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

Percentage Accurate: 71.5% → 99.9%

Time: 15.0s

Alternatives: 14

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: 71.5% 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: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(x, y)
	return log(Float64(Float64(exp(1) / Float64(1.0 - x)) * Float64(1.0 - y)))
end

MATLAB

function tmp = code(x, y)
	tmp = log(((2.71828182845904523536 / (1.0 - x)) * (1.0 - y)));
end

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. sub-neg 74.1%

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

   2. log1p-def 74.1%

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

   3. distribute-neg-frac 74.1%

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

   4. sub-neg 74.1%

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

   5. distribute-neg-in 74.1%

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

   6. remove-double-neg 74.1%

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

   7. +-commutative 74.1%

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

   8. sub-neg 74.1%

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

3. Simplified 74.1%

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

   1. add-log-exp 74.1%

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

   2. exp-diff 74.1%

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

   3. exp-1-e 74.1%

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

   4. log1p-udef 74.1%

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

   5. add-exp-log 74.1%

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

5. Applied egg-rr 74.1%

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

6. Taylor expanded in y around 0 83.7%

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

   1. +-commutative 83.7%

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

   2. mul-1-neg 83.7%

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

   3. unsub-neg 83.7%

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

   4. associate-/l* 83.7%

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

   5. *-commutative 83.7%

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

   6. associate-/r* 89.9%

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

   7. sub-neg 89.9%

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

   8. mul-1-neg 89.9%

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

   9. unpow2 89.9%

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

   10. associate-/l* 100.0%

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

   11. mul-1-neg 100.0%

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

   12. sub-neg 100.0%

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

   13. *-inverses 100.0%

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

8. Simplified 100.0%

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

9. Taylor expanded in y around 0 99.6%

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

    1. *-rgt-identity 99.6%

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

    2. mul-1-neg 99.6%

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

    3. associate-*l/ 100.0%

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

    4. distribute-rgt-neg-out 100.0%

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

    5. distribute-lft-out 100.0%

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

    6. unsub-neg 100.0%

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

11. Simplified 100.0%

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

12. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.998)
   (- 1.0 (log1p (* (/ 1.0 (- 1.0 y)) (- y x))))
   (log (/ E (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = log((((double) M_E) / ((x + -1.0) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - Math.log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = Math.log((Math.E / ((x + -1.0) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.998:
		tmp = 1.0 - math.log1p(((1.0 / (1.0 - y)) * (y - x)))
	else:
		tmp = math.log((math.e / ((x + -1.0) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.998)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 / Float64(1.0 - y)) * Float64(y - x))));
	else
		tmp = log(Float64(exp(1) / Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.998], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.998

   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 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. Step-by-step derivation

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 0.998 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 5.4%

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

      1. sub-neg 5.4%

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

      2. log1p-def 5.4%

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

      3. distribute-neg-frac 5.4%

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

      4. sub-neg 5.4%

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

      5. distribute-neg-in 5.4%

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

      6. remove-double-neg 5.4%

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

      7. +-commutative 5.4%

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

      8. sub-neg 5.4%

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

   3. Simplified 5.4%

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

      1. add-log-exp 5.4%

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

      2. exp-diff 5.4%

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

      3. exp-1-e 5.4%

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

      4. log1p-udef 5.4%

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

      5. add-exp-log 5.4%

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

   5. Applied egg-rr 5.4%

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

   6. Taylor expanded in y around -inf 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y}}\right)\\ \end{array} \]

Alternative 3: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999999999)
   (- 1.0 (log1p (* (/ 1.0 (- 1.0 y)) (- y x))))
   (+ 1.0 (- (/ -1.0 y) (log (/ -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999999999) {
		tmp = 1.0 - log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - log((-1.0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999999999) {
		tmp = 1.0 - Math.log1p(((1.0 / (1.0 - y)) * (y - x)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - Math.log((-1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999999999:
		tmp = 1.0 - math.log1p(((1.0 / (1.0 - y)) * (y - x)))
	else:
		tmp = 1.0 + ((-1.0 / y) - math.log((-1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999999999)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 / Float64(1.0 - y)) * Float64(y - x))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - log(Float64(-1.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999999999999], N[(1.0 - N[Log[1 + N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-1.0 / y), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999999999999000022

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. log1p-def 99.6%

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

      3. distribute-neg-frac 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-neg-in 99.6%

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

      6. remove-double-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. sub-neg 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. associate-/r/ 99.6%

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

   5. Applied egg-rr 99.6%

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

   if 0.999999999999000022 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 4.8%

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

      1. sub-neg 4.8%

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

      2. log1p-def 4.8%

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

      3. distribute-neg-frac 4.8%

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

      4. sub-neg 4.8%

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

      5. distribute-neg-in 4.8%

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

      6. remove-double-neg 4.8%

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

      7. +-commutative 4.8%

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

      8. sub-neg 4.8%

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

   3. Simplified 4.8%

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

   4. Taylor expanded in x around 0 4.8%

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

      1. log1p-def 4.8%

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

   6. Simplified 4.8%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. sub-neg 0.0%

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

      4. log-div 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{1}{1 - y} \cdot \left(y - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \]

Alternative 4: 90.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999999999999)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (+ 1.0 (- (/ -1.0 y) (log (/ -1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999999999) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - log((-1.0 / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999999999999) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 + ((-1.0 / y) - Math.log((-1.0 / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999999999999:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 + ((-1.0 / y) - math.log((-1.0 / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999999999999)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - log(Float64(-1.0 / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999999999999], 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[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999999999999000022

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. log1p-def 99.6%

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

      3. distribute-neg-frac 99.6%

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

      4. sub-neg 99.6%

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

      5. distribute-neg-in 99.6%

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

      6. remove-double-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. sub-neg 99.6%

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

   3. Simplified 99.6%

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

   if 0.999999999999000022 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 4.8%

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

      1. sub-neg 4.8%

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

      2. log1p-def 4.8%

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

      3. distribute-neg-frac 4.8%

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

      4. sub-neg 4.8%

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

      5. distribute-neg-in 4.8%

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

      6. remove-double-neg 4.8%

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

      7. +-commutative 4.8%

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

      8. sub-neg 4.8%

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

   3. Simplified 4.8%

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

   4. Taylor expanded in x around 0 4.8%

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

      1. log1p-def 4.8%

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

   6. Simplified 4.8%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. sub-neg 0.0%

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

      4. log-div 67.4%

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

   9. Simplified 67.4%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999999999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{-1}{y} - \log \left(\frac{-1}{y}\right)\right)\\ \end{array} \]

Alternative 5: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3900000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 50000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3900000000000.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 50000000000.0)
     (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
     (- 1.0 (log (/ x (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3900000000000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 50000000000.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3900000000000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 50000000000.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3900000000000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 50000000000.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3900000000000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 50000000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3900000000000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 50000000000.0], 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[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9e12

   1. Initial program 22.3%

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

      1. sub-neg 22.3%

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

      2. log1p-def 22.3%

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

      3. distribute-neg-frac 22.3%

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

      4. sub-neg 22.3%

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

      5. distribute-neg-in 22.3%

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

      6. remove-double-neg 22.3%

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

      7. +-commutative 22.3%

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

      8. sub-neg 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in x around 0 4.3%

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

      1. log1p-def 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 64.7%

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

   9. Simplified 64.7%

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

   if -3.9e12 < y < 5e10

   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 5e10 < y

   1. Initial program 60.3%

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

      1. sub-neg 60.3%

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

      2. log1p-def 60.3%

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

      3. distribute-neg-frac 60.3%

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

      4. sub-neg 60.3%

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

      5. distribute-neg-in 60.3%

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

      6. remove-double-neg 60.3%

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

      7. +-commutative 60.3%

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

      8. sub-neg 60.3%

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

   3. Simplified 60.3%

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

   4. Taylor expanded in x around inf 55.4%

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

      1. neg-mul-1 55.4%

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

      2. distribute-neg-frac 55.4%

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

   6. Simplified 55.4%

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

      1. frac-2neg 55.4%

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

      2. div-inv 55.4%

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

      3. remove-double-neg 55.4%

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

   8. Applied egg-rr 55.4%

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

      1. associate-*r/ 55.4%

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

      2. *-rgt-identity 55.4%

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

      3. neg-sub0 55.4%

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

      4. associate--r- 55.4%

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

      5. metadata-eval 55.4%

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

   10. Simplified 55.4%

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

   11. Taylor expanded in x around inf 97.9%

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

       1. log-rec 97.9%

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

       2. unsub-neg 97.9%

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

       3. mul-1-neg 97.9%

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

       4. log-rec 97.9%

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

       5. remove-double-neg 97.9%

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

       6. sub-neg 97.9%

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

       7. metadata-eval 97.9%

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

       8. log-div 99.7%

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

   13. Simplified 99.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3900000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 50000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 6: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -55:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.38:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -55.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 0.38)
     (- 1.0 (+ y (log1p (- x))))
     (- 1.0 (log (/ x (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -55.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 0.38) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -55.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 0.38) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -55.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 0.38:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -55.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 0.38)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -55.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.38], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -55

   1. Initial program 24.3%

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

      1. sub-neg 24.3%

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

      2. log1p-def 24.3%

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

      3. distribute-neg-frac 24.3%

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

      4. sub-neg 24.3%

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

      5. distribute-neg-in 24.3%

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

      6. remove-double-neg 24.3%

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

      7. +-commutative 24.3%

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

      8. sub-neg 24.3%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in x around 0 5.5%

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

      1. log1p-def 5.5%

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

   6. Simplified 5.5%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 63.4%

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

   9. Simplified 63.4%

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

   if -55 < y < 0.38

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-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 99.2%

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

      1. div-sub 99.2%

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

      2. mul-1-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. *-inverses 99.2%

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

      5. *-rgt-identity 99.2%

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

      6. log1p-def 99.2%

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

      7. mul-1-neg 99.2%

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

   6. Simplified 99.2%

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

   if 0.38 < y

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. log1p-def 61.8%

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

      3. distribute-neg-frac 61.8%

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

      4. sub-neg 61.8%

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

      5. distribute-neg-in 61.8%

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

      6. remove-double-neg 61.8%

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

      7. +-commutative 61.8%

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

      8. sub-neg 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around inf 54.5%

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

      1. neg-mul-1 54.5%

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

      2. distribute-neg-frac 54.5%

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

   6. Simplified 54.5%

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

      1. frac-2neg 54.5%

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

      2. div-inv 54.5%

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

      3. remove-double-neg 54.5%

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

   8. Applied egg-rr 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-rgt-identity 54.5%

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

      3. neg-sub0 54.5%

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

      4. associate--r- 54.5%

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

      5. metadata-eval 54.5%

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

   10. Simplified 54.5%

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

   11. Taylor expanded in x around inf 97.0%

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

       1. log-rec 97.0%

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

       2. unsub-neg 97.0%

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

       3. mul-1-neg 97.0%

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

       4. log-rec 97.0%

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

       5. remove-double-neg 97.0%

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

       6. sub-neg 97.0%

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

       7. metadata-eval 97.0%

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

       8. log-div 98.8%

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

   13. Simplified 98.8%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -55:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.38:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 7: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + -1}\\ \mathbf{if}\;y \leq -3800000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(t_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y -1.0))))
   (if (<= y -3800000000000.0)
     (- 1.0 (log (/ -1.0 y)))
     (if (<= y 1.0) (- 1.0 (log1p t_0)) (- 1.0 (log t_0))))))

C

double code(double x, double y) {
	double t_0 = x / (y + -1.0);
	double tmp;
	if (y <= -3800000000000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(t_0);
	} else {
		tmp = 1.0 - log(t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = x / (y + -1.0);
	double tmp;
	if (y <= -3800000000000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(t_0);
	} else {
		tmp = 1.0 - Math.log(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + -1.0)
	tmp = 0
	if y <= -3800000000000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(t_0)
	else:
		tmp = 1.0 - math.log(t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + -1.0))
	tmp = 0.0
	if (y <= -3800000000000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(t_0));
	else
		tmp = Float64(1.0 - log(t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3800000000000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e12

   1. Initial program 22.3%

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

      1. sub-neg 22.3%

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

      2. log1p-def 22.3%

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

      3. distribute-neg-frac 22.3%

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

      4. sub-neg 22.3%

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

      5. distribute-neg-in 22.3%

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

      6. remove-double-neg 22.3%

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

      7. +-commutative 22.3%

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

      8. sub-neg 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in x around 0 4.3%

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

      1. log1p-def 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 64.7%

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

   9. Simplified 64.7%

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

   if -3.8e12 < 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. Taylor expanded in x around inf 98.4%

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

      1. neg-mul-1 98.4%

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

      2. distribute-neg-frac 98.4%

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

   6. Simplified 98.4%

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

      1. frac-2neg 98.4%

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

      2. div-inv 98.4%

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

      3. remove-double-neg 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. associate-*r/ 98.4%

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

      2. *-rgt-identity 98.4%

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

      3. neg-sub0 98.4%

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

      4. associate--r- 98.4%

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

      5. metadata-eval 98.4%

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

   10. Simplified 98.4%

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

   if 1 < y

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. log1p-def 61.8%

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

      3. distribute-neg-frac 61.8%

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

      4. sub-neg 61.8%

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

      5. distribute-neg-in 61.8%

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

      6. remove-double-neg 61.8%

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

      7. +-commutative 61.8%

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

      8. sub-neg 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around inf 54.5%

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

      1. neg-mul-1 54.5%

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

      2. distribute-neg-frac 54.5%

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

   6. Simplified 54.5%

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

      1. frac-2neg 54.5%

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

      2. div-inv 54.5%

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

      3. remove-double-neg 54.5%

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

   8. Applied egg-rr 54.5%

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

      1. associate-*r/ 54.5%

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

      2. *-rgt-identity 54.5%

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

      3. neg-sub0 54.5%

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

      4. associate--r- 54.5%

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

      5. metadata-eval 54.5%

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

   10. Simplified 54.5%

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

   11. Taylor expanded in x around inf 97.0%

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

       1. log-rec 97.0%

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

       2. unsub-neg 97.0%

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

       3. mul-1-neg 97.0%

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

       4. log-rec 97.0%

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

       5. remove-double-neg 97.0%

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

       6. sub-neg 97.0%

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

       7. metadata-eval 97.0%

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

       8. log-div 98.8%

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

   13. Simplified 98.8%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3800000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 8: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -29:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -29.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -29.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log1p((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -29.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log1p((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -29.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log1p((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -29.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log1p(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -29.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -29

   1. Initial program 24.3%

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

      1. sub-neg 24.3%

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

      2. log1p-def 24.3%

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

      3. distribute-neg-frac 24.3%

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

      4. sub-neg 24.3%

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

      5. distribute-neg-in 24.3%

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

      6. remove-double-neg 24.3%

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

      7. +-commutative 24.3%

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

      8. sub-neg 24.3%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in x around 0 5.5%

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

      1. log1p-def 5.5%

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

   6. Simplified 5.5%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 63.4%

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

   9. Simplified 63.4%

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

   if -29 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-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 99.2%

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

      1. div-sub 99.2%

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

      2. mul-1-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. *-inverses 99.2%

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

      5. *-rgt-identity 99.2%

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

      6. log1p-def 99.2%

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

      7. mul-1-neg 99.2%

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

   6. Simplified 99.2%

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

   if 1 < y

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. log1p-def 61.8%

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

      3. distribute-neg-frac 61.8%

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

      4. sub-neg 61.8%

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

      5. distribute-neg-in 61.8%

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

      6. remove-double-neg 61.8%

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

      7. +-commutative 61.8%

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

      8. sub-neg 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around inf 54.5%

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

      1. neg-mul-1 54.5%

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

      2. distribute-neg-frac 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in y around inf 54.1%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -29:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 9: 84.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3800000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3800000000000.0)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (- 1.0 (log1p (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3800000000000.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		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 <= -3800000000000.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		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 <= -3800000000000.0:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		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 <= -3800000000000.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		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, -3800000000000.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], 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 < -3.8e12

   1. Initial program 22.3%

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

      1. sub-neg 22.3%

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

      2. log1p-def 22.3%

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

      3. distribute-neg-frac 22.3%

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

      4. sub-neg 22.3%

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

      5. distribute-neg-in 22.3%

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

      6. remove-double-neg 22.3%

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

      7. +-commutative 22.3%

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

      8. sub-neg 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in x around 0 4.3%

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

      1. log1p-def 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 64.7%

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

   9. Simplified 64.7%

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

   if -3.8e12 < 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. Taylor expanded in y around 0 96.9%

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

      1. log1p-def 96.9%

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

      2. mul-1-neg 96.9%

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

   6. Simplified 96.9%

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

   if 1 < y

   1. Initial program 61.8%

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

      1. sub-neg 61.8%

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

      2. log1p-def 61.8%

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

      3. distribute-neg-frac 61.8%

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

      4. sub-neg 61.8%

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

      5. distribute-neg-in 61.8%

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

      6. remove-double-neg 61.8%

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

      7. +-commutative 61.8%

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

      8. sub-neg 61.8%

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

   3. Simplified 61.8%

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

   4. Taylor expanded in x around inf 54.5%

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

      1. neg-mul-1 54.5%

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

      2. distribute-neg-frac 54.5%

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

   6. Simplified 54.5%

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

   7. Taylor expanded in y around inf 54.1%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3800000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 10: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3800000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3800000000000.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3800000000000.0) {
		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 <= -3800000000000.0) {
		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 <= -3800000000000.0:
		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 <= -3800000000000.0)
		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, -3800000000000.0], 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 < -3.8e12

   1. Initial program 22.3%

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

      1. sub-neg 22.3%

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

      2. log1p-def 22.3%

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

      3. distribute-neg-frac 22.3%

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

      4. sub-neg 22.3%

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

      5. distribute-neg-in 22.3%

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

      6. remove-double-neg 22.3%

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

      7. +-commutative 22.3%

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

      8. sub-neg 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in x around 0 4.3%

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

      1. log1p-def 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. log-rec 0.0%

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

      2. sub-neg 0.0%

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

      3. log-div 64.7%

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

   9. Simplified 64.7%

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

   if -3.8e12 < y

   1. Initial program 94.3%

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

      1. sub-neg 94.3%

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

      2. log1p-def 94.4%

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

      3. distribute-neg-frac 94.4%

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

      4. sub-neg 94.4%

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

      5. distribute-neg-in 94.4%

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

      6. remove-double-neg 94.4%

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

      7. +-commutative 94.4%

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

      8. sub-neg 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in y around 0 82.7%

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

      1. log1p-def 82.7%

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

      2. mul-1-neg 82.7%

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

   6. Simplified 82.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3800000000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 11: 61.8% 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 74.1%

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

   1. sub-neg 74.1%

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

   2. log1p-def 74.1%

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

   3. distribute-neg-frac 74.1%

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

   4. sub-neg 74.1%

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

   5. distribute-neg-in 74.1%

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

   6. remove-double-neg 74.1%

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

   7. +-commutative 74.1%

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

   8. sub-neg 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in y around 0 62.9%

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

   1. log1p-def 63.0%

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

   2. mul-1-neg 63.0%

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

6. Simplified 63.0%

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

7. Final simplification 63.0%

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

Alternative 12: 44.0% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3800000000000:\\ \;\;\;\;1 + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3800000000000.0) (+ 1.0 (/ -1.0 y)) (+ 1.0 (- x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3800000000000.0) {
		tmp = 1.0 + (-1.0 / y);
	} else {
		tmp = 1.0 + (x - y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3800000000000.0d0)) then
        tmp = 1.0d0 + ((-1.0d0) / y)
    else
        tmp = 1.0d0 + (x - y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3800000000000.0) {
		tmp = 1.0 + (-1.0 / y);
	} else {
		tmp = 1.0 + (x - y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3800000000000.0:
		tmp = 1.0 + (-1.0 / y)
	else:
		tmp = 1.0 + (x - y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3800000000000.0)
		tmp = Float64(1.0 + Float64(-1.0 / y));
	else
		tmp = Float64(1.0 + Float64(x - y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3800000000000.0)
		tmp = 1.0 + (-1.0 / y);
	else
		tmp = 1.0 + (x - y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3800000000000.0], N[(1.0 + N[(-1.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x - y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8e12

   1. Initial program 22.3%

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

      1. sub-neg 22.3%

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

      2. log1p-def 22.3%

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

      3. distribute-neg-frac 22.3%

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

      4. sub-neg 22.3%

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

      5. distribute-neg-in 22.3%

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

      6. remove-double-neg 22.3%

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

      7. +-commutative 22.3%

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

      8. sub-neg 22.3%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in x around 0 4.3%

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

      1. log1p-def 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in y around inf 0.0%

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

      1. +-commutative 0.0%

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

      2. log-rec 0.0%

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

      3. sub-neg 0.0%

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

      4. log-div 64.8%

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

   9. Simplified 64.8%

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

   10. Taylor expanded in y around 0 11.9%

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

   if -3.8e12 < y

   1. Initial program 94.3%

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

      1. sub-neg 94.3%

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

      2. log1p-def 94.4%

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

      3. distribute-neg-frac 94.4%

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

      4. sub-neg 94.4%

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

      5. distribute-neg-in 94.4%

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

      6. remove-double-neg 94.4%

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

      7. +-commutative 94.4%

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

      8. sub-neg 94.4%

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

   3. Simplified 94.4%

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

   4. Taylor expanded in x around 0 61.0%

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

      1. +-commutative 61.0%

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

      2. mul-1-neg 61.0%

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

      3. unsub-neg 61.0%

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

      4. log1p-def 61.0%

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

      5. *-commutative 61.0%

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

   6. Simplified 61.0%

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

   7. Taylor expanded in y around 0 60.7%

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

      1. mul-1-neg 60.7%

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

      2. sub-neg 60.7%

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

   9. Simplified 60.7%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3800000000000:\\ \;\;\;\;1 + \frac{-1}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x - y\right)\\ \end{array} \]

Alternative 13: 41.8% accurate, 22.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. sub-neg 74.1%

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

   2. log1p-def 74.1%

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

   3. distribute-neg-frac 74.1%

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

   4. sub-neg 74.1%

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

   5. distribute-neg-in 74.1%

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

   6. remove-double-neg 74.1%

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

   7. +-commutative 74.1%

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

   8. sub-neg 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in x around 0 44.6%

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

   1. +-commutative 44.6%

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

   2. mul-1-neg 44.6%

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

   3. unsub-neg 44.6%

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

   4. log1p-def 44.6%

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

   5. *-commutative 44.6%

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

6. Simplified 44.6%

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

7. Taylor expanded in y around 0 45.0%

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

   1. mul-1-neg 45.0%

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

   2. sub-neg 45.0%

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

9. Simplified 45.0%

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

10. Final simplification 45.0%

    \[\leadsto 1 + \left(x - y\right) \]

Alternative 14: 40.8% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. sub-neg 74.1%

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

   2. log1p-def 74.1%

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

   3. distribute-neg-frac 74.1%

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

   4. sub-neg 74.1%

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

   5. distribute-neg-in 74.1%

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

   6. remove-double-neg 74.1%

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

   7. +-commutative 74.1%

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

   8. sub-neg 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in x around 0 43.9%

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

   1. log1p-def 43.9%

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

6. Simplified 43.9%

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

7. Taylor expanded in y around 0 43.8%

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

8. Final simplification 43.8%

   \[\leadsto 1 - y \]

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 2023279 
(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))))))