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

Percentage Accurate: 71.8% → 98.8%

Time: 10.6s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y \cdot y} + \frac{x + -1}{y}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 2e-9)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (log (/ E (+ (/ (+ x -1.0) (* y y)) (/ (+ x -1.0) y))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 2e-9) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = log((((double) M_E) / (((x + -1.0) / (y * y)) + ((x + -1.0) / y))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 2e-9) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = Math.log((Math.E / (((x + -1.0) / (y * y)) + ((x + -1.0) / y))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 2e-9:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = math.log((math.e / (((x + -1.0) / (y * y)) + ((x + -1.0) / y))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 2e-9)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = log(Float64(exp(1) / Float64(Float64(Float64(x + -1.0) / Float64(y * y)) + 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], 2e-9], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[(E / N[(N[(N[(x + -1.0), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 2.00000000000000012e-9

   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. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   if 2.00000000000000012e-9 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 7.2%

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

      1. sub-neg 7.2%

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

      2. log1p-def 7.2%

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

      3. neg-sub0 7.2%

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

      4. div-sub 7.2%

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

      5. associate--r- 7.2%

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

      6. neg-sub0 7.2%

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

      7. +-commutative 7.2%

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

      8. sub-neg 7.2%

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

      9. div-sub 7.2%

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

   3. Simplified 7.2%

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

      1. add-log-exp 7.2%

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

      2. exp-diff 7.2%

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

      3. exp-1-e 7.2%

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

      4. log1p-udef 7.2%

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

      5. add-exp-log 7.2%

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

   5. Applied egg-rr 7.2%

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

   6. Taylor expanded in y around -inf 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. associate-*r/ 99.9%

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

      5. sub-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. mul-1-neg 99.9%

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

      8. neg-mul-1 99.9%

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

      9. sub-neg 99.9%

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

      10. unpow2 99.9%

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

      11. sub-neg 99.9%

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

      12. div-sub 99.9%

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

      13. sub-neg 99.9%

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

      14. metadata-eval 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 2 \cdot 10^{-9}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e}{\frac{x + -1}{y \cdot y} + \frac{x + -1}{y}}\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.9999999)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9999999) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.9999999) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.9999999:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.9999999)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(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], 0.9999999], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999999900000000053

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. log1p-def 99.8%

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

      3. neg-sub0 99.8%

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

      4. div-sub 99.8%

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

      5. associate--r- 99.8%

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

      6. neg-sub0 99.8%

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

      7. +-commutative 99.8%

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

      8. sub-neg 99.8%

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

      9. div-sub 99.8%

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

   3. Simplified 99.8%

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

   if 0.999999900000000053 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 5.3%

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

      1. sub-neg 5.3%

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

      2. log1p-def 5.3%

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

      3. neg-sub0 5.3%

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

      4. div-sub 5.3%

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

      5. associate--r- 5.3%

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

      6. neg-sub0 5.3%

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

      7. +-commutative 5.3%

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

      8. sub-neg 5.3%

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

      9. div-sub 5.3%

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

   3. Simplified 5.3%

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

   4. Taylor expanded in y around inf 5.2%

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

      1. *-un-lft-identity 5.2%

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

      2. associate--r+ 5.2%

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

      3. sub-neg 5.2%

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

      4. sub-div 5.2%

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

      5. metadata-eval 5.2%

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

   6. Applied egg-rr 5.2%

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

      1. *-lft-identity 5.2%

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

      2. log1p-def 5.2%

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

      3. +-commutative 5.2%

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

      4. associate-+l+ 99.6%

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

      5. sub-neg 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   8. Simplified 99.6%

      \[\leadsto 1 - \color{blue}{\log \left(\frac{x + -1}{y} + 0\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.9999999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternative 3: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -22000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 880000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))) (t_1 (- 1.0 (log (/ x y)))))
   (if (<= y -4.5e+180)
     t_0
     (if (<= y -1.25e+109)
       t_1
       (if (<= y -22000.0)
         t_0
         (if (<= y 880000000000.0) (- 1.0 (log1p (/ x (+ y -1.0)))) t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((-1.0 / y));
	double t_1 = 1.0 - log((x / y));
	double tmp;
	if (y <= -4.5e+180) {
		tmp = t_0;
	} else if (y <= -1.25e+109) {
		tmp = t_1;
	} else if (y <= -22000.0) {
		tmp = t_0;
	} else if (y <= 880000000000.0) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((-1.0 / y));
	double t_1 = 1.0 - Math.log((x / y));
	double tmp;
	if (y <= -4.5e+180) {
		tmp = t_0;
	} else if (y <= -1.25e+109) {
		tmp = t_1;
	} else if (y <= -22000.0) {
		tmp = t_0;
	} else if (y <= 880000000000.0) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((-1.0 / y))
	t_1 = 1.0 - math.log((x / y))
	tmp = 0
	if y <= -4.5e+180:
		tmp = t_0
	elif y <= -1.25e+109:
		tmp = t_1
	elif y <= -22000.0:
		tmp = t_0
	elif y <= 880000000000.0:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(-1.0 / y)))
	t_1 = Float64(1.0 - log(Float64(x / y)))
	tmp = 0.0
	if (y <= -4.5e+180)
		tmp = t_0;
	elseif (y <= -1.25e+109)
		tmp = t_1;
	elseif (y <= -22000.0)
		tmp = t_0;
	elseif (y <= 880000000000.0)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.5e+180], t$95$0, If[LessEqual[y, -1.25e+109], t$95$1, If[LessEqual[y, -22000.0], t$95$0, If[LessEqual[y, 880000000000.0], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.49999999999999981e180 or -1.25e109 < y < -22000

   1. Initial program 17.2%

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

      1. sub-neg 17.2%

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

      2. log1p-def 17.2%

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

      3. neg-sub0 17.2%

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

      4. div-sub 17.2%

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

      5. associate--r- 17.2%

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

      6. neg-sub0 17.2%

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

      7. +-commutative 17.2%

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

      8. sub-neg 17.2%

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

      9. div-sub 17.2%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in y around inf 16.9%

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

   5. Taylor expanded in x around 0 68.5%

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

      1. distribute-neg-frac 68.5%

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

      2. metadata-eval 68.5%

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

   7. Simplified 68.5%

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

   if -4.49999999999999981e180 < y < -1.25e109 or 8.8e11 < y

   1. Initial program 38.0%

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

      1. sub-neg 38.0%

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

      2. log1p-def 38.0%

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

      3. neg-sub0 38.0%

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

      4. div-sub 38.1%

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

      5. associate--r- 38.1%

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

      6. neg-sub0 38.1%

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

      7. +-commutative 38.1%

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

      8. sub-neg 38.1%

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

      9. div-sub 38.0%

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

   3. Simplified 38.0%

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

   4. Taylor expanded in y around -inf 35.6%

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

      1. sub-neg 35.6%

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

      2. metadata-eval 35.6%

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

      3. distribute-lft-in 35.6%

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

      4. metadata-eval 35.6%

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

      5. +-commutative 35.6%

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

      6. log1p-def 35.6%

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

      7. mul-1-neg 35.6%

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

   6. Simplified 35.6%

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

      1. add-log-exp 35.6%

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

      2. exp-diff 35.6%

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

      3. exp-1-e 35.6%

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

      4. exp-sum 35.6%

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

      5. neg-mul-1 35.6%

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

      6. log1p-def 35.6%

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

      7. add-exp-log 35.6%

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

      8. add-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

      10. +-commutative 100.0%

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

      11. add-sqr-sqrt 30.8%

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

      12. sqrt-unprod 28.9%

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

      13. sqr-neg 28.9%

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

      14. sqrt-unprod 5.1%

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

      15. add-sqr-sqrt 10.3%

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

      16. add-sqr-sqrt 10.3%

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

      17. sqrt-unprod 38.0%

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

      18. frac-times 36.8%

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

      19. metadata-eval 36.8%

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

      20. metadata-eval 36.8%

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

      21. frac-times 38.0%

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

      22. sqrt-unprod 59.1%

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

      23. add-sqr-sqrt 84.7%

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

   8. Applied egg-rr 84.7%

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

   9. Taylor expanded in x around inf 85.5%

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

       1. *-un-lft-identity 85.5%

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

       2. log-prod 85.5%

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

       3. metadata-eval 85.5%

          \[\leadsto \color{blue}{0} + \log \left(\frac{e \cdot y}{x}\right) \]

       4. associate-/l* 85.4%

          \[\leadsto 0 + \log \color{blue}{\left(\frac{e}{\frac{x}{y}}\right)} \]

       5. log-div 85.5%

          \[\leadsto 0 + \color{blue}{\left(\log e - \log \left(\frac{x}{y}\right)\right)} \]

       6. e-exp-1 85.5%

          \[\leadsto 0 + \left(\log \color{blue}{\left(e^{1}\right)} - \log \left(\frac{x}{y}\right)\right) \]

       7. add-log-exp 85.5%

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

   11. Applied egg-rr 85.5%

       \[\leadsto \color{blue}{0 + \left(1 - \log \left(\frac{x}{y}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lft-identity 85.5%

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

   13. Simplified 85.5%

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

   if -22000 < y < 8.8e11

   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. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. distribute-neg-frac 98.7%

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

   6. Simplified 98.7%

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

      1. frac-2neg 98.7%

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

      2. div-inv 98.7%

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

      3. remove-double-neg 98.7%

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

   8. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.7%

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

      2. *-rgt-identity 98.7%

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

      3. sub-neg 98.7%

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

      4. distribute-neg-in 98.7%

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

      5. metadata-eval 98.7%

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

      6. remove-double-neg 98.7%

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

   10. Simplified 98.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+180}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -22000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 880000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 4: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+180}:\\ \;\;\;\;1 - t_0\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4800000:\\ \;\;\;\;\left(x + 1\right) - t_0\\ \mathbf{elif}\;y \leq 105000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (log (/ -1.0 y))) (t_1 (- 1.0 (log (/ x y)))))
   (if (<= y -4.5e+180)
     (- 1.0 t_0)
     (if (<= y -1.25e+109)
       t_1
       (if (<= y -4800000.0)
         (- (+ x 1.0) t_0)
         (if (<= y 105000000000.0) (- 1.0 (log1p (/ x (+ y -1.0)))) t_1))))))

C

double code(double x, double y) {
	double t_0 = log((-1.0 / y));
	double t_1 = 1.0 - log((x / y));
	double tmp;
	if (y <= -4.5e+180) {
		tmp = 1.0 - t_0;
	} else if (y <= -1.25e+109) {
		tmp = t_1;
	} else if (y <= -4800000.0) {
		tmp = (x + 1.0) - t_0;
	} else if (y <= 105000000000.0) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = Math.log((-1.0 / y));
	double t_1 = 1.0 - Math.log((x / y));
	double tmp;
	if (y <= -4.5e+180) {
		tmp = 1.0 - t_0;
	} else if (y <= -1.25e+109) {
		tmp = t_1;
	} else if (y <= -4800000.0) {
		tmp = (x + 1.0) - t_0;
	} else if (y <= 105000000000.0) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.log((-1.0 / y))
	t_1 = 1.0 - math.log((x / y))
	tmp = 0
	if y <= -4.5e+180:
		tmp = 1.0 - t_0
	elif y <= -1.25e+109:
		tmp = t_1
	elif y <= -4800000.0:
		tmp = (x + 1.0) - t_0
	elif y <= 105000000000.0:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = log(Float64(-1.0 / y))
	t_1 = Float64(1.0 - log(Float64(x / y)))
	tmp = 0.0
	if (y <= -4.5e+180)
		tmp = Float64(1.0 - t_0);
	elseif (y <= -1.25e+109)
		tmp = t_1;
	elseif (y <= -4800000.0)
		tmp = Float64(Float64(x + 1.0) - t_0);
	elseif (y <= 105000000000.0)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.5e+180], N[(1.0 - t$95$0), $MachinePrecision], If[LessEqual[y, -1.25e+109], t$95$1, If[LessEqual[y, -4800000.0], N[(N[(x + 1.0), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[y, 105000000000.0], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.49999999999999981e180

   1. Initial program 13.6%

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

      1. sub-neg 13.6%

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

      2. log1p-def 13.6%

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

      3. neg-sub0 13.6%

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

      4. div-sub 13.6%

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

      5. associate--r- 13.6%

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

      6. neg-sub0 13.6%

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

      7. +-commutative 13.6%

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

      8. sub-neg 13.6%

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

      9. div-sub 13.6%

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

   3. Simplified 13.6%

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

   4. Taylor expanded in y around inf 13.6%

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

   5. Taylor expanded in x around 0 61.3%

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

      1. distribute-neg-frac 61.3%

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

      2. metadata-eval 61.3%

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

   7. Simplified 61.3%

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

   if -4.49999999999999981e180 < y < -1.25e109 or 1.05e11 < y

   1. Initial program 38.0%

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

      1. sub-neg 38.0%

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

      2. log1p-def 38.0%

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

      3. neg-sub0 38.0%

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

      4. div-sub 38.1%

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

      5. associate--r- 38.1%

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

      6. neg-sub0 38.1%

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

      7. +-commutative 38.1%

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

      8. sub-neg 38.1%

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

      9. div-sub 38.0%

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

   3. Simplified 38.0%

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

   4. Taylor expanded in y around -inf 35.6%

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

      1. sub-neg 35.6%

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

      2. metadata-eval 35.6%

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

      3. distribute-lft-in 35.6%

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

      4. metadata-eval 35.6%

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

      5. +-commutative 35.6%

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

      6. log1p-def 35.6%

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

      7. mul-1-neg 35.6%

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

   6. Simplified 35.6%

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

      1. add-log-exp 35.6%

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

      2. exp-diff 35.6%

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

      3. exp-1-e 35.6%

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

      4. exp-sum 35.6%

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

      5. neg-mul-1 35.6%

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

      6. log1p-def 35.6%

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

      7. add-exp-log 35.6%

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

      8. add-exp-log 100.0%

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

      9. neg-mul-1 100.0%

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

      10. +-commutative 100.0%

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

      11. add-sqr-sqrt 30.8%

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

      12. sqrt-unprod 28.9%

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

      13. sqr-neg 28.9%

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

      14. sqrt-unprod 5.1%

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

      15. add-sqr-sqrt 10.3%

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

      16. add-sqr-sqrt 10.3%

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

      17. sqrt-unprod 38.0%

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

      18. frac-times 36.8%

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

      19. metadata-eval 36.8%

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

      20. metadata-eval 36.8%

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

      21. frac-times 38.0%

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

      22. sqrt-unprod 59.1%

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

      23. add-sqr-sqrt 84.7%

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

   8. Applied egg-rr 84.7%

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

   9. Taylor expanded in x around inf 85.5%

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

       1. *-un-lft-identity 85.5%

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

       2. log-prod 85.5%

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

       3. metadata-eval 85.5%

          \[\leadsto \color{blue}{0} + \log \left(\frac{e \cdot y}{x}\right) \]

       4. associate-/l* 85.4%

          \[\leadsto 0 + \log \color{blue}{\left(\frac{e}{\frac{x}{y}}\right)} \]

       5. log-div 85.5%

          \[\leadsto 0 + \color{blue}{\left(\log e - \log \left(\frac{x}{y}\right)\right)} \]

       6. e-exp-1 85.5%

          \[\leadsto 0 + \left(\log \color{blue}{\left(e^{1}\right)} - \log \left(\frac{x}{y}\right)\right) \]

       7. add-log-exp 85.5%

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

   11. Applied egg-rr 85.5%

       \[\leadsto \color{blue}{0 + \left(1 - \log \left(\frac{x}{y}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lft-identity 85.5%

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

   13. Simplified 85.5%

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

   if -1.25e109 < y < -4.8e6

   1. Initial program 22.0%

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

      1. sub-neg 22.0%

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

      2. log1p-def 22.0%

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

      3. neg-sub0 22.0%

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

      4. div-sub 22.0%

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

      5. associate--r- 22.0%

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

      6. neg-sub0 22.0%

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

      7. +-commutative 22.0%

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

      8. sub-neg 22.0%

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

      9. div-sub 22.0%

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

   3. Simplified 22.0%

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

   4. Taylor expanded in y around inf 21.2%

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

   5. Taylor expanded in x around 0 78.5%

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

      1. +-commutative 78.5%

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

      2. distribute-neg-frac 78.5%

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

      3. metadata-eval 78.5%

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

   7. Simplified 78.5%

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

   if -4.8e6 < y < 1.05e11

   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. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. distribute-neg-frac 98.7%

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

   6. Simplified 98.7%

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

      1. frac-2neg 98.7%

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

      2. div-inv 98.7%

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

      3. remove-double-neg 98.7%

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

   8. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.7%

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

      2. *-rgt-identity 98.7%

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

      3. sub-neg 98.7%

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

      4. distribute-neg-in 98.7%

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

      5. metadata-eval 98.7%

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

      6. remove-double-neg 98.7%

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

   10. Simplified 98.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+180}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -4800000:\\ \;\;\;\;\left(x + 1\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 105000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 5: 86.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -31:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))) (t_1 (- 1.0 (log (/ x y)))))
   (if (<= y -5.8e+180)
     t_0
     (if (<= y -1.25e+109)
       t_1
       (if (<= y -31.0)
         t_0
         (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((-1.0 / y));
	double t_1 = 1.0 - log((x / y));
	double tmp;
	if (y <= -5.8e+180) {
		tmp = t_0;
	} else if (y <= -1.25e+109) {
		tmp = t_1;
	} else if (y <= -31.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((-1.0 / y));
	double t_1 = 1.0 - Math.log((x / y));
	double tmp;
	if (y <= -5.8e+180) {
		tmp = t_0;
	} else if (y <= -1.25e+109) {
		tmp = t_1;
	} else if (y <= -31.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((-1.0 / y))
	t_1 = 1.0 - math.log((x / y))
	tmp = 0
	if y <= -5.8e+180:
		tmp = t_0
	elif y <= -1.25e+109:
		tmp = t_1
	elif y <= -31.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(-1.0 / y)))
	t_1 = Float64(1.0 - log(Float64(x / y)))
	tmp = 0.0
	if (y <= -5.8e+180)
		tmp = t_0;
	elseif (y <= -1.25e+109)
		tmp = t_1;
	elseif (y <= -31.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e+180], t$95$0, If[LessEqual[y, -1.25e+109], t$95$1, If[LessEqual[y, -31.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.80000000000000015e180 or -1.25e109 < y < -31

   1. Initial program 18.5%

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

      1. sub-neg 18.5%

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

      2. log1p-def 18.5%

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

      3. neg-sub0 18.5%

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

      4. div-sub 18.5%

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

      5. associate--r- 18.5%

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

      6. neg-sub0 18.5%

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

      7. +-commutative 18.5%

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

      8. sub-neg 18.5%

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

      9. div-sub 18.5%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in y around inf 17.3%

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

   5. Taylor expanded in x around 0 67.5%

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

      1. distribute-neg-frac 67.5%

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

      2. metadata-eval 67.5%

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

   7. Simplified 67.5%

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

   if -5.80000000000000015e180 < y < -1.25e109 or 1 < y

   1. Initial program 39.5%

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

      1. sub-neg 39.5%

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

      2. log1p-def 39.5%

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

      3. neg-sub0 39.5%

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

      4. div-sub 39.6%

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

      5. associate--r- 39.6%

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

      6. neg-sub0 39.6%

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

      7. +-commutative 39.6%

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

      8. sub-neg 39.6%

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

      9. div-sub 39.5%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in y around -inf 34.7%

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

      1. sub-neg 34.7%

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

      2. metadata-eval 34.7%

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

      3. distribute-lft-in 34.7%

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

      4. metadata-eval 34.7%

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

      5. +-commutative 34.7%

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

      6. log1p-def 34.7%

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

      7. mul-1-neg 34.7%

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

   6. Simplified 34.7%

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

      1. add-log-exp 34.7%

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

      2. exp-diff 34.7%

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

      3. exp-1-e 34.7%

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

      4. exp-sum 34.7%

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

      5. neg-mul-1 34.7%

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

      6. log1p-def 34.7%

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

      7. add-exp-log 34.7%

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

      8. add-exp-log 98.9%

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

      9. neg-mul-1 98.9%

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

      10. +-commutative 98.9%

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

      11. add-sqr-sqrt 30.0%

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

      12. sqrt-unprod 28.2%

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

      13. sqr-neg 28.2%

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

      14. sqrt-unprod 5.0%

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

      15. add-sqr-sqrt 10.0%

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

      16. add-sqr-sqrt 10.0%

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

      17. sqrt-unprod 38.5%

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

      18. frac-times 37.3%

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

      19. metadata-eval 37.3%

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

      20. metadata-eval 37.3%

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

      21. frac-times 38.5%

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

      22. sqrt-unprod 59.1%

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

      23. add-sqr-sqrt 84.0%

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

   8. Applied egg-rr 84.0%

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

   9. Taylor expanded in x around inf 84.8%

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

       1. *-un-lft-identity 84.8%

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

       2. log-prod 84.8%

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

       3. metadata-eval 84.8%

          \[\leadsto \color{blue}{0} + \log \left(\frac{e \cdot y}{x}\right) \]

       4. associate-/l* 84.8%

          \[\leadsto 0 + \log \color{blue}{\left(\frac{e}{\frac{x}{y}}\right)} \]

       5. log-div 84.8%

          \[\leadsto 0 + \color{blue}{\left(\log e - \log \left(\frac{x}{y}\right)\right)} \]

       6. e-exp-1 84.8%

          \[\leadsto 0 + \left(\log \color{blue}{\left(e^{1}\right)} - \log \left(\frac{x}{y}\right)\right) \]

       7. add-log-exp 84.8%

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

   11. Applied egg-rr 84.8%

       \[\leadsto \color{blue}{0 + \left(1 - \log \left(\frac{x}{y}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lft-identity 84.8%

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

   13. Simplified 84.8%

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

   if -31 < 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. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 98.7%

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

      1. div-sub 98.7%

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

      2. mul-1-neg 98.7%

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

      3. sub-neg 98.7%

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

      4. *-inverses 98.7%

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

      5. *-rgt-identity 98.7%

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

      6. log1p-def 98.7%

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

      7. mul-1-neg 98.7%

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

   6. Simplified 98.7%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+180}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -31:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 6: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -4.5 \cdot 10^{+180}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1050:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))) (t_1 (- 1.0 (log (/ x y)))))
   (if (<= y -4.5e+180)
     t_0
     (if (<= y -1.25e+109)
       t_1
       (if (<= y -1050.0) t_0 (if (<= y 1.0) (- 1.0 (log1p (- x))) t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((-1.0 / y));
	double t_1 = 1.0 - log((x / y));
	double tmp;
	if (y <= -4.5e+180) {
		tmp = t_0;
	} else if (y <= -1.25e+109) {
		tmp = t_1;
	} else if (y <= -1050.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((-1.0 / y));
	double t_1 = 1.0 - Math.log((x / y));
	double tmp;
	if (y <= -4.5e+180) {
		tmp = t_0;
	} else if (y <= -1.25e+109) {
		tmp = t_1;
	} else if (y <= -1050.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((-1.0 / y))
	t_1 = 1.0 - math.log((x / y))
	tmp = 0
	if y <= -4.5e+180:
		tmp = t_0
	elif y <= -1.25e+109:
		tmp = t_1
	elif y <= -1050.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(-1.0 / y)))
	t_1 = Float64(1.0 - log(Float64(x / y)))
	tmp = 0.0
	if (y <= -4.5e+180)
		tmp = t_0;
	elseif (y <= -1.25e+109)
		tmp = t_1;
	elseif (y <= -1050.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.5e+180], t$95$0, If[LessEqual[y, -1.25e+109], t$95$1, If[LessEqual[y, -1050.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.49999999999999981e180 or -1.25e109 < y < -1050

   1. Initial program 17.2%

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

      1. sub-neg 17.2%

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

      2. log1p-def 17.2%

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

      3. neg-sub0 17.2%

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

      4. div-sub 17.2%

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

      5. associate--r- 17.2%

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

      6. neg-sub0 17.2%

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

      7. +-commutative 17.2%

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

      8. sub-neg 17.2%

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

      9. div-sub 17.2%

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

   3. Simplified 17.2%

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

   4. Taylor expanded in y around inf 16.9%

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

   5. Taylor expanded in x around 0 68.5%

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

      1. distribute-neg-frac 68.5%

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

      2. metadata-eval 68.5%

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

   7. Simplified 68.5%

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

   if -4.49999999999999981e180 < y < -1.25e109 or 1 < y

   1. Initial program 39.5%

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

      1. sub-neg 39.5%

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

      2. log1p-def 39.5%

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

      3. neg-sub0 39.5%

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

      4. div-sub 39.6%

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

      5. associate--r- 39.6%

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

      6. neg-sub0 39.6%

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

      7. +-commutative 39.6%

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

      8. sub-neg 39.6%

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

      9. div-sub 39.5%

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

   3. Simplified 39.5%

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

   4. Taylor expanded in y around -inf 34.7%

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

      1. sub-neg 34.7%

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

      2. metadata-eval 34.7%

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

      3. distribute-lft-in 34.7%

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

      4. metadata-eval 34.7%

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

      5. +-commutative 34.7%

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

      6. log1p-def 34.7%

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

      7. mul-1-neg 34.7%

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

   6. Simplified 34.7%

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

      1. add-log-exp 34.7%

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

      2. exp-diff 34.7%

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

      3. exp-1-e 34.7%

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

      4. exp-sum 34.7%

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

      5. neg-mul-1 34.7%

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

      6. log1p-def 34.7%

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

      7. add-exp-log 34.7%

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

      8. add-exp-log 98.9%

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

      9. neg-mul-1 98.9%

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

      10. +-commutative 98.9%

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

      11. add-sqr-sqrt 30.0%

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

      12. sqrt-unprod 28.2%

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

      13. sqr-neg 28.2%

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

      14. sqrt-unprod 5.0%

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

      15. add-sqr-sqrt 10.0%

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

      16. add-sqr-sqrt 10.0%

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

      17. sqrt-unprod 38.5%

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

      18. frac-times 37.3%

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

      19. metadata-eval 37.3%

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

      20. metadata-eval 37.3%

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

      21. frac-times 38.5%

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

      22. sqrt-unprod 59.1%

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

      23. add-sqr-sqrt 84.0%

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

   8. Applied egg-rr 84.0%

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

   9. Taylor expanded in x around inf 84.8%

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

       1. *-un-lft-identity 84.8%

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

       2. log-prod 84.8%

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

       3. metadata-eval 84.8%

          \[\leadsto \color{blue}{0} + \log \left(\frac{e \cdot y}{x}\right) \]

       4. associate-/l* 84.8%

          \[\leadsto 0 + \log \color{blue}{\left(\frac{e}{\frac{x}{y}}\right)} \]

       5. log-div 84.8%

          \[\leadsto 0 + \color{blue}{\left(\log e - \log \left(\frac{x}{y}\right)\right)} \]

       6. e-exp-1 84.8%

          \[\leadsto 0 + \left(\log \color{blue}{\left(e^{1}\right)} - \log \left(\frac{x}{y}\right)\right) \]

       7. add-log-exp 84.8%

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

   11. Applied egg-rr 84.8%

       \[\leadsto \color{blue}{0 + \left(1 - \log \left(\frac{x}{y}\right)\right)} \]
   12. Step-by-step derivation

       1. +-lft-identity 84.8%

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

   13. Simplified 84.8%

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

   if -1050 < 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. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.2%

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

      1. log1p-def 97.2%

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

      2. mul-1-neg 97.2%

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

   6. Simplified 97.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+180}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.25 \cdot 10^{+109}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -1050:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]

Alternative 7: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2000 \lor \neg \left(y \leq 1950000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2000.0) (not (<= y 1950000000000.0)))
   (- 1.0 (log (/ (+ x -1.0) y)))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2000.0) || !(y <= 1950000000000.0)) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2000.0) || !(y <= 1950000000000.0)) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2000.0) or not (y <= 1950000000000.0):
		tmp = 1.0 - math.log(((x + -1.0) / y))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2000.0) || !(y <= 1950000000000.0))
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2000.0], N[Not[LessEqual[y, 1950000000000.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2e3 or 1.95e12 < y

   1. Initial program 25.0%

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

      1. sub-neg 25.0%

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

      2. log1p-def 25.0%

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

      3. neg-sub0 25.0%

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

      4. div-sub 25.1%

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

      5. associate--r- 25.1%

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

      6. neg-sub0 25.1%

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

      7. +-commutative 25.1%

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

      8. sub-neg 25.1%

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

      9. div-sub 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in y around inf 24.8%

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

      1. *-un-lft-identity 24.8%

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

      2. associate--r+ 24.8%

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

      3. sub-neg 24.8%

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

      4. sub-div 24.8%

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

      5. metadata-eval 24.8%

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

   6. Applied egg-rr 24.8%

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

      1. *-lft-identity 24.8%

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

      2. log1p-def 24.8%

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

      3. +-commutative 24.8%

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

      4. associate-+l+ 99.3%

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

      5. sub-neg 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

   if -2e3 < y < 1.95e12

   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. neg-sub0 100.0%

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

      4. div-sub 100.0%

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

      5. associate--r- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

      9. div-sub 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. distribute-neg-frac 98.7%

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

   6. Simplified 98.7%

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

      1. frac-2neg 98.7%

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

      2. div-inv 98.7%

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

      3. remove-double-neg 98.7%

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

   8. Applied egg-rr 98.7%

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

      1. associate-*r/ 98.7%

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

      2. *-rgt-identity 98.7%

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

      3. sub-neg 98.7%

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

      4. distribute-neg-in 98.7%

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

      5. metadata-eval 98.7%

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

      6. remove-double-neg 98.7%

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

   10. Simplified 98.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2000 \lor \neg \left(y \leq 1950000000000\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]

Alternative 8: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1050:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1050.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1050.0) {
		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 <= -1050.0) {
		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 <= -1050.0:
		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 <= -1050.0)
		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, -1050.0], 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 < -1050

   1. Initial program 19.0%

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

      1. sub-neg 19.0%

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

      2. log1p-def 19.0%

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

      3. neg-sub0 19.0%

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

      4. div-sub 19.0%

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

      5. associate--r- 19.0%

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

      6. neg-sub0 19.0%

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

      7. +-commutative 19.0%

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

      8. sub-neg 19.0%

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

      9. div-sub 19.0%

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

   3. Simplified 19.0%

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

   4. Taylor expanded in y around inf 18.7%

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

   5. Taylor expanded in x around 0 63.2%

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

      1. distribute-neg-frac 63.2%

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

      2. metadata-eval 63.2%

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

   7. Simplified 63.2%

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

   if -1050 < y

   1. Initial program 92.1%

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

      1. sub-neg 92.1%

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

      2. log1p-def 92.1%

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

      3. neg-sub0 92.1%

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

      4. div-sub 92.1%

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

      5. associate--r- 92.1%

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

      6. neg-sub0 92.1%

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

      7. +-commutative 92.1%

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

      8. sub-neg 92.1%

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

      9. div-sub 92.1%

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

   3. Simplified 92.1%

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

   4. Taylor expanded in y around 0 82.9%

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

      1. log1p-def 82.9%

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

      2. mul-1-neg 82.9%

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

   6. Simplified 82.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1050:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 9: 62.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. sub-neg 69.5%

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

   2. log1p-def 69.5%

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

   3. neg-sub0 69.5%

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

   4. div-sub 69.5%

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

   5. associate--r- 69.5%

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

   6. neg-sub0 69.5%

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

   7. +-commutative 69.5%

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

   8. sub-neg 69.5%

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

   9. div-sub 69.5%

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

3. Simplified 69.5%

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

4. Taylor expanded in y around 0 60.8%

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

   1. log1p-def 60.8%

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

   2. mul-1-neg 60.8%

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

6. Simplified 60.8%

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

7. Final simplification 60.8%

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

Alternative 10: 45.2% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.5%

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

   1. sub-neg 69.5%

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

   2. log1p-def 69.5%

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

   3. neg-sub0 69.5%

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

   4. div-sub 69.5%

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

   5. associate--r- 69.5%

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

   6. neg-sub0 69.5%

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

   7. +-commutative 69.5%

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

   8. sub-neg 69.5%

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

   9. div-sub 69.5%

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

3. Simplified 69.5%

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

4. Taylor expanded in x around inf 70.8%

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

   1. neg-mul-1 70.8%

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

   2. distribute-neg-frac 70.8%

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

6. Simplified 70.8%

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

7. Taylor expanded in x around 0 46.6%

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

8. Final simplification 46.6%

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

Alternative 11: 43.6% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 69.5%

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

   1. sub-neg 69.5%

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

   2. log1p-def 69.5%

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

   3. neg-sub0 69.5%

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

   4. div-sub 69.5%

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

   5. associate--r- 69.5%

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

   6. neg-sub0 69.5%

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

   7. +-commutative 69.5%

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

   8. sub-neg 69.5%

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

   9. div-sub 69.5%

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

3. Simplified 69.5%

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

4. Taylor expanded in x around inf 70.8%

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

   1. neg-mul-1 70.8%

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

   2. distribute-neg-frac 70.8%

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

6. Simplified 70.8%

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

7. Taylor expanded in x around 0 45.2%

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

8. Final simplification 45.2%

   \[\leadsto 1 \]

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 2023238 
(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))))))