e18 < y

   1. Initial program 51.6%

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

      1. sub-neg 51.6%

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

      2. log1p-define 51.6%

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

      3. distribute-neg-frac2 51.6%

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

      4. neg-sub0 51.6%

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

      5. associate--r- 51.6%

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

      6. metadata-eval 51.6%

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

      7. +-commutative 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in y around inf 98.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1550:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5 + \frac{-0.3333333333333333}{y}}{y} + -1}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+18}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2400000000:\\ \;\;\;\;\left(\frac{-1}{y} + \left(1 - \mathsf{log1p}\left(-x\right)\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 230000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\left(x - y\right) \cdot \frac{-1}{x \cdot \left(1 - y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2400000000.0)
   (- (+ (/ -1.0 y) (- 1.0 (log1p (- x)))) (log (/ -1.0 y)))
   (if (<= y 230000000000.0)
     (- 1.0 (log1p (* x (* (- x y) (/ -1.0 (* x (- 1.0 y)))))))
     (- 1.0 (+ (log (+ x -1.0)) (log (/ 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2400000000.0) {
		tmp = ((-1.0 / y) + (1.0 - log1p(-x))) - log((-1.0 / y));
	} else if (y <= 230000000000.0) {
		tmp = 1.0 - log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))));
	} else {
		tmp = 1.0 - (log((x + -1.0)) + log((1.0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2400000000.0) {
		tmp = ((-1.0 / y) + (1.0 - Math.log1p(-x))) - Math.log((-1.0 / y));
	} else if (y <= 230000000000.0) {
		tmp = 1.0 - Math.log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) + Math.log((1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2400000000.0:
		tmp = ((-1.0 / y) + (1.0 - math.log1p(-x))) - math.log((-1.0 / y))
	elif y <= 230000000000.0:
		tmp = 1.0 - math.log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) + math.log((1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2400000000.0)
		tmp = Float64(Float64(Float64(-1.0 / y) + Float64(1.0 - log1p(Float64(-x)))) - log(Float64(-1.0 / y)));
	elseif (y <= 230000000000.0)
		tmp = Float64(1.0 - log1p(Float64(x * Float64(Float64(x - y) * Float64(-1.0 / Float64(x * Float64(1.0 - y)))))));
	else
		tmp = Float64(1.0 - Float64(log(Float64(x + -1.0)) + log(Float64(1.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2400000000.0], N[(N[(N[(-1.0 / y), $MachinePrecision] + N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 230000000000.0], N[(1.0 - N[Log[1 + N[(x * N[(N[(x - y), $MachinePrecision] * N[(-1.0 / N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] + N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e9

   1. Initial program 21.5%

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

      1. sub-neg 21.5%

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

      2. log1p-define 21.5%

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

      3. distribute-neg-frac2 21.5%

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

      4. neg-sub0 21.5%

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

      5. associate--r- 21.5%

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

      6. metadata-eval 21.5%

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

      7. +-commutative 21.5%

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

   3. Simplified 21.5%

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

   5. Taylor expanded in y around -inf 99.6%

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

   7. Simplified 99.6%

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

   if -2.4e9 < y < 2.3e11

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate--r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. mul-1-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

      1. frac-sub 99.9%

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

      2. clear-num 99.9%

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

      3. *-un-lft-identity 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. associate-/r/ 99.9%

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

       2. *-commutative 99.9%

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

       3. +-commutative 99.9%

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

       4. sub-neg 99.9%

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

       5. distribute-rgt-neg-out 99.9%

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

       6. cancel-sign-sub 99.9%

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

       7. distribute-rgt-neg-out 99.9%

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

       8. distribute-lft-neg-in 99.9%

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

       9. remove-double-neg 99.9%

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

       10. +-commutative 99.9%

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

       11. +-commutative 99.9%

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

   11. Simplified 99.9%

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

       1. associate-*l/ 99.9%

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

       2. *-un-lft-identity 99.9%

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

       3. clear-num 99.9%

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

       4. +-commutative 99.9%

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

       5. clear-num 99.9%

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

       6. un-div-inv 99.9%

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

       7. +-commutative 99.9%

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

       8. +-commutative 99.9%

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

   13. Applied egg-rr 99.9%

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

       1. associate-/r/ 99.9%

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

       2. metadata-eval 99.9%

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

       3. distribute-neg-frac 99.9%

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

       4. distribute-neg-frac2 99.9%

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

       5. distribute-rgt-neg-in 99.9%

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

       6. distribute-neg-in 99.9%

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

       7. metadata-eval 99.9%

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

       8. +-commutative 99.9%

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

       9. unsub-neg 99.9%

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

       10. associate-/r/ 99.9%

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

       11. *-inverses 99.9%

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

       12. cancel-sign-sub-inv 99.9%

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

       13. metadata-eval 99.9%

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

       14. neg-mul-1 99.9%

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

       15. sub-neg 99.9%

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

   15. Simplified 99.9%

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

   if 2.3e11 < y

   1. Initial program 51.6%

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

      1. sub-neg 51.6%

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

      2. log1p-define 51.6%

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

      3. distribute-neg-frac2 51.6%

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

      4. neg-sub0 51.6%

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

      5. associate--r- 51.6%

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

      6. metadata-eval 51.6%

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

      7. +-commutative 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in y around inf 98.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2400000000:\\ \;\;\;\;\left(\frac{-1}{y} + \left(1 - \mathsf{log1p}\left(-x\right)\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 230000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\left(x - y\right) \cdot \frac{-1}{x \cdot \left(1 - y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2500000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\left(x - y\right) \cdot \frac{-1}{x \cdot \left(1 - y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2500000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 1.2e+14)
     (- 1.0 (log1p (* x (* (- x y) (/ -1.0 (* x (- 1.0 y)))))))
     (- 1.0 (+ (log (+ x -1.0)) (log (/ 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2500000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 1.2e+14) {
		tmp = 1.0 - log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))));
	} else {
		tmp = 1.0 - (log((x + -1.0)) + log((1.0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2500000000.0) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 1.2e+14) {
		tmp = 1.0 - Math.log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) + Math.log((1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2500000000.0:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 1.2e+14:
		tmp = 1.0 - math.log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) + math.log((1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2500000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 1.2e+14)
		tmp = Float64(1.0 - log1p(Float64(x * Float64(Float64(x - y) * Float64(-1.0 / Float64(x * Float64(1.0 - y)))))));
	else
		tmp = Float64(1.0 - Float64(log(Float64(x + -1.0)) + log(Float64(1.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2500000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+14], N[(1.0 - N[Log[1 + N[(x * N[(N[(x - y), $MachinePrecision] * N[(-1.0 / N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] + N[Log[N[(1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e9

   1. Initial program 21.5%

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

      1. sub-neg 21.5%

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

      2. log1p-define 21.5%

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

      3. distribute-neg-frac2 21.5%

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

      4. neg-sub0 21.5%

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

      5. associate--r- 21.5%

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

      6. metadata-eval 21.5%

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

      7. +-commutative 21.5%

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

   3. Simplified 21.5%

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

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

   if -2.5e9 < y < 1.2e14

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate--r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. mul-1-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

      1. frac-sub 99.9%

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

      2. clear-num 99.9%

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

      3. *-un-lft-identity 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. associate-/r/ 99.9%

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

       2. *-commutative 99.9%

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

       3. +-commutative 99.9%

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

       4. sub-neg 99.9%

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

       5. distribute-rgt-neg-out 99.9%

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

       6. cancel-sign-sub 99.9%

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

       7. distribute-rgt-neg-out 99.9%

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

       8. distribute-lft-neg-in 99.9%

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

       9. remove-double-neg 99.9%

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

       10. +-commutative 99.9%

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

       11. +-commutative 99.9%

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

   11. Simplified 99.9%

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

       1. associate-*l/ 99.9%

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

       2. *-un-lft-identity 99.9%

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

       3. clear-num 99.9%

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

       4. +-commutative 99.9%

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

       5. clear-num 99.9%

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

       6. un-div-inv 99.9%

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

       7. +-commutative 99.9%

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

       8. +-commutative 99.9%

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

   13. Applied egg-rr 99.9%

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

       1. associate-/r/ 99.9%

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

       2. metadata-eval 99.9%

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

       3. distribute-neg-frac 99.9%

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

       4. distribute-neg-frac2 99.9%

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

       5. distribute-rgt-neg-in 99.9%

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

       6. distribute-neg-in 99.9%

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

       7. metadata-eval 99.9%

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

       8. +-commutative 99.9%

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

       9. unsub-neg 99.9%

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

       10. associate-/r/ 99.9%

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

       11. *-inverses 99.9%

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

       12. cancel-sign-sub-inv 99.9%

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

       13. metadata-eval 99.9%

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

       14. neg-mul-1 99.9%

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

       15. sub-neg 99.9%

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

   15. Simplified 99.9%

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

   if 1.2e14 < y

   1. Initial program 51.6%

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

      1. sub-neg 51.6%

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

      2. log1p-define 51.6%

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

      3. distribute-neg-frac2 51.6%

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

      4. neg-sub0 51.6%

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

      5. associate--r- 51.6%

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

      6. metadata-eval 51.6%

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

      7. +-commutative 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in y around inf 98.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2500000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\left(x - y\right) \cdot \frac{-1}{x \cdot \left(1 - y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) + \log \left(\frac{1}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2500000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 6500000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\left(x - y\right) \cdot \frac{-1}{x \cdot \left(1 - y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log y\right) - \log \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2500000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 6500000000000.0)
     (- 1.0 (log1p (* x (* (- x y) (/ -1.0 (* x (- 1.0 y)))))))
     (- (+ 1.0 (log y)) (log (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2500000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 6500000000000.0) {
		tmp = 1.0 - log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))));
	} else {
		tmp = (1.0 + log(y)) - log((x + -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2500000000.0) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 6500000000000.0) {
		tmp = 1.0 - Math.log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))));
	} else {
		tmp = (1.0 + Math.log(y)) - Math.log((x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2500000000.0:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 6500000000000.0:
		tmp = 1.0 - math.log1p((x * ((x - y) * (-1.0 / (x * (1.0 - y))))))
	else:
		tmp = (1.0 + math.log(y)) - math.log((x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2500000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 6500000000000.0)
		tmp = Float64(1.0 - log1p(Float64(x * Float64(Float64(x - y) * Float64(-1.0 / Float64(x * Float64(1.0 - y)))))));
	else
		tmp = Float64(Float64(1.0 + log(y)) - log(Float64(x + -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2500000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6500000000000.0], N[(1.0 - N[Log[1 + N[(x * N[(N[(x - y), $MachinePrecision] * N[(-1.0 / N[(x * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Log[y], $MachinePrecision]), $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.5e9

   1. Initial program 21.5%

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

      1. sub-neg 21.5%

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

      2. log1p-define 21.5%

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

      3. distribute-neg-frac2 21.5%

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

      4. neg-sub0 21.5%

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

      5. associate--r- 21.5%

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

      6. metadata-eval 21.5%

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

      7. +-commutative 21.5%

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

   3. Simplified 21.5%

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

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

   if -2.5e9 < y < 6.5e12

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 99.9%

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

      3. distribute-neg-frac2 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate--r- 99.9%

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

      6. metadata-eval 99.9%

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

      7. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 99.9%

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

      1. +-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. mul-1-neg 99.9%

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

      5. unsub-neg 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

      8. *-commutative 99.9%

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

      9. associate-/r* 99.9%

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

   7. Simplified 99.9%

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

      1. frac-sub 99.9%

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

      2. clear-num 99.9%

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

      3. *-un-lft-identity 99.9%

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

   9. Applied egg-rr 99.9%

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

       1. associate-/r/ 99.9%

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

       2. *-commutative 99.9%

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

       3. +-commutative 99.9%

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

       4. sub-neg 99.9%

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

       5. distribute-rgt-neg-out 99.9%

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

       6. cancel-sign-sub 99.9%

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

       7. distribute-rgt-neg-out 99.9%

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

       8. distribute-lft-neg-in 99.9%

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

       9. remove-double-neg 99.9%

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

       10. +-commutative 99.9%

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

       11. +-commutative 99.9%

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

   11. Simplified 99.9%

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

       1. associate-*l/ 99.9%

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

       2. *-un-lft-identity 99.9%

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

       3. clear-num 99.9%

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

       4. +-commutative 99.9%

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

       5. clear-num 99.9%

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

       6. un-div-inv 99.9%

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

       7. +-commutative 99.9%

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

       8. +-commutative 99.9%

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

   13. Applied egg-rr 99.9%

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

       1. associate-/r/ 99.9%

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

       2. metadata-eval 99.9%

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

       3. distribute-neg-frac 99.9%

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

       4. distribute-neg-frac2 99.9%

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

       5. distribute-rgt-neg-in 99.9%

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

       6. distribute-neg-in 99.9%

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

       7. metadata-eval 99.9%

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

       8. +-commutative 99.9%

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

       9. unsub-neg 99.9%

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

       10. associate-/r/ 99.9%

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

       11. *-inverses 99.9%

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

       12. cancel-sign-sub-inv 99.9%

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

       13. metadata-eval 99.9%

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

       14. neg-mul-1 99.9%

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

       15. sub-neg 99.9%

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

   15. Simplified 99.9%

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

   if 6.5e12 < y

   1. Initial program 51.6%

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

      1. sub-neg 51.6%

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

      2. log1p-define 51.6%

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

      3. distribute-neg-frac2 51.6%

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

      4. neg-sub0 51.6%

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

      5. associate--r- 51.6%

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

      6. metadata-eval 51.6%

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

      7. +-commutative 51.6%

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

   3. Simplified 51.6%

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

   5. Taylor expanded in y around inf 98.9%

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

      1. +-commutative 98.9%

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

      2. associate--r+ 98.9%

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

      3. sub-neg 98.9%

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

      4. log-rec 98.9%

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

      5. remove-double-neg 98.9%

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

      6. sub-neg 98.9%

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

      7. metadata-eval 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2500000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 6500000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\left(x - y\right) \cdot \frac{-1}{x \cdot \left(1 - y\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log y\right) - \log \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{y + -1}\\ \mathbf{if}\;1 + t\_0 \leq 10^{-13}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (+ y -1.0))))
   (if (<= (+ 1.0 t_0) 1e-13) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p t_0)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (y + -1.0);
	double tmp;
	if ((1.0 + t_0) <= 1e-13) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (y + -1.0);
	double tmp;
	if ((1.0 + t_0) <= 1e-13) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (y + -1.0)
	tmp = 0
	if (1.0 + t_0) <= 1e-13:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(y + -1.0))
	tmp = 0.0
	if (Float64(1.0 + t_0) <= 1e-13)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 + t$95$0), $MachinePrecision], 1e-13], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 1e-13

   1. Initial program 3.8%

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

      1. sub-neg 3.8%

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

      2. log1p-define 3.8%

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

      3. distribute-neg-frac2 3.8%

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

      4. neg-sub0 3.8%

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

      5. associate--r- 3.8%

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

      6. metadata-eval 3.8%

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

      7. +-commutative 3.8%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in y around -inf 84.2%

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

      1. sub-neg 84.2%

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

      2. metadata-eval 84.2%

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

      3. distribute-lft-in 84.2%

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

      4. metadata-eval 84.2%

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

      5. +-commutative 84.2%

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

      6. log1p-define 84.2%

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

      7. mul-1-neg 84.2%

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

   7. Simplified 84.2%

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

   8. Taylor expanded in x around 0 75.3%

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

   if 1e-13 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.3%

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

      1. sub-neg 99.3%

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

      2. log1p-define 99.3%

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

      3. distribute-neg-frac2 99.3%

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

      4. neg-sub0 99.3%

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

      5. associate--r- 99.3%

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

      6. metadata-eval 99.3%

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

      7. +-commutative 99.3%

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

   3. Simplified 99.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 10^{-13}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+32}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.6e+32)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.6e+32) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.6e+32) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.6e+32:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.6e+32)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.6e+32], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5999999999999999e32

   1. Initial program 17.4%

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

      1. sub-neg 17.4%

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

      2. log1p-define 17.4%

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

      3. distribute-neg-frac2 17.4%

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

      4. neg-sub0 17.4%

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

      5. associate--r- 17.4%

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

      6. metadata-eval 17.4%

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

      7. +-commutative 17.4%

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

   3. Simplified 17.4%

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

   5. Taylor expanded in y around -inf 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. log1p-define 99.6%

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

      7. mul-1-neg 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 75.7%

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

   if -1.5999999999999999e32 < y

   1. Initial program 94.4%

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

      1. sub-neg 94.4%

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

      2. log1p-define 94.4%

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

      3. distribute-neg-frac2 94.4%

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

      4. neg-sub0 94.4%

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

      5. associate--r- 94.4%

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

      6. metadata-eval 94.4%

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

      7. +-commutative 94.4%

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

   3. Simplified 94.4%

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

   5. Taylor expanded in x around inf 92.4%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+32}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 - y\right) - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.2) (- 1.0 (log (/ -1.0 y))) (- (- 1.0 y) (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = (1.0 - y) - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.2) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = (1.0 - y) - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.2:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = (1.0 - y) - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.2)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(Float64(1.0 - y) - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.2], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - y), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.1999999999999993

   1. Initial program 24.7%

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

      1. sub-neg 24.7%

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

      2. log1p-define 24.7%

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

      3. distribute-neg-frac2 24.7%

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

      4. neg-sub0 24.7%

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

      5. associate--r- 24.7%

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

      6. metadata-eval 24.7%

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

      7. +-commutative 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in y around -inf 97.3%

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

      1. sub-neg 97.3%

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

      2. metadata-eval 97.3%

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

      3. distribute-lft-in 97.3%

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

      4. metadata-eval 97.3%

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

      5. +-commutative 97.3%

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

      6. log1p-define 97.3%

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

      7. mul-1-neg 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in x around 0 70.5%

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

   if -9.1999999999999993 < y

   1. Initial program 94.5%

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

      1. sub-neg 94.5%

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

      2. log1p-define 94.5%

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

      3. distribute-neg-frac2 94.5%

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

      4. neg-sub0 94.5%

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

      5. associate--r- 94.5%

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

      6. metadata-eval 94.5%

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

      7. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around 0 86.8%

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

   7. Simplified 86.8%

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

Alternative 8: 79.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -5.8) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -5.8) {
		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 <= -5.8) {
		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 <= -5.8:
		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 <= -5.8)
		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, -5.8], 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 < -5.79999999999999982

   1. Initial program 24.7%

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

      1. sub-neg 24.7%

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

      2. log1p-define 24.7%

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

      3. distribute-neg-frac2 24.7%

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

      4. neg-sub0 24.7%

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

      5. associate--r- 24.7%

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

      6. metadata-eval 24.7%

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

      7. +-commutative 24.7%

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

   3. Simplified 24.7%

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

   5. Taylor expanded in y around -inf 97.3%

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

      1. sub-neg 97.3%

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

      2. metadata-eval 97.3%

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

      3. distribute-lft-in 97.3%

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

      4. metadata-eval 97.3%

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

      5. +-commutative 97.3%

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

      6. log1p-define 97.3%

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

      7. mul-1-neg 97.3%

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

   7. Simplified 97.3%

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

   8. Taylor expanded in x around 0 70.5%

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

   if -5.79999999999999982 < y

   1. Initial program 94.5%

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

      1. sub-neg 94.5%

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

      2. log1p-define 94.5%

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

      3. distribute-neg-frac2 94.5%

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

      4. neg-sub0 94.5%

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

      5. associate--r- 94.5%

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

      6. metadata-eval 94.5%

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

      7. +-commutative 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around 0 86.4%

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

      1. log1p-define 86.4%

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

      2. mul-1-neg 86.4%

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

   7. Simplified 86.4%

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

Alternative 9: 63.4% 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 75.4%

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

   1. sub-neg 75.4%

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

   2. log1p-define 75.4%

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

   3. distribute-neg-frac2 75.4%

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

   4. neg-sub0 75.4%

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

   5. associate--r- 75.4%

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

   6. metadata-eval 75.4%

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

   7. +-commutative 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in y around 0 66.5%

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

   1. log1p-define 66.5%

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

   2. mul-1-neg 66.5%

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

7. Simplified 66.5%

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

Alternative 10: 43.7% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.4%

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

   1. sub-neg 75.4%

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

   2. log1p-define 75.4%

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

   3. distribute-neg-frac2 75.4%

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

   4. neg-sub0 75.4%

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

   5. associate--r- 75.4%

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

   6. metadata-eval 75.4%

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

   7. +-commutative 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in y around 0 66.5%

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

   1. log1p-define 66.5%

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

   2. mul-1-neg 66.5%

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

7. Simplified 66.5%

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

8. Taylor expanded in x around 0 45.9%

   \[\leadsto \color{blue}{1 + x} \]
9. Add Preprocessing

Alternative 11: 43.4% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 75.4%

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

   1. sub-neg 75.4%

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

   2. log1p-define 75.4%

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

   3. distribute-neg-frac2 75.4%

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

   4. neg-sub0 75.4%

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

   5. associate--r- 75.4%

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

   6. metadata-eval 75.4%

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

   7. +-commutative 75.4%

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

3. Simplified 75.4%

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

5. Taylor expanded in y around 0 66.5%

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

   1. log1p-define 66.5%

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

   2. mul-1-neg 66.5%

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

7. Simplified 66.5%

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

8. Taylor expanded in x around 0 45.4%

   \[\leadsto \color{blue}{1} \]
9. 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 2024096 
(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))))))