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

Percentage Accurate: 72.8% → 99.7%

Time: 11.6s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.8% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1750:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5 + \frac{-0.3333333333333333}{y}}{y} + -1}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+23}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1750.0)
   (-
    (+ 1.0 (/ (+ (/ (+ -0.5 (/ -0.3333333333333333 y)) y) -1.0) y))
    (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 2.75e+23)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (- (log (+ x -1.0)) (log y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1750.0) {
		tmp = (1.0 + ((((-0.5 + (-0.3333333333333333 / y)) / y) + -1.0) / y)) - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 2.75e+23) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (log((x + -1.0)) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1750.0) {
		tmp = (1.0 + ((((-0.5 + (-0.3333333333333333 / y)) / y) + -1.0) / y)) - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 2.75e+23) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1750.0:
		tmp = (1.0 + ((((-0.5 + (-0.3333333333333333 / y)) / y) + -1.0) / y)) - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 2.75e+23:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) - math.log(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1750.0)
		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 <= 2.75e+23)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - Float64(log(Float64(x + -1.0)) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1750.0], 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, 2.75e+23], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1750

   1. Initial program 16.9%

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

      1. sub-neg 16.9%

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

      2. log1p-define 16.9%

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

      3. distribute-neg-frac2 16.9%

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

      4. neg-sub0 16.9%

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

      5. associate--r- 16.9%

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

      6. metadata-eval 16.9%

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

      7. +-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in y around -inf 87.5%

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

   7. Simplified 99.7%

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

   if -1750 < y < 2.75000000000000002e23

   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 2.75000000000000002e23 < y

   1. Initial program 65.1%

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

      1. sub-neg 65.1%

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

      2. log1p-define 65.1%

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

      3. distribute-neg-frac2 65.1%

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

      4. neg-sub0 65.1%

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

      5. associate--r- 65.1%

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

      6. metadata-eval 65.1%

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

      7. +-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in y around inf 98.2%

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

      1. log-rec 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1750:\\ \;\;\;\;\left(1 + \frac{\frac{-0.5 + \frac{-0.3333333333333333}{y}}{y} + -1}{y}\right) - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 2.75 \cdot 10^{+23}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2300000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 12500000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2300000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 12500000000000.0)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (- (log (+ x -1.0)) (log y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2300000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 12500000000000.0) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (log((x + -1.0)) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2300000000.0) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 12500000000000.0) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2300000000.0:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 12500000000000.0:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) - math.log(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2300000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 12500000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - Float64(log(Float64(x + -1.0)) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2300000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 12500000000000.0], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.3e9

   1. Initial program 14.5%

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

      1. sub-neg 14.5%

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

      2. log1p-define 14.5%

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

      3. distribute-neg-frac2 14.5%

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

      4. neg-sub0 14.5%

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

      5. associate--r- 14.5%

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

      6. metadata-eval 14.5%

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

      7. +-commutative 14.5%

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

   3. Simplified 14.5%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 -2.3e9 < y < 1.25e13

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. log1p-define 99.6%

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

      3. distribute-neg-frac2 99.6%

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

      4. neg-sub0 99.6%

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

      5. associate--r- 99.6%

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

      6. metadata-eval 99.6%

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

      7. +-commutative 99.6%

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

   3. Simplified 99.6%

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

   if 1.25e13 < y

   1. Initial program 65.1%

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

      1. sub-neg 65.1%

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

      2. log1p-define 65.1%

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

      3. distribute-neg-frac2 65.1%

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

      4. neg-sub0 65.1%

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

      5. associate--r- 65.1%

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

      6. metadata-eval 65.1%

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

      7. +-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in y around inf 98.2%

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

      1. log-rec 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2300000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 12500000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -460000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 48000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -460000.0)
   (+ 1.0 (- (- (/ -1.0 y) (log (/ -1.0 y))) (log1p (- x))))
   (if (<= y 48000000000000.0)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (- (log (+ x -1.0)) (log y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -460000.0) {
		tmp = 1.0 + (((-1.0 / y) - log((-1.0 / y))) - log1p(-x));
	} else if (y <= 48000000000000.0) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (log((x + -1.0)) - log(y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -460000.0) {
		tmp = 1.0 + (((-1.0 / y) - Math.log((-1.0 / y))) - Math.log1p(-x));
	} else if (y <= 48000000000000.0) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - (Math.log((x + -1.0)) - Math.log(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -460000.0:
		tmp = 1.0 + (((-1.0 / y) - math.log((-1.0 / y))) - math.log1p(-x))
	elif y <= 48000000000000.0:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - (math.log((x + -1.0)) - math.log(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -460000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 / y) - log(Float64(-1.0 / y))) - log1p(Float64(-x))));
	elseif (y <= 48000000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - Float64(log(Float64(x + -1.0)) - log(y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -460000.0], N[(1.0 + N[(N[(N[(-1.0 / y), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 48000000000000.0], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision] - N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6e5

   1. Initial program 16.0%

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

      1. sub-neg 16.0%

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

      2. log1p-define 16.0%

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

      3. distribute-neg-frac2 16.0%

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

      4. neg-sub0 16.0%

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

      5. associate--r- 16.0%

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

      6. metadata-eval 16.0%

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

      7. +-commutative 16.0%

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

   3. Simplified 16.0%

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

   5. Taylor expanded in y around -inf 99.8%

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

   7. Simplified 99.8%

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

   if -4.6e5 < y < 4.8e13

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 4.8e13 < y

   1. Initial program 65.1%

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

      1. sub-neg 65.1%

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

      2. log1p-define 65.1%

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

      3. distribute-neg-frac2 65.1%

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

      4. neg-sub0 65.1%

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

      5. associate--r- 65.1%

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

      6. metadata-eval 65.1%

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

      7. +-commutative 65.1%

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

   3. Simplified 65.1%

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

   5. Taylor expanded in y around inf 98.2%

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

      1. log-rec 98.2%

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

      2. unsub-neg 98.2%

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

      3. sub-neg 98.2%

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

      4. metadata-eval 98.2%

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

   7. Simplified 98.2%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -460000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 48000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(x + -1\right) - \log y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{y + -1}\\ \mathbf{if}\;1 + t\_0 \leq 1.66 \cdot 10^{-8}:\\ \;\;\;\;\left(1 - \log \left(\frac{-1}{y}\right)\right) + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (+ y -1.0))))
   (if (<= (+ 1.0 t_0) 1.66e-8)
     (+ (- 1.0 (log (/ -1.0 y))) (/ -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) <= 1.66e-8) {
		tmp = (1.0 - log((-1.0 / y))) + (-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) <= 1.66e-8) {
		tmp = (1.0 - Math.log((-1.0 / y))) + (-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) <= 1.66e-8:
		tmp = (1.0 - math.log((-1.0 / y))) + (-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) <= 1.66e-8)
		tmp = Float64(Float64(1.0 - log(Float64(-1.0 / y))) + 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], 1.66e-8], N[(N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / y), $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))) < 1.65999999999999998e-8

   1. Initial program 6.6%

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

      1. sub-neg 6.6%

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

      2. log1p-define 6.6%

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

      3. distribute-neg-frac2 6.6%

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

      4. neg-sub0 6.6%

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

      5. associate--r- 6.6%

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

      6. metadata-eval 6.6%

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

      7. +-commutative 6.6%

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

   3. Simplified 6.6%

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

   5. Taylor expanded in x around 0 6.6%

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

      1. sub-neg 6.6%

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

      2. mul-1-neg 6.6%

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

      3. sub-neg 6.6%

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

      4. metadata-eval 6.6%

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

      5. neg-mul-1 6.6%

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

      6. log1p-define 6.6%

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

      7. distribute-neg-frac2 6.6%

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

      8. +-commutative 6.6%

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

      9. distribute-neg-in 6.6%

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

      10. metadata-eval 6.6%

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

      11. unsub-neg 6.6%

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

   7. Simplified 6.6%

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

   8. Taylor expanded in y around -inf 79.4%

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

      1. associate--r+ 79.5%

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

   10. Simplified 79.5%

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

   if 1.65999999999999998e-8 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. log1p-define 99.8%

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

      3. distribute-neg-frac2 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate--r- 99.8%

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

      6. metadata-eval 99.8%

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

      7. +-commutative 99.8%

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

   3. Simplified 99.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 1.66 \cdot 10^{-8}:\\ \;\;\;\;\left(1 - \log \left(\frac{-1}{y}\right)\right) + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8400000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8400000000.0)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8400000000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8400000000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8400000000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8400000000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8400000000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.4e9

   1. Initial program 14.5%

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

      1. sub-neg 14.5%

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

      2. log1p-define 14.5%

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

      3. distribute-neg-frac2 14.5%

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

      4. neg-sub0 14.5%

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

      5. associate--r- 14.5%

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

      6. metadata-eval 14.5%

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

      7. +-commutative 14.5%

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

   3. Simplified 14.5%

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

   5. Taylor expanded in x around 0 4.8%

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

      1. sub-neg 4.8%

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

      2. mul-1-neg 4.8%

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

      3. sub-neg 4.8%

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

      4. metadata-eval 4.8%

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

      5. neg-mul-1 4.8%

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

      6. log1p-define 4.8%

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

      7. distribute-neg-frac2 4.8%

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

      8. +-commutative 4.8%

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

      9. distribute-neg-in 4.8%

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

      10. metadata-eval 4.8%

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

      11. unsub-neg 4.8%

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

   7. Simplified 4.8%

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

   8. Taylor expanded in y around -inf 79.9%

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

   if -8.4e9 < y

   1. Initial program 94.7%

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

      1. sub-neg 94.7%

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

      2. log1p-define 94.7%

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

      3. distribute-neg-frac2 94.7%

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

      4. neg-sub0 94.7%

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

      5. associate--r- 94.7%

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

      6. metadata-eval 94.7%

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

      7. +-commutative 94.7%

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

   3. Simplified 94.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8400000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -800:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -800.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -800.0) {
		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 <= -800.0) {
		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 <= -800.0:
		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 <= -800.0)
		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, -800.0], 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 < -800

   1. Initial program 16.9%

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

      1. sub-neg 16.9%

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

      2. log1p-define 16.9%

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

      3. distribute-neg-frac2 16.9%

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

      4. neg-sub0 16.9%

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

      5. associate--r- 16.9%

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

      6. metadata-eval 16.9%

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

      7. +-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in x around 0 7.5%

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

      1. sub-neg 7.5%

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

      2. mul-1-neg 7.5%

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

      3. sub-neg 7.5%

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

      4. metadata-eval 7.5%

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

      5. neg-mul-1 7.5%

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

      6. log1p-define 7.5%

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

      7. distribute-neg-frac2 7.5%

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

      8. +-commutative 7.5%

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

      9. distribute-neg-in 7.5%

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

      10. metadata-eval 7.5%

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

      11. unsub-neg 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in y around -inf 79.1%

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

   if -800 < y

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      2. log1p-define 95.0%

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

      3. distribute-neg-frac2 95.0%

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

      4. neg-sub0 95.0%

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

      5. associate--r- 95.0%

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

      6. metadata-eval 95.0%

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

      7. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in x around inf 93.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -800:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -31:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -31.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -31.0) {
		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 <= -31.0) {
		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 <= -31.0:
		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 <= -31.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -31.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -31

   1. Initial program 16.9%

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

      1. sub-neg 16.9%

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

      2. log1p-define 16.9%

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

      3. distribute-neg-frac2 16.9%

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

      4. neg-sub0 16.9%

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

      5. associate--r- 16.9%

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

      6. metadata-eval 16.9%

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

      7. +-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in x around 0 7.5%

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

      1. sub-neg 7.5%

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

      2. mul-1-neg 7.5%

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

      3. sub-neg 7.5%

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

      4. metadata-eval 7.5%

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

      5. neg-mul-1 7.5%

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

      6. log1p-define 7.5%

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

      7. distribute-neg-frac2 7.5%

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

      8. +-commutative 7.5%

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

      9. distribute-neg-in 7.5%

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

      10. metadata-eval 7.5%

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

      11. unsub-neg 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in y around -inf 79.1%

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

   if -31 < y

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      2. log1p-define 95.0%

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

      3. distribute-neg-frac2 95.0%

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

      4. neg-sub0 95.0%

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

      5. associate--r- 95.0%

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

      6. metadata-eval 95.0%

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

      7. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in y around 0 84.0%

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

      1. +-commutative 84.0%

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

      2. div-sub 84.0%

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

      3. mul-1-neg 84.0%

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

      4. sub-neg 84.0%

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

      5. *-inverses 84.0%

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

      6. *-rgt-identity 84.0%

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

      7. log1p-define 84.0%

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

      8. mul-1-neg 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -31:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -27:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -27.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -27.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -27.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -27.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -27.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -27.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -27

   1. Initial program 16.9%

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

      1. sub-neg 16.9%

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

      2. log1p-define 16.9%

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

      3. distribute-neg-frac2 16.9%

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

      4. neg-sub0 16.9%

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

      5. associate--r- 16.9%

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

      6. metadata-eval 16.9%

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

      7. +-commutative 16.9%

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

   3. Simplified 16.9%

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

   5. Taylor expanded in x around 0 7.5%

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

      1. sub-neg 7.5%

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

      2. mul-1-neg 7.5%

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

      3. sub-neg 7.5%

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

      4. metadata-eval 7.5%

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

      5. neg-mul-1 7.5%

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

      6. log1p-define 7.5%

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

      7. distribute-neg-frac2 7.5%

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

      8. +-commutative 7.5%

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

      9. distribute-neg-in 7.5%

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

      10. metadata-eval 7.5%

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

      11. unsub-neg 7.5%

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

   7. Simplified 7.5%

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

   8. Taylor expanded in y around -inf 79.1%

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

   if -27 < y

   1. Initial program 95.0%

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

      1. sub-neg 95.0%

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

      2. log1p-define 95.0%

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

      3. distribute-neg-frac2 95.0%

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

      4. neg-sub0 95.0%

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

      5. associate--r- 95.0%

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

      6. metadata-eval 95.0%

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

      7. +-commutative 95.0%

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

   3. Simplified 95.0%

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

   5. Taylor expanded in y around 0 82.8%

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

      1. log1p-define 82.8%

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

      2. mul-1-neg 82.8%

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

   7. Simplified 82.8%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -27:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{log1p}\left(-x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log1p (- x))))

C

double code(double x, double y) {
	return 1.0 - log1p(-x);
}

Java

public static double code(double x, double y) {
	return 1.0 - Math.log1p(-x);
}

Python

def code(x, y):
	return 1.0 - math.log1p(-x)

Julia

function code(x, y)
	return Float64(1.0 - log1p(Float64(-x)))
end

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. sub-neg 70.3%

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

   2. log1p-define 70.3%

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

   3. distribute-neg-frac2 70.3%

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

   4. neg-sub0 70.3%

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

   5. associate--r- 70.3%

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

   6. metadata-eval 70.3%

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

   7. +-commutative 70.3%

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

3. Simplified 70.3%

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

5. Taylor expanded in y around 0 60.7%

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

   1. log1p-define 60.7%

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

   2. mul-1-neg 60.7%

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

7. Simplified 60.7%

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

8. Final simplification 60.7%

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

Alternative 10: 43.9% 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 70.3%

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

   1. sub-neg 70.3%

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

   2. log1p-define 70.3%

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

   3. distribute-neg-frac2 70.3%

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

   4. neg-sub0 70.3%

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

   5. associate--r- 70.3%

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

   6. metadata-eval 70.3%

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

   7. +-commutative 70.3%

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

3. Simplified 70.3%

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

5. Taylor expanded in y around 0 60.7%

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

   1. log1p-define 60.7%

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

   2. mul-1-neg 60.7%

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

7. Simplified 60.7%

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

8. Taylor expanded in x around 0 42.8%

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

9. Final simplification 42.8%

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

Alternative 11: 43.6% 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 70.3%

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

   1. sub-neg 70.3%

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

   2. log1p-define 70.3%

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

   3. distribute-neg-frac2 70.3%

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

   4. neg-sub0 70.3%

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

   5. associate--r- 70.3%

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

   6. metadata-eval 70.3%

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

   7. +-commutative 70.3%

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

3. Simplified 70.3%

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

5. Taylor expanded in y around 0 60.7%

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

   1. log1p-define 60.7%

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

   2. mul-1-neg 60.7%

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

7. Simplified 60.7%

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

8. Taylor expanded in x around 0 42.5%

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

9. Final simplification 42.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024077 
(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))))))