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

Percentage Accurate: 73.1% → 99.7%

Time: 12.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12000:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \left(\mathsf{log1p}\left(-x\right) + \left(\frac{0.5}{y \cdot y} + \log \left(\frac{-1}{y}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 40000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{y} \cdot \left(1 + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -12000.0)
   (+
    1.0
    (- (/ -1.0 y) (+ (log1p (- x)) (+ (/ 0.5 (* y y)) (log (/ -1.0 y))))))
   (if (<= y 40000000000.0)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (- 1.0 (log (* (/ 1.0 y) (+ 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -12000.0) {
		tmp = 1.0 + ((-1.0 / y) - (log1p(-x) + ((0.5 / (y * y)) + log((-1.0 / y)))));
	} else if (y <= 40000000000.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((1.0 / y) * (1.0 + x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -12000.0) {
		tmp = 1.0 + ((-1.0 / y) - (Math.log1p(-x) + ((0.5 / (y * y)) + Math.log((-1.0 / y)))));
	} else if (y <= 40000000000.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((1.0 / y) * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -12000.0:
		tmp = 1.0 + ((-1.0 / y) - (math.log1p(-x) + ((0.5 / (y * y)) + math.log((-1.0 / y)))))
	elif y <= 40000000000.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((1.0 / y) * (1.0 + x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -12000.0)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - Float64(log1p(Float64(-x)) + Float64(Float64(0.5 / Float64(y * y)) + log(Float64(-1.0 / y))))));
	elseif (y <= 40000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(1.0 / y) * Float64(1.0 + x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -12000.0], N[(1.0 + N[(N[(-1.0 / y), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[(N[(0.5 / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 40000000000.0], 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[(N[(1.0 / y), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -12000

   1. Initial program 21.0%

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

      1. sub-neg 21.0%

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

      2. log1p-def 21.0%

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

      3. distribute-neg-frac 21.0%

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

      4. sub-neg 21.0%

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

      5. distribute-neg-in 21.0%

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

      6. remove-double-neg 21.0%

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

      7. +-commutative 21.0%

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

      8. sub-neg 21.0%

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

   3. Simplified 21.0%

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

      1. flip-- 21.4%

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

      2. associate-/r/ 21.8%

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

      3. metadata-eval 21.8%

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

      4. +-commutative 21.8%

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

   5. Applied egg-rr 21.8%

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

   6. Taylor expanded in y around -inf 99.6%

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{y} + \left(0.5 \cdot \frac{1}{{y}^{2}} + \left(\log \left(\frac{-1}{y}\right) + \log \left(1 + -1 \cdot x\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 99.6%

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

      2. associate-*r/ 99.6%

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

      3. metadata-eval 99.6%

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

      4. unpow2 99.6%

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

      5. log1p-def 99.6%

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

      6. mul-1-neg 99.6%

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

   8. Simplified 99.6%

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

   if -12000 < y < 4e10

   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%

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

   if 4e10 < y

   1. Initial program 68.5%

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

      1. sub-neg 68.5%

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

      2. log1p-def 68.5%

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

      3. distribute-neg-frac 68.5%

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

      4. sub-neg 68.5%

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

      5. distribute-neg-in 68.5%

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

      6. remove-double-neg 68.5%

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

      7. +-commutative 68.5%

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

      8. sub-neg 68.5%

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

   3. Simplified 68.5%

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

      1. add-cube-cbrt 67.4%

         \[\leadsto 1 - \color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}} \]

      2. pow3 67.4%

         \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right)}^{3}} \]

   5. Applied egg-rr 67.4%

      \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right)}^{3}} \]

   6. Taylor expanded in y around -inf 0.0%

      \[\leadsto 1 - {\color{blue}{\left({\left(\log \left(-1 \cdot \left(x - 1\right)\right) + \log \left(\frac{-1}{y}\right)\right)}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 0.0%

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

      2. *-lft-identity 0.0%

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

      3. *-lft-identity 0.0%

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

      4. mul-1-neg 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(-\left(x - 1\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      5. sub-neg 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \left(-\color{blue}{\left(x + \left(-1\right)\right)}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      6. metadata-eval 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \left(-\left(x + \color{blue}{-1}\right)\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      7. distribute-neg-in 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(\left(-x\right) + \left(--1\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      8. mul-1-neg 0.0%

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

      9. metadata-eval 0.0%

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

      10. +-commutative 0.0%

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

      11. mul-1-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \left(1 + \color{blue}{\left(-x\right)}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      12. sub-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(1 - x\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      13. sub-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(1 + \left(-x\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      14. mul-1-neg 0.0%

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

      15. log1p-def 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      16. mul-1-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{-x}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

   8. Simplified 0.0%

      \[\leadsto 1 - {\color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow3 0.0%

         \[\leadsto 1 - \color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}} \]

      2. add-cube-cbrt 0.0%

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

      3. +-commutative 0.0%

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

      4. log1p-udef 0.0%

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

      5. sum-log 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.1%

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

      8. frac-times 1.2%

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

      9. metadata-eval 1.2%

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

      10. metadata-eval 1.2%

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

      11. frac-times 1.1%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 53.2%

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

      16. sqr-neg 53.2%

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

      17. sqrt-unprod 100.0%

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

      18. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12000:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \left(\mathsf{log1p}\left(-x\right) + \left(\frac{0.5}{y \cdot y} + \log \left(\frac{-1}{y}\right)\right)\right)\right)\\ \mathbf{elif}\;y \leq 40000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{y} \cdot \left(1 + x\right)\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+15}:\\ \;\;\;\;1 - \left(\log \left(\frac{-1}{y}\right) + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 40000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{y} \cdot \left(1 + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.05e+15)
   (- 1.0 (+ (log (/ -1.0 y)) (log1p (- x))))
   (if (<= y 40000000000.0)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (- 1.0 (log (* (/ 1.0 y) (+ 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.05e+15) {
		tmp = 1.0 - (log((-1.0 / y)) + log1p(-x));
	} else if (y <= 40000000000.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((1.0 / y) * (1.0 + x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.05e+15) {
		tmp = 1.0 - (Math.log((-1.0 / y)) + Math.log1p(-x));
	} else if (y <= 40000000000.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((1.0 / y) * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.05e+15:
		tmp = 1.0 - (math.log((-1.0 / y)) + math.log1p(-x))
	elif y <= 40000000000.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((1.0 / y) * (1.0 + x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.05e+15)
		tmp = Float64(1.0 - Float64(log(Float64(-1.0 / y)) + log1p(Float64(-x))));
	elseif (y <= 40000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(1.0 / y) * Float64(1.0 + x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.05e+15], N[(1.0 - N[(N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision] + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 40000000000.0], 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[(N[(1.0 / y), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.05e15

   1. Initial program 17.6%

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

      1. sub-neg 17.6%

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

      2. log1p-def 17.6%

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

      3. distribute-neg-frac 17.6%

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

      4. sub-neg 17.6%

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

      5. distribute-neg-in 17.6%

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

      6. remove-double-neg 17.6%

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

      7. +-commutative 17.6%

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

      8. sub-neg 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in y around -inf 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. log1p-def 99.6%

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

      7. mul-1-neg 99.6%

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

   6. Simplified 99.6%

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

   if -2.05e15 < y < 4e10

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. log1p-def 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   if 4e10 < y

   1. Initial program 68.5%

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

      1. sub-neg 68.5%

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

      2. log1p-def 68.5%

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

      3. distribute-neg-frac 68.5%

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

      4. sub-neg 68.5%

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

      5. distribute-neg-in 68.5%

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

      6. remove-double-neg 68.5%

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

      7. +-commutative 68.5%

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

      8. sub-neg 68.5%

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

   3. Simplified 68.5%

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

      1. add-cube-cbrt 67.4%

         \[\leadsto 1 - \color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}} \]

      2. pow3 67.4%

         \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right)}^{3}} \]

   5. Applied egg-rr 67.4%

      \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right)}^{3}} \]

   6. Taylor expanded in y around -inf 0.0%

      \[\leadsto 1 - {\color{blue}{\left({\left(\log \left(-1 \cdot \left(x - 1\right)\right) + \log \left(\frac{-1}{y}\right)\right)}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 0.0%

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

      2. *-lft-identity 0.0%

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

      3. *-lft-identity 0.0%

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

      4. mul-1-neg 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(-\left(x - 1\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      5. sub-neg 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \left(-\color{blue}{\left(x + \left(-1\right)\right)}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      6. metadata-eval 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \left(-\left(x + \color{blue}{-1}\right)\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      7. distribute-neg-in 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(\left(-x\right) + \left(--1\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      8. mul-1-neg 0.0%

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

      9. metadata-eval 0.0%

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

      10. +-commutative 0.0%

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

      11. mul-1-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \left(1 + \color{blue}{\left(-x\right)}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      12. sub-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(1 - x\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      13. sub-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(1 + \left(-x\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      14. mul-1-neg 0.0%

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

      15. log1p-def 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      16. mul-1-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{-x}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

   8. Simplified 0.0%

      \[\leadsto 1 - {\color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow3 0.0%

         \[\leadsto 1 - \color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}} \]

      2. add-cube-cbrt 0.0%

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

      3. +-commutative 0.0%

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

      4. log1p-udef 0.0%

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

      5. sum-log 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.1%

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

      8. frac-times 1.2%

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

      9. metadata-eval 1.2%

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

      10. metadata-eval 1.2%

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

      11. frac-times 1.1%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 53.2%

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

      16. sqr-neg 53.2%

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

      17. sqrt-unprod 100.0%

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

      18. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+15}:\\ \;\;\;\;1 - \left(\log \left(\frac{-1}{y}\right) + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{elif}\;y \leq 40000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{y} \cdot \left(1 + x\right)\right)\\ \end{array} \]

Alternative 3: 73.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 1.0)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (+ 1.0 (- (/ -1.0 y) (+ (/ 0.5 (* y y)) (log (/ -1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 1.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - ((0.5 / (y * y)) + log((-1.0 / y))));
	}
	return tmp;
}

Java

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

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 1.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + ((-1.0 / y) - ((0.5 / (y * y)) + math.log((-1.0 / y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - Float64(Float64(0.5 / Float64(y * y)) + log(Float64(-1.0 / y)))));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 75.9%

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

      1. sub-neg 75.9%

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

      2. log1p-def 75.9%

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

      3. distribute-neg-frac 75.9%

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

      4. sub-neg 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. remove-double-neg 75.9%

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

      7. +-commutative 75.9%

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

      8. sub-neg 75.9%

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

   3. Simplified 75.9%

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

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

   1. Initial program 75.9%

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

      1. sub-neg 75.9%

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

      2. log1p-def 75.9%

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

      3. distribute-neg-frac 75.9%

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

      4. sub-neg 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. remove-double-neg 75.9%

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

      7. +-commutative 75.9%

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

      8. sub-neg 75.9%

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

   3. Simplified 75.9%

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

      1. flip-- 74.6%

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

      2. associate-/r/ 74.6%

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

      3. metadata-eval 74.6%

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

      4. +-commutative 74.6%

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

   5. Applied egg-rr 74.6%

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

   6. Taylor expanded in y around -inf 27.5%

      \[\leadsto 1 - \color{blue}{\left(\frac{1}{y} + \left(0.5 \cdot \frac{1}{{y}^{2}} + \left(\log \left(\frac{-1}{y}\right) + \log \left(1 + -1 \cdot x\right)\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-+r+ 27.5%

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

      2. associate-*r/ 27.5%

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

      3. metadata-eval 27.5%

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

      4. unpow2 27.5%

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

      5. log1p-def 27.5%

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

      6. mul-1-neg 27.5%

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

   8. Simplified 27.5%

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

   9. Taylor expanded in x around 0 19.8%

      \[\leadsto 1 - \left(\frac{1}{y} + \color{blue}{\left(0.5 \cdot \frac{1}{{y}^{2}} + \log \left(\frac{-1}{y}\right)\right)}\right) \]
   10. Step-by-step derivation

       1. associate-*r/ 19.8%

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

       2. metadata-eval 19.8%

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

       3. unpow2 19.8%

          \[\leadsto 1 - \left(\frac{1}{y} + \left(\frac{0.5}{\color{blue}{y \cdot y}} + \log \left(\frac{-1}{y}\right)\right)\right) \]

       4. associate-/l/ 19.8%

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

       5. +-commutative 19.8%

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

       6. associate-/l/ 19.8%

          \[\leadsto 1 - \left(\frac{1}{y} + \left(\log \left(\frac{-1}{y}\right) + \color{blue}{\frac{0.5}{y \cdot y}}\right)\right) \]

   11. Simplified 19.8%

       \[\leadsto 1 - \left(\frac{1}{y} + \color{blue}{\left(\log \left(\frac{-1}{y}\right) + \frac{0.5}{y \cdot y}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \left(\frac{0.5}{y \cdot y} + \log \left(\frac{-1}{y}\right)\right)\right)\\ \end{array} \]

Alternative 4: 73.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 1.0)
   (- 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)) <= 1.0) {
		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)) <= 1.0) {
		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)) <= 1.0:
		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)) <= 1.0)
		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], 1.0], 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)) < 1

   1. Initial program 75.9%

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

      1. sub-neg 75.9%

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

      2. log1p-def 75.9%

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

      3. distribute-neg-frac 75.9%

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

      4. sub-neg 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. remove-double-neg 75.9%

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

      7. +-commutative 75.9%

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

      8. sub-neg 75.9%

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

   3. Simplified 75.9%

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

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

   1. Initial program 75.9%

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

      1. sub-neg 75.9%

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

      2. log1p-def 75.9%

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

      3. distribute-neg-frac 75.9%

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

      4. sub-neg 75.9%

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

      5. distribute-neg-in 75.9%

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

      6. remove-double-neg 75.9%

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

      7. +-commutative 75.9%

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

      8. sub-neg 75.9%

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

   3. Simplified 75.9%

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

   4. Taylor expanded in y around -inf 28.6%

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

      1. sub-neg 28.6%

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

      2. metadata-eval 28.6%

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

      3. distribute-lft-in 28.6%

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

      4. metadata-eval 28.6%

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

      5. +-commutative 28.6%

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

      6. log1p-def 28.6%

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

      7. mul-1-neg 28.6%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in x around 0 21.1%

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

4. Final simplification 75.9%

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

Alternative 5: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+15}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 48000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(-\frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{y} \cdot \left(1 + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+15)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 48000000000000.0)
     (- 1.0 (log1p (- (/ x (- 1.0 y)))))
     (- 1.0 (log (* (/ 1.0 y) (+ 1.0 x)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+15) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 48000000000000.0) {
		tmp = 1.0 - log1p(-(x / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((1.0 / y) * (1.0 + x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+15) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 48000000000000.0) {
		tmp = 1.0 - Math.log1p(-(x / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((1.0 / y) * (1.0 + x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+15:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 48000000000000.0:
		tmp = 1.0 - math.log1p(-(x / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((1.0 / y) * (1.0 + x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+15)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 48000000000000.0)
		tmp = Float64(1.0 - log1p(Float64(-Float64(x / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(1.0 / y) * Float64(1.0 + x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+15], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 48000000000000.0], N[(1.0 - N[Log[1 + (-N[(x / N[(1.0 - y), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(1.0 / y), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e15

   1. Initial program 17.6%

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

      1. sub-neg 17.6%

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

      2. log1p-def 17.6%

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

      3. distribute-neg-frac 17.6%

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

      4. sub-neg 17.6%

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

      5. distribute-neg-in 17.6%

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

      6. remove-double-neg 17.6%

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

      7. +-commutative 17.6%

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

      8. sub-neg 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in y around -inf 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. log1p-def 99.6%

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

      7. mul-1-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 73.0%

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

   if -3.8e15 < y < 4.8e13

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. log1p-def 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. sub-neg 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 97.1%

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

      1. neg-mul-1 97.1%

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

      2. distribute-neg-frac 97.1%

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

   6. Simplified 97.1%

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

   if 4.8e13 < y

   1. Initial program 68.5%

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

      1. sub-neg 68.5%

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

      2. log1p-def 68.5%

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

      3. distribute-neg-frac 68.5%

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

      4. sub-neg 68.5%

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

      5. distribute-neg-in 68.5%

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

      6. remove-double-neg 68.5%

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

      7. +-commutative 68.5%

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

      8. sub-neg 68.5%

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

   3. Simplified 68.5%

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

      1. add-cube-cbrt 67.4%

         \[\leadsto 1 - \color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}} \]

      2. pow3 67.4%

         \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right)}^{3}} \]

   5. Applied egg-rr 67.4%

      \[\leadsto 1 - \color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(\frac{y - x}{1 - y}\right)}\right)}^{3}} \]

   6. Taylor expanded in y around -inf 0.0%

      \[\leadsto 1 - {\color{blue}{\left({\left(\log \left(-1 \cdot \left(x - 1\right)\right) + \log \left(\frac{-1}{y}\right)\right)}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 0.0%

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

      2. *-lft-identity 0.0%

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

      3. *-lft-identity 0.0%

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

      4. mul-1-neg 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(-\left(x - 1\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      5. sub-neg 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \left(-\color{blue}{\left(x + \left(-1\right)\right)}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      6. metadata-eval 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \left(-\left(x + \color{blue}{-1}\right)\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      7. distribute-neg-in 0.0%

         \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(\left(-x\right) + \left(--1\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      8. mul-1-neg 0.0%

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

      9. metadata-eval 0.0%

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

      10. +-commutative 0.0%

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

      11. mul-1-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \left(1 + \color{blue}{\left(-x\right)}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      12. sub-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(1 - x\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      13. sub-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\log \color{blue}{\left(1 + \left(-x\right)\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      14. mul-1-neg 0.0%

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

      15. log1p-def 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\color{blue}{\mathsf{log1p}\left(-1 \cdot x\right)} + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

      16. mul-1-neg 0.0%

          \[\leadsto 1 - {\left(\sqrt[3]{\mathsf{log1p}\left(\color{blue}{-x}\right) + \log \left(\frac{-1}{y}\right)}\right)}^{3} \]

   8. Simplified 0.0%

      \[\leadsto 1 - {\color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}\right)}}^{3} \]
   9. Step-by-step derivation

      1. unpow3 0.0%

         \[\leadsto 1 - \color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)}} \]

      2. add-cube-cbrt 0.0%

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

      3. +-commutative 0.0%

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

      4. log1p-udef 0.0%

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

      5. sum-log 100.0%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 1.1%

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

      8. frac-times 1.2%

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

      9. metadata-eval 1.2%

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

      10. metadata-eval 1.2%

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

      11. frac-times 1.1%

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

      12. sqrt-unprod 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. add-sqr-sqrt 0.0%

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

      15. sqrt-unprod 53.2%

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

      16. sqr-neg 53.2%

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

      17. sqrt-unprod 100.0%

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

      18. add-sqr-sqrt 100.0%

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

   10. Applied egg-rr 100.0%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+15}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 48000000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(-\frac{x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{1}{y} \cdot \left(1 + x\right)\right)\\ \end{array} \]

Alternative 6: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -15.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -15.5)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -15.5) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -15.5) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -15.5:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -15.5)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -15.5], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -15.5

   1. Initial program 22.1%

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

      1. sub-neg 22.1%

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

      2. log1p-def 22.1%

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

      3. distribute-neg-frac 22.1%

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

      4. sub-neg 22.1%

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

      5. distribute-neg-in 22.1%

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

      6. remove-double-neg 22.1%

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

      7. +-commutative 22.1%

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

      8. sub-neg 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in y around -inf 97.5%

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

      1. sub-neg 97.5%

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

      2. metadata-eval 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. metadata-eval 97.5%

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

      5. +-commutative 97.5%

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

      6. log1p-def 97.5%

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

      7. mul-1-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in x around 0 70.1%

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

   if -15.5 < y < 1

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-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. Step-by-step derivation

      1. flip-- 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 98.4%

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

      1. +-commutative 98.4%

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

      2. log1p-def 98.5%

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

      3. mul-1-neg 98.5%

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

   8. Simplified 98.5%

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

   if 1 < y

   1. Initial program 69.7%

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

      1. sub-neg 69.7%

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

      2. log1p-def 69.7%

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

      3. distribute-neg-frac 69.7%

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

      4. sub-neg 69.7%

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

      5. distribute-neg-in 69.7%

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

      6. remove-double-neg 69.7%

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

      7. +-commutative 69.7%

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

      8. sub-neg 69.7%

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

   3. Simplified 69.7%

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

   4. Taylor expanded in x around inf 63.6%

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

      1. neg-mul-1 63.6%

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

      2. distribute-neg-frac 63.6%

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

   6. Simplified 63.6%

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

   7. Taylor expanded in y around inf 63.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -15.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 7: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-\frac{x}{1 - y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e+15)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (- (/ x (- 1.0 y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e+15) {
		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 <= -2.4e+15) {
		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 <= -2.4e+15:
		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 <= -2.4e+15)
		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, -2.4e+15], 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 < -2.4e15

   1. Initial program 17.6%

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

      1. sub-neg 17.6%

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

      2. log1p-def 17.6%

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

      3. distribute-neg-frac 17.6%

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

      4. sub-neg 17.6%

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

      5. distribute-neg-in 17.6%

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

      6. remove-double-neg 17.6%

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

      7. +-commutative 17.6%

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

      8. sub-neg 17.6%

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

   3. Simplified 17.6%

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

   4. Taylor expanded in y around -inf 99.6%

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

      1. sub-neg 99.6%

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

      2. metadata-eval 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. metadata-eval 99.6%

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

      5. +-commutative 99.6%

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

      6. log1p-def 99.6%

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

      7. mul-1-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around 0 73.0%

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

   if -2.4e15 < y

   1. Initial program 95.7%

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

      1. sub-neg 95.7%

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

      2. log1p-def 95.8%

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

      3. distribute-neg-frac 95.8%

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

      4. sub-neg 95.8%

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

      5. distribute-neg-in 95.8%

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

      6. remove-double-neg 95.8%

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

      7. +-commutative 95.8%

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

      8. sub-neg 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in x around inf 92.5%

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

      1. neg-mul-1 92.5%

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

      2. distribute-neg-frac 92.5%

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

   6. Simplified 92.5%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-\frac{x}{1 - y}\right)\\ \end{array} \]

Alternative 8: 84.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -21:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -21.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.8e-17) (- 1.0 (log1p (- x))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -21.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.8e-17) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -21.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.8e-17) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -21.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.8e-17:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -21.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.8e-17)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -21.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-17], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -21

   1. Initial program 22.1%

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

      1. sub-neg 22.1%

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

      2. log1p-def 22.1%

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

      3. distribute-neg-frac 22.1%

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

      4. sub-neg 22.1%

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

      5. distribute-neg-in 22.1%

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

      6. remove-double-neg 22.1%

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

      7. +-commutative 22.1%

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

      8. sub-neg 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in y around -inf 97.5%

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

      1. sub-neg 97.5%

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

      2. metadata-eval 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. metadata-eval 97.5%

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

      5. +-commutative 97.5%

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

      6. log1p-def 97.5%

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

      7. mul-1-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in x around 0 70.1%

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

   if -21 < y < 1.79999999999999997e-17

   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. Taylor expanded in y around 0 98.5%

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

      1. log1p-def 98.5%

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

      2. mul-1-neg 98.5%

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

   6. Simplified 98.5%

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

   if 1.79999999999999997e-17 < y

   1. Initial program 74.6%

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

      1. sub-neg 74.6%

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

      2. log1p-def 74.6%

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

      3. distribute-neg-frac 74.6%

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

      4. sub-neg 74.6%

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

      5. distribute-neg-in 74.6%

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

      6. remove-double-neg 74.6%

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

      7. +-commutative 74.6%

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

      8. sub-neg 74.6%

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

   3. Simplified 74.6%

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

   4. Taylor expanded in x around inf 63.7%

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

      1. neg-mul-1 63.7%

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

      2. distribute-neg-frac 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in y around inf 63.2%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -21:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 9: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -18.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -18.0) {
		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 <= -18.0) {
		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 <= -18.0:
		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 <= -18.0)
		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, -18.0], 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 < -18

   1. Initial program 22.1%

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

      1. sub-neg 22.1%

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

      2. log1p-def 22.1%

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

      3. distribute-neg-frac 22.1%

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

      4. sub-neg 22.1%

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

      5. distribute-neg-in 22.1%

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

      6. remove-double-neg 22.1%

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

      7. +-commutative 22.1%

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

      8. sub-neg 22.1%

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

   3. Simplified 22.1%

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

   4. Taylor expanded in y around -inf 97.5%

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

      1. sub-neg 97.5%

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

      2. metadata-eval 97.5%

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

      3. distribute-lft-in 97.5%

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

      4. metadata-eval 97.5%

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

      5. +-commutative 97.5%

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

      6. log1p-def 97.5%

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

      7. mul-1-neg 97.5%

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

   6. Simplified 97.5%

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

   7. Taylor expanded in x around 0 70.1%

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

   if -18 < y

   1. Initial program 95.7%

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

      1. sub-neg 95.7%

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

      2. log1p-def 95.8%

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

      3. distribute-neg-frac 95.8%

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

      4. sub-neg 95.8%

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

      5. distribute-neg-in 95.8%

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

      6. remove-double-neg 95.8%

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

      7. +-commutative 95.8%

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

      8. sub-neg 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in y around 0 83.9%

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

      1. log1p-def 83.9%

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

      2. mul-1-neg 83.9%

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

   6. Simplified 83.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 10: 63.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. sub-neg 75.9%

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

   2. log1p-def 75.9%

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

   3. distribute-neg-frac 75.9%

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

   4. sub-neg 75.9%

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

   5. distribute-neg-in 75.9%

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

   6. remove-double-neg 75.9%

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

   7. +-commutative 75.9%

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

   8. sub-neg 75.9%

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

3. Simplified 75.9%

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

4. Taylor expanded in y around 0 64.7%

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

   1. log1p-def 64.8%

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

   2. mul-1-neg 64.8%

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

6. Simplified 64.8%

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

7. Final simplification 64.8%

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

Alternative 11: 42.6% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. sub-neg 75.9%

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

   2. log1p-def 75.9%

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

   3. distribute-neg-frac 75.9%

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

   4. sub-neg 75.9%

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

   5. distribute-neg-in 75.9%

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

   6. remove-double-neg 75.9%

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

   7. +-commutative 75.9%

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

   8. sub-neg 75.9%

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

3. Simplified 75.9%

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

4. Taylor expanded in x around inf 75.2%

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

   1. neg-mul-1 75.2%

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

   2. distribute-neg-frac 75.2%

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

6. Simplified 75.2%

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

   1. add-sqr-sqrt 45.5%

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

   2. sqrt-unprod 56.6%

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

   3. sqr-neg 56.6%

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

   4. sqrt-unprod 23.4%

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

   5. add-sqr-sqrt 47.7%

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

   6. div-inv 47.7%

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

8. Applied egg-rr 47.7%

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

   1. associate-*r/ 47.7%

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

   2. *-rgt-identity 47.7%

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

10. Simplified 47.7%

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

11. Taylor expanded in x around 0 48.0%

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

12. Final simplification 48.0%

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

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 2023268 
(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))))))