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

Percentage Accurate: 72.4% → 99.6%

Time: 14.6s

Alternatives: 15

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: 72.4% 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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -550000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -550000.0)
   (+ 1.0 (- (- (/ -1.0 y) (log (/ -1.0 y))) (log1p (- x))))
   (if (<= y 8.5e+37)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -550000.0) {
		tmp = 1.0 + (((-1.0 / y) - log((-1.0 / y))) - log1p(-x));
	} else if (y <= 8.5e+37) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -550000.0) {
		tmp = 1.0 + (((-1.0 / y) - Math.log((-1.0 / y))) - Math.log1p(-x));
	} else if (y <= 8.5e+37) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -550000.0:
		tmp = 1.0 + (((-1.0 / y) - math.log((-1.0 / y))) - math.log1p(-x))
	elif y <= 8.5e+37:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -550000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 / y) - log(Float64(-1.0 / y))) - log1p(Float64(-x))));
	elseif (y <= 8.5e+37)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -550000.0], 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, 8.5e+37], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.5e5

   1. Initial program 21.4%

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

      1. sub-neg 21.4%

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

      2. log1p-def 21.4%

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

      3. distribute-neg-frac 21.4%

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

      4. sub-neg 21.4%

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

      5. distribute-neg-in 21.4%

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

      6. remove-double-neg 21.4%

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

      7. +-commutative 21.4%

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

      8. sub-neg 21.4%

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

   3. Simplified 21.4%

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

   5. Taylor expanded in y around -inf 99.4%

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

      1. sub-neg 99.4%

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

      2. metadata-eval 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. +-commutative 99.4%

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

      6. log1p-def 99.4%

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

      7. mul-1-neg 99.4%

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

      8. mul-1-neg 99.4%

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

      9. unsub-neg 99.4%

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

      10. div-sub 99.4%

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

      11. associate-/l/ 99.4%

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

      12. sub-neg 99.4%

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

      13. metadata-eval 99.4%

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

      14. +-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0 99.4%

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

   if -5.5e5 < y < 8.4999999999999999e37

   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-def 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. sub-neg 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   if 8.4999999999999999e37 < y

   1. Initial program 23.3%

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

      1. sub-neg 23.3%

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

      2. log1p-def 23.3%

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

      3. distribute-neg-frac 23.3%

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

      4. sub-neg 23.3%

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

      5. distribute-neg-in 23.3%

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

      6. remove-double-neg 23.3%

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

      7. +-commutative 23.3%

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

      8. sub-neg 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in y around inf 99.0%

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

      1. log-rec 99.0%

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

      2. unsub-neg 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. +-commutative 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -550000:\\ \;\;\;\;1 + \left(\left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right) - \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -118000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y} - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -118000000000.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 8.5e+37)
     (- 1.0 (log1p (- (/ y (- 1.0 y)) (/ x (- 1.0 y)))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -118000000000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 8.5e+37) {
		tmp = 1.0 - log1p(((y / (1.0 - y)) - (x / (1.0 - y))));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -118000000000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 8.5e+37) {
		tmp = 1.0 - Math.log1p(((y / (1.0 - y)) - (x / (1.0 - y))));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -118000000000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 8.5e+37:
		tmp = 1.0 - math.log1p(((y / (1.0 - y)) - (x / (1.0 - y))))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -118000000000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 8.5e+37)
		tmp = Float64(1.0 - log1p(Float64(Float64(y / Float64(1.0 - y)) - Float64(x / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -118000000000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+37], N[(1.0 - N[Log[1 + N[(N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] - N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.18e11

   1. Initial program 20.8%

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

      1. sub-neg 20.8%

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

      2. log1p-def 20.8%

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

      3. distribute-neg-frac 20.8%

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

      4. sub-neg 20.8%

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

      5. distribute-neg-in 20.8%

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

      6. remove-double-neg 20.8%

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

      7. +-commutative 20.8%

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

      8. sub-neg 20.8%

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

   3. Simplified 20.8%

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

   5. Taylor expanded in x around 0 3.1%

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

      1. log1p-def 3.1%

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

   7. Simplified 3.1%

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

   8. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 69.2%

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

   10. Simplified 69.2%

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

   if -1.18e11 < y < 8.4999999999999999e37

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-def 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. sub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. div-sub 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 8.4999999999999999e37 < y

   1. Initial program 23.3%

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

      1. sub-neg 23.3%

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

      2. log1p-def 23.3%

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

      3. distribute-neg-frac 23.3%

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

      4. sub-neg 23.3%

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

      5. distribute-neg-in 23.3%

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

      6. remove-double-neg 23.3%

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

      7. +-commutative 23.3%

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

      8. sub-neg 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in y around inf 99.0%

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

      1. log-rec 99.0%

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

      2. unsub-neg 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. +-commutative 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -118000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y} - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4500000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y} - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4500000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 8.5e+37)
     (- 1.0 (log1p (- (/ y (- 1.0 y)) (/ x (- 1.0 y)))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4500000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 8.5e+37) {
		tmp = 1.0 - log1p(((y / (1.0 - y)) - (x / (1.0 - y))));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4500000000.0) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 8.5e+37) {
		tmp = 1.0 - Math.log1p(((y / (1.0 - y)) - (x / (1.0 - y))));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4500000000.0:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 8.5e+37:
		tmp = 1.0 - math.log1p(((y / (1.0 - y)) - (x / (1.0 - y))))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4500000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 8.5e+37)
		tmp = Float64(1.0 - log1p(Float64(Float64(y / Float64(1.0 - y)) - Float64(x / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4500000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+37], N[(1.0 - N[Log[1 + N[(N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] - N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.5e9

   1. Initial program 20.8%

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

      1. sub-neg 20.8%

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

      2. log1p-def 20.8%

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

      3. distribute-neg-frac 20.8%

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

      4. sub-neg 20.8%

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

      5. distribute-neg-in 20.8%

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

      6. remove-double-neg 20.8%

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

      7. +-commutative 20.8%

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

      8. sub-neg 20.8%

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

   3. Simplified 20.8%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \]
   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-def 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 -4.5e9 < y < 8.4999999999999999e37

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-def 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. sub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. div-sub 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 8.4999999999999999e37 < y

   1. Initial program 23.3%

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

      1. sub-neg 23.3%

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

      2. log1p-def 23.3%

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

      3. distribute-neg-frac 23.3%

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

      4. sub-neg 23.3%

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

      5. distribute-neg-in 23.3%

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

      6. remove-double-neg 23.3%

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

      7. +-commutative 23.3%

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

      8. sub-neg 23.3%

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

   3. Simplified 23.3%

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

   5. Taylor expanded in y around inf 99.0%

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

      1. log-rec 99.0%

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

      2. unsub-neg 99.0%

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

      3. sub-neg 99.0%

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

      4. metadata-eval 99.0%

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

      5. +-commutative 99.0%

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

   7. Simplified 99.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4500000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+37}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y} - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y} - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999995)
   (- 1.0 (log1p (- (/ y (- 1.0 y)) (/ x (- 1.0 y)))))
   (- 1.0 (log (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999995) {
		tmp = 1.0 - log1p(((y / (1.0 - y)) - (x / (1.0 - y))));
	} else {
		tmp = 1.0 - log((-1.0 / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999995) {
		tmp = 1.0 - Math.log1p(((y / (1.0 - y)) - (x / (1.0 - y))));
	} else {
		tmp = 1.0 - Math.log((-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999995:
		tmp = 1.0 - math.log1p(((y / (1.0 - y)) - (x / (1.0 - y))))
	else:
		tmp = 1.0 - math.log((-1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999995)
		tmp = Float64(1.0 - log1p(Float64(Float64(y / Float64(1.0 - y)) - Float64(x / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999999995], N[(1.0 - N[Log[1 + N[(N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] - N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.99999999500000003

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-def 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. sub-neg 99.8%

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

   3. Simplified 99.8%

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

      1. div-sub 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 0.99999999500000003 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 3.5%

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

      1. sub-neg 3.5%

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

      2. log1p-def 3.5%

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

      3. distribute-neg-frac 3.5%

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

      4. sub-neg 3.5%

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

      5. distribute-neg-in 3.5%

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

      6. remove-double-neg 3.5%

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

      7. +-commutative 3.5%

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

      8. sub-neg 3.5%

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

   3. Simplified 3.5%

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

   5. Taylor expanded in x around 0 3.5%

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

      1. log1p-def 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 63.7%

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

   10. Simplified 63.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y} - \frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999995)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999995) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log((-1.0 / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999995) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log((-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999995:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log((-1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999995)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999999995], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.99999999500000003

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-def 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. sub-neg 99.8%

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

      5. distribute-neg-in 99.8%

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

      6. remove-double-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. sub-neg 99.8%

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

   3. Simplified 99.8%

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

   if 0.99999999500000003 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 3.5%

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

      1. sub-neg 3.5%

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

      2. log1p-def 3.5%

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

      3. distribute-neg-frac 3.5%

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

      4. sub-neg 3.5%

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

      5. distribute-neg-in 3.5%

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

      6. remove-double-neg 3.5%

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

      7. +-commutative 3.5%

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

      8. sub-neg 3.5%

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

   3. Simplified 3.5%

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

   5. Taylor expanded in x around 0 3.5%

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

      1. log1p-def 3.5%

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

   7. Simplified 3.5%

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

   8. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 63.7%

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

   10. Simplified 63.7%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -17.0)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ (- x) (- 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -17.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -17.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p((-x / (1.0 - y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -17.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p((-x / (1.0 - y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -17.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(-x) / Float64(1.0 - y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -17.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[((-x) / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -17

   1. Initial program 23.4%

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

      1. sub-neg 23.4%

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

      2. log1p-def 23.4%

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

      3. distribute-neg-frac 23.4%

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

      4. sub-neg 23.4%

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

      5. distribute-neg-in 23.4%

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

      6. remove-double-neg 23.4%

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

      7. +-commutative 23.4%

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

      8. sub-neg 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in x around 0 6.2%

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

      1. log1p-def 6.2%

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

   7. Simplified 6.2%

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

   8. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 67.9%

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

   10. Simplified 67.9%

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

   if -17 < y

   1. Initial program 89.7%

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

      1. sub-neg 89.7%

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

      2. log1p-def 89.8%

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

      3. distribute-neg-frac 89.8%

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

      4. sub-neg 89.8%

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

      5. distribute-neg-in 89.8%

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

      6. remove-double-neg 89.8%

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

      7. +-commutative 89.8%

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

      8. sub-neg 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in x around inf 89.4%

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

      1. neg-mul-1 89.4%

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

      2. distribute-neg-frac 89.4%

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

   7. Simplified 89.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -17:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-x}{1 - y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -9.5

   1. Initial program 23.4%

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

      1. sub-neg 23.4%

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

      2. log1p-def 23.4%

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

      3. distribute-neg-frac 23.4%

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

      4. sub-neg 23.4%

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

      5. distribute-neg-in 23.4%

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

      6. remove-double-neg 23.4%

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

      7. +-commutative 23.4%

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

      8. sub-neg 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in x around 0 6.2%

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

      1. log1p-def 6.2%

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

   7. Simplified 6.2%

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

   8. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 67.9%

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

   10. Simplified 67.9%

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

   if -9.5 < y

   1. Initial program 89.7%

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

      1. sub-neg 89.7%

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

      2. log1p-def 89.8%

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

      3. distribute-neg-frac 89.8%

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

      4. sub-neg 89.8%

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

      5. distribute-neg-in 89.8%

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

      6. remove-double-neg 89.8%

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

      7. +-commutative 89.8%

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

      8. sub-neg 89.8%

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

   3. Simplified 89.8%

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

      1. div-sub 89.8%

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

   6. Applied egg-rr 89.8%

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

   7. Taylor expanded in y around 0 83.5%

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

      1. sub-neg 83.5%

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

      2. mul-1-neg 83.5%

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

      3. log1p-def 83.5%

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

      4. mul-1-neg 83.5%

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

      5. +-commutative 83.5%

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

      6. mul-1-neg 83.5%

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

      7. unsub-neg 83.5%

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

      8. sub-neg 83.5%

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

      9. mul-1-neg 83.5%

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

      10. sub-neg 83.5%

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

      11. mul-1-neg 83.5%

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

      12. div-sub 83.5%

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

      13. sub-neg 83.5%

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

      14. mul-1-neg 83.5%

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

      15. *-inverses 83.5%

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

      16. *-rgt-identity 83.5%

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

   9. Simplified 83.5%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13.0) {
		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 <= -13.0) {
		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 <= -13.0:
		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 <= -13.0)
		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, -13.0], 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 < -13

   1. Initial program 23.4%

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

      1. sub-neg 23.4%

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

      2. log1p-def 23.4%

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

      3. distribute-neg-frac 23.4%

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

      4. sub-neg 23.4%

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

      5. distribute-neg-in 23.4%

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

      6. remove-double-neg 23.4%

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

      7. +-commutative 23.4%

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

      8. sub-neg 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in x around 0 6.2%

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

      1. log1p-def 6.2%

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

   7. Simplified 6.2%

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

   8. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 67.9%

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

   10. Simplified 67.9%

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

   if -13 < y

   1. Initial program 89.7%

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

      1. sub-neg 89.7%

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

      2. log1p-def 89.8%

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

      3. distribute-neg-frac 89.8%

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

      4. sub-neg 89.8%

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

      5. distribute-neg-in 89.8%

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

      6. remove-double-neg 89.8%

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

      7. +-commutative 89.8%

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

      8. sub-neg 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in y around 0 82.4%

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

      1. log1p-def 82.5%

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

      2. mul-1-neg 82.5%

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

   7. Simplified 82.5%

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

4. Final simplification 78.2%

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

Alternative 9: 44.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.65:\\ \;\;\;\;1 + \frac{\left(-1 - x\right) - \frac{x}{x + -1}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.65)
   (+ 1.0 (/ (- (- -1.0 x) (/ x (+ x -1.0))) y))
   (- 1.0 (log1p y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.65) {
		tmp = 1.0 + (((-1.0 - x) - (x / (x + -1.0))) / y);
	} else {
		tmp = 1.0 - log1p(y);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.65) {
		tmp = 1.0 + (((-1.0 - x) - (x / (x + -1.0))) / y);
	} else {
		tmp = 1.0 - Math.log1p(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.65:
		tmp = 1.0 + (((-1.0 - x) - (x / (x + -1.0))) / y)
	else:
		tmp = 1.0 - math.log1p(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.65)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 - x) - Float64(x / Float64(x + -1.0))) / y));
	else
		tmp = Float64(1.0 - log1p(y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.65], N[(1.0 + N[(N[(N[(-1.0 - x), $MachinePrecision] - N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + y], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.650000000000000022

   1. Initial program 23.4%

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

      1. sub-neg 23.4%

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

      2. log1p-def 23.4%

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

      3. distribute-neg-frac 23.4%

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

      4. sub-neg 23.4%

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

      5. distribute-neg-in 23.4%

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

      6. remove-double-neg 23.4%

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

      7. +-commutative 23.4%

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

      8. sub-neg 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in y around -inf 97.9%

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

      1. sub-neg 97.9%

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

      2. metadata-eval 97.9%

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

      3. distribute-lft-in 97.9%

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

      4. metadata-eval 97.9%

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

      5. +-commutative 97.9%

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

      6. log1p-def 97.9%

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

      7. mul-1-neg 97.9%

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

      8. mul-1-neg 97.9%

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

      9. unsub-neg 97.9%

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

      10. div-sub 97.9%

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

      11. associate-/l/ 97.9%

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

      12. sub-neg 97.9%

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

      13. metadata-eval 97.9%

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

      14. +-commutative 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in y around 0 12.1%

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

   9. Taylor expanded in x around 0 12.9%

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

   if -0.650000000000000022 < y

   1. Initial program 89.7%

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

      1. sub-neg 89.7%

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

      2. log1p-def 89.8%

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

      3. distribute-neg-frac 89.8%

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

      4. sub-neg 89.8%

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

      5. distribute-neg-in 89.8%

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

      6. remove-double-neg 89.8%

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

      7. +-commutative 89.8%

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

      8. sub-neg 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in x around 0 56.5%

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

      1. log1p-def 56.5%

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

   7. Simplified 56.5%

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

   8. Taylor expanded in y around 0 56.2%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.65:\\ \;\;\;\;1 + \frac{\left(-1 - x\right) - \frac{x}{x + -1}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. sub-neg 70.0%

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

   2. log1p-def 70.1%

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

   3. distribute-neg-frac 70.1%

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

   4. sub-neg 70.1%

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

   5. distribute-neg-in 70.1%

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

   6. remove-double-neg 70.1%

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

   7. +-commutative 70.1%

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

   8. sub-neg 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in y around 0 61.9%

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

   1. log1p-def 61.9%

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

   2. mul-1-neg 61.9%

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

7. Simplified 61.9%

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

8. Final simplification 61.9%

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

Alternative 11: 43.6% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.78:\\ \;\;\;\;1 + \frac{\left(-1 - x\right) - \frac{x}{x + -1}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.78) (+ 1.0 (/ (- (- -1.0 x) (/ x (+ x -1.0))) y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.78) {
		tmp = 1.0 + (((-1.0 - x) - (x / (x + -1.0))) / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-0.78d0)) then
        tmp = 1.0d0 + ((((-1.0d0) - x) - (x / (x + (-1.0d0)))) / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -0.78) {
		tmp = 1.0 + (((-1.0 - x) - (x / (x + -1.0))) / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -0.78:
		tmp = 1.0 + (((-1.0 - x) - (x / (x + -1.0))) / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.78)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 - x) - Float64(x / Float64(x + -1.0))) / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -0.78)
		tmp = 1.0 + (((-1.0 - x) - (x / (x + -1.0))) / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.78], N[(1.0 + N[(N[(N[(-1.0 - x), $MachinePrecision] - N[(x / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.78000000000000003

   1. Initial program 23.4%

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

      1. sub-neg 23.4%

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

      2. log1p-def 23.4%

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

      3. distribute-neg-frac 23.4%

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

      4. sub-neg 23.4%

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

      5. distribute-neg-in 23.4%

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

      6. remove-double-neg 23.4%

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

      7. +-commutative 23.4%

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

      8. sub-neg 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in y around -inf 97.9%

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

      1. sub-neg 97.9%

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

      2. metadata-eval 97.9%

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

      3. distribute-lft-in 97.9%

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

      4. metadata-eval 97.9%

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

      5. +-commutative 97.9%

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

      6. log1p-def 97.9%

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

      7. mul-1-neg 97.9%

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

      8. mul-1-neg 97.9%

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

      9. unsub-neg 97.9%

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

      10. div-sub 97.9%

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

      11. associate-/l/ 97.9%

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

      12. sub-neg 97.9%

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

      13. metadata-eval 97.9%

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

      14. +-commutative 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in y around 0 12.1%

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

   9. Taylor expanded in x around 0 12.9%

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

   if -0.78000000000000003 < y

   1. Initial program 89.7%

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

      1. sub-neg 89.7%

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

      2. log1p-def 89.8%

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

      3. distribute-neg-frac 89.8%

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

      4. sub-neg 89.8%

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

      5. distribute-neg-in 89.8%

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

      6. remove-double-neg 89.8%

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

      7. +-commutative 89.8%

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

      8. sub-neg 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in x around 0 56.5%

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

      1. log1p-def 56.5%

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

   7. Simplified 56.5%

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

      1. clear-num 56.5%

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

      2. associate-/r/ 56.8%

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

   9. Applied egg-rr 56.8%

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

   10. Taylor expanded in y around 0 55.7%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.78:\\ \;\;\;\;1 + \frac{\left(-1 - x\right) - \frac{x}{x + -1}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;1 + \frac{-1 + \frac{1}{x + -1}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.4) (+ 1.0 (/ (+ -1.0 (/ 1.0 (+ x -1.0))) y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.4) {
		tmp = 1.0 + ((-1.0 + (1.0 / (x + -1.0))) / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-1.4d0)) then
        tmp = 1.0d0 + (((-1.0d0) + (1.0d0 / (x + (-1.0d0)))) / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.4) {
		tmp = 1.0 + ((-1.0 + (1.0 / (x + -1.0))) / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.4:
		tmp = 1.0 + ((-1.0 + (1.0 / (x + -1.0))) / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.4)
		tmp = Float64(1.0 + Float64(Float64(-1.0 + Float64(1.0 / Float64(x + -1.0))) / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.4)
		tmp = 1.0 + ((-1.0 + (1.0 / (x + -1.0))) / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.4], N[(1.0 + N[(N[(-1.0 + N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3999999999999999

   1. Initial program 23.4%

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

      1. sub-neg 23.4%

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

      2. log1p-def 23.4%

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

      3. distribute-neg-frac 23.4%

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

      4. sub-neg 23.4%

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

      5. distribute-neg-in 23.4%

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

      6. remove-double-neg 23.4%

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

      7. +-commutative 23.4%

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

      8. sub-neg 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in y around -inf 97.9%

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

      1. sub-neg 97.9%

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

      2. metadata-eval 97.9%

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

      3. distribute-lft-in 97.9%

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

      4. metadata-eval 97.9%

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

      5. +-commutative 97.9%

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

      6. log1p-def 97.9%

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

      7. mul-1-neg 97.9%

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

      8. mul-1-neg 97.9%

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

      9. unsub-neg 97.9%

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

      10. div-sub 97.9%

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

      11. associate-/l/ 97.9%

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

      12. sub-neg 97.9%

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

      13. metadata-eval 97.9%

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

      14. +-commutative 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in y around 0 12.1%

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

   9. Taylor expanded in x around inf 12.1%

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

   if -1.3999999999999999 < y

   1. Initial program 89.7%

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

      1. sub-neg 89.7%

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

      2. log1p-def 89.8%

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

      3. distribute-neg-frac 89.8%

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

      4. sub-neg 89.8%

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

      5. distribute-neg-in 89.8%

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

      6. remove-double-neg 89.8%

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

      7. +-commutative 89.8%

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

      8. sub-neg 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in x around 0 56.5%

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

      1. log1p-def 56.5%

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

   7. Simplified 56.5%

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

      1. clear-num 56.5%

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

      2. associate-/r/ 56.8%

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

   9. Applied egg-rr 56.8%

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

   10. Taylor expanded in y around 0 55.7%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;1 + \frac{-1 + \frac{1}{x + -1}}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.3% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13.5:\\ \;\;\;\;1 - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13.5) (- 1.0 (/ -1.0 y)) (- 1.0 y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13.5) {
		tmp = 1.0 - (-1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-13.5d0)) then
        tmp = 1.0d0 - ((-1.0d0) / y)
    else
        tmp = 1.0d0 - y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -13.5) {
		tmp = 1.0 - (-1.0 / y);
	} else {
		tmp = 1.0 - y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -13.5:
		tmp = 1.0 - (-1.0 / y)
	else:
		tmp = 1.0 - y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -13.5)
		tmp = Float64(1.0 - Float64(-1.0 / y));
	else
		tmp = Float64(1.0 - y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -13.5)
		tmp = 1.0 - (-1.0 / y);
	else
		tmp = 1.0 - y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -13.5], N[(1.0 - N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 - y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -13.5

   1. Initial program 23.4%

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

      1. sub-neg 23.4%

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

      2. log1p-def 23.4%

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

      3. distribute-neg-frac 23.4%

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

      4. sub-neg 23.4%

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

      5. distribute-neg-in 23.4%

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

      6. remove-double-neg 23.4%

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

      7. +-commutative 23.4%

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

      8. sub-neg 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in y around -inf 97.9%

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

      1. sub-neg 97.9%

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

      2. metadata-eval 97.9%

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

      3. distribute-lft-in 97.9%

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

      4. metadata-eval 97.9%

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

      5. +-commutative 97.9%

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

      6. log1p-def 97.9%

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

      7. mul-1-neg 97.9%

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

      8. mul-1-neg 97.9%

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

      9. unsub-neg 97.9%

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

      10. div-sub 97.9%

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

      11. associate-/l/ 97.9%

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

      12. sub-neg 97.9%

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

      13. metadata-eval 97.9%

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

      14. +-commutative 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in y around 0 12.1%

      \[\leadsto 1 - \color{blue}{\frac{\frac{x}{x - 1} - \frac{1}{x - 1}}{y}} \]
   9. Step-by-step derivation

      1. sub-neg 12.1%

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

      2. +-commutative 12.1%

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

   10. Applied egg-rr 12.1%

       \[\leadsto \color{blue}{\frac{1}{y} + 1} \]

   if -13.5 < y

   1. Initial program 89.7%

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

      1. sub-neg 89.7%

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

      2. log1p-def 89.8%

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

      3. distribute-neg-frac 89.8%

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

      4. sub-neg 89.8%

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

      5. distribute-neg-in 89.8%

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

      6. remove-double-neg 89.8%

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

      7. +-commutative 89.8%

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

      8. sub-neg 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in x around 0 56.5%

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

      1. log1p-def 56.5%

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

   7. Simplified 56.5%

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

      1. clear-num 56.5%

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

      2. associate-/r/ 56.8%

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

   9. Applied egg-rr 56.8%

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

   10. Taylor expanded in y around 0 55.7%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13.5:\\ \;\;\;\;1 - \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.3% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if y < -1

   1. Initial program 23.4%

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

      1. sub-neg 23.4%

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

      2. log1p-def 23.4%

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

      3. distribute-neg-frac 23.4%

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

      4. sub-neg 23.4%

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

      5. distribute-neg-in 23.4%

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

      6. remove-double-neg 23.4%

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

      7. +-commutative 23.4%

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

      8. sub-neg 23.4%

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

   3. Simplified 23.4%

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

   5. Taylor expanded in y around -inf 97.9%

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

      1. sub-neg 97.9%

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

      2. metadata-eval 97.9%

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

      3. distribute-lft-in 97.9%

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

      4. metadata-eval 97.9%

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

      5. +-commutative 97.9%

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

      6. log1p-def 97.9%

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

      7. mul-1-neg 97.9%

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

      8. mul-1-neg 97.9%

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

      9. unsub-neg 97.9%

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

      10. div-sub 97.9%

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

      11. associate-/l/ 97.9%

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

      12. sub-neg 97.9%

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

      13. metadata-eval 97.9%

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

      14. +-commutative 97.9%

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

   7. Simplified 97.9%

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

   8. Taylor expanded in y around 0 12.1%

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

   9. Taylor expanded in x around 0 12.1%

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

   if -1 < y

   1. Initial program 89.7%

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

      1. sub-neg 89.7%

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

      2. log1p-def 89.8%

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

      3. distribute-neg-frac 89.8%

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

      4. sub-neg 89.8%

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

      5. distribute-neg-in 89.8%

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

      6. remove-double-neg 89.8%

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

      7. +-commutative 89.8%

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

      8. sub-neg 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in x around 0 56.5%

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

      1. log1p-def 56.5%

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

   7. Simplified 56.5%

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

      1. clear-num 56.5%

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

      2. associate-/r/ 56.8%

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

   9. Applied egg-rr 56.8%

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

   10. Taylor expanded in y around 0 55.7%

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

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 - y\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.1% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.0%

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

   1. sub-neg 70.0%

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

   2. log1p-def 70.1%

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

   3. distribute-neg-frac 70.1%

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

   4. sub-neg 70.1%

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

   5. distribute-neg-in 70.1%

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

   6. remove-double-neg 70.1%

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

   7. +-commutative 70.1%

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

   8. sub-neg 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in x around 0 41.5%

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

   1. log1p-def 41.5%

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

7. Simplified 41.5%

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

   1. clear-num 41.5%

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

   2. associate-/r/ 42.2%

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

9. Applied egg-rr 42.2%

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

10. Taylor expanded in y around 0 40.6%

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

11. Final simplification 40.6%

    \[\leadsto 1 - y \]
12. 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 2024021 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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