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

Percentage Accurate: 73.2% → 99.6%

Time: 14.2s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -460000:\\ \;\;\;\;1 - \left(\frac{\frac{1 - x}{y}}{1 - x} + \left(\mathsf{log1p}\left(-x\right) - \log \left(\frac{y}{-1}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -460000.0)
   (- 1.0 (+ (/ (/ (- 1.0 x) y) (- 1.0 x)) (- (log1p (- x)) (log (/ y -1.0)))))
   (if (<= y 2.9e+61)
     (- 1.0 (log1p (* (/ (- x y) (fma y y -1.0)) (+ y 1.0))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -460000.0) {
		tmp = 1.0 - ((((1.0 - x) / y) / (1.0 - x)) + (log1p(-x) - log((y / -1.0))));
	} else if (y <= 2.9e+61) {
		tmp = 1.0 - log1p((((x - y) / fma(y, y, -1.0)) * (y + 1.0)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -460000.0)
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(1.0 - x) / y) / Float64(1.0 - x)) + Float64(log1p(Float64(-x)) - log(Float64(y / -1.0)))));
	elseif (y <= 2.9e+61)
		tmp = Float64(1.0 - log1p(Float64(Float64(Float64(x - y) / fma(y, y, -1.0)) * Float64(y + 1.0))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -460000.0], N[(1.0 - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[(N[Log[1 + (-x)], $MachinePrecision] - N[Log[N[(y / -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+61], N[(1.0 - N[Log[1 + N[(N[(N[(x - y), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.6e5

   1. Initial program 24.2%

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

      1. sub-neg 24.2%

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

      2. log1p-define 24.2%

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

      3. distribute-neg-frac2 24.2%

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

      4. neg-sub0 24.2%

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

      5. associate--r- 24.2%

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

      6. metadata-eval 24.2%

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

      7. +-commutative 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in y around -inf 99.6%

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

   7. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. log-div 99.6%

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

      3. metadata-eval 99.6%

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

   9. Applied egg-rr 99.6%

      \[\leadsto 1 - \left(\frac{\frac{1 - x}{y}}{1 - x} + \left(\mathsf{log1p}\left(-x\right) + \color{blue}{\left(0 - \log \left(\frac{y}{-1}\right)\right)}\right)\right) \]
   10. Step-by-step derivation

       1. neg-sub0 99.6%

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

   11. Simplified 99.6%

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

   if -4.6e5 < y < 2.9000000000000001e61

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 2.9000000000000001e61 < y

   1. Initial program 36.4%

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

      1. sub-neg 36.4%

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

      2. log1p-define 36.4%

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

      3. distribute-neg-frac2 36.4%

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

      4. neg-sub0 36.4%

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

      5. associate--r- 36.4%

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

      6. metadata-eval 36.4%

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

      7. +-commutative 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. unsub-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -460000:\\ \;\;\;\;1 - \left(\frac{\frac{1 - x}{y}}{1 - x} + \left(\mathsf{log1p}\left(-x\right) - \log \left(\frac{y}{-1}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3050000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3050000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 2.6e+61)
     (- 1.0 (log1p (* (/ (- x y) (fma y y -1.0)) (+ y 1.0))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3050000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 2.6e+61) {
		tmp = 1.0 - log1p((((x - y) / fma(y, y, -1.0)) * (y + 1.0)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3050000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 2.6e+61)
		tmp = Float64(1.0 - log1p(Float64(Float64(Float64(x - y) / fma(y, y, -1.0)) * Float64(y + 1.0))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3050000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+61], N[(1.0 - N[Log[1 + N[(N[(N[(x - y), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.05e9

   1. Initial program 24.2%

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

      1. sub-neg 24.2%

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

      2. log1p-define 24.2%

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

      3. distribute-neg-frac2 24.2%

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

      4. neg-sub0 24.2%

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

      5. associate--r- 24.2%

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

      6. metadata-eval 24.2%

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

      7. +-commutative 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in y around -inf 98.8%

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

      1. sub-neg 98.8%

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

      2. metadata-eval 98.8%

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

      3. distribute-lft-in 98.8%

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

      4. metadata-eval 98.8%

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

      5. +-commutative 98.8%

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

      6. log1p-define 98.8%

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

      7. mul-1-neg 98.8%

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

   7. Simplified 98.8%

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

   if -3.05e9 < y < 2.59999999999999973e61

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 2.59999999999999973e61 < y

   1. Initial program 36.4%

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

      1. sub-neg 36.4%

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

      2. log1p-define 36.4%

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

      3. distribute-neg-frac2 36.4%

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

      4. neg-sub0 36.4%

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

      5. associate--r- 36.4%

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

      6. metadata-eval 36.4%

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

      7. +-commutative 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. unsub-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3050000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -510000:\\ \;\;\;\;1 + \left(\frac{\frac{1 - x}{y}}{x + -1} - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -510000.0)
   (+
    1.0
    (- (/ (/ (- 1.0 x) y) (+ x -1.0)) (+ (log1p (- x)) (log (/ -1.0 y)))))
   (if (<= y 2.6e+61)
     (- 1.0 (log1p (* (/ (- x y) (fma y y -1.0)) (+ y 1.0))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -510000.0) {
		tmp = 1.0 + ((((1.0 - x) / y) / (x + -1.0)) - (log1p(-x) + log((-1.0 / y))));
	} else if (y <= 2.6e+61) {
		tmp = 1.0 - log1p((((x - y) / fma(y, y, -1.0)) * (y + 1.0)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -510000.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(Float64(1.0 - x) / y) / Float64(x + -1.0)) - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y)))));
	elseif (y <= 2.6e+61)
		tmp = Float64(1.0 - log1p(Float64(Float64(Float64(x - y) / fma(y, y, -1.0)) * Float64(y + 1.0))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -510000.0], N[(1.0 + N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+61], N[(1.0 - N[Log[1 + N[(N[(N[(x - y), $MachinePrecision] / N[(y * y + -1.0), $MachinePrecision]), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.1e5

   1. Initial program 24.2%

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

      1. sub-neg 24.2%

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

      2. log1p-define 24.2%

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

      3. distribute-neg-frac2 24.2%

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

      4. neg-sub0 24.2%

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

      5. associate--r- 24.2%

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

      6. metadata-eval 24.2%

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

      7. +-commutative 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in y around -inf 99.6%

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

   7. Simplified 99.6%

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

   if -5.1e5 < y < 2.59999999999999973e61

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-/r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. fma-neg 100.0%

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

      5. metadata-eval 100.0%

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

      6. sub-neg 100.0%

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

      7. metadata-eval 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 2.59999999999999973e61 < y

   1. Initial program 36.4%

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

      1. sub-neg 36.4%

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

      2. log1p-define 36.4%

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

      3. distribute-neg-frac2 36.4%

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

      4. neg-sub0 36.4%

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

      5. associate--r- 36.4%

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

      6. metadata-eval 36.4%

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

      7. +-commutative 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. unsub-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -510000:\\ \;\;\;\;1 + \left(\frac{\frac{1 - x}{y}}{x + -1} - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\mathsf{fma}\left(y, y, -1\right)} \cdot \left(y + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(x - y\right) \cdot \frac{-1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.3e+39)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 2.9e+61)
     (- 1.0 (log1p (* (- x y) (/ -1.0 (- 1.0 y)))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.3e+39) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 2.9e+61) {
		tmp = 1.0 - log1p(((x - y) * (-1.0 / (1.0 - y))));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.3e+39) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 2.9e+61) {
		tmp = 1.0 - Math.log1p(((x - y) * (-1.0 / (1.0 - y))));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.3e+39:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 2.9e+61:
		tmp = 1.0 - math.log1p(((x - y) * (-1.0 / (1.0 - y))))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.3e+39)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 2.9e+61)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) * Float64(-1.0 / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.3e+39], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+61], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] * N[(-1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.3e39

   1. Initial program 16.3%

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

      1. sub-neg 16.3%

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

      2. log1p-define 16.3%

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

      3. distribute-neg-frac2 16.3%

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

      4. neg-sub0 16.3%

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

      5. associate--r- 16.3%

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

      6. metadata-eval 16.3%

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

      7. +-commutative 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in x around 0 2.8%

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

      1. sub-neg 2.8%

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

      2. metadata-eval 2.8%

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

      3. neg-mul-1 2.8%

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

      4. distribute-neg-frac 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in y around -inf 68.8%

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

   if -4.3e39 < y < 2.9000000000000001e61

   1. Initial program 97.7%

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

      1. sub-neg 97.7%

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

      2. log1p-define 97.8%

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

      3. distribute-neg-frac2 97.8%

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

      4. neg-sub0 97.8%

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

      5. associate--r- 97.8%

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

      6. metadata-eval 97.8%

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

      7. +-commutative 97.8%

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

   3. Simplified 97.8%

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

      1. clear-num 97.7%

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

      2. associate-/r/ 97.8%

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

   6. Applied egg-rr 97.8%

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

   if 2.9000000000000001e61 < y

   1. Initial program 36.4%

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

      1. sub-neg 36.4%

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

      2. log1p-define 36.4%

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

      3. distribute-neg-frac2 36.4%

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

      4. neg-sub0 36.4%

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

      5. associate--r- 36.4%

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

      6. metadata-eval 36.4%

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

      7. +-commutative 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. unsub-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(x - y\right) \cdot \frac{-1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6900000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6900000000.0)
   (- 1.0 (+ (log1p (- x)) (log (/ -1.0 y))))
   (if (<= y 2.6e+61)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (+ 1.0 (- (log y) (log (+ x -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6900000000.0) {
		tmp = 1.0 - (log1p(-x) + log((-1.0 / y)));
	} else if (y <= 2.6e+61) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + (log(y) - log((x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6900000000.0) {
		tmp = 1.0 - (Math.log1p(-x) + Math.log((-1.0 / y)));
	} else if (y <= 2.6e+61) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + (Math.log(y) - Math.log((x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6900000000.0:
		tmp = 1.0 - (math.log1p(-x) + math.log((-1.0 / y)))
	elif y <= 2.6e+61:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + (math.log(y) - math.log((x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6900000000.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + log(Float64(-1.0 / y))));
	elseif (y <= 2.6e+61)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + Float64(log(y) - log(Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6900000000.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+61], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Log[y], $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.9e9

   1. Initial program 24.2%

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

      1. sub-neg 24.2%

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

      2. log1p-define 24.2%

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

      3. distribute-neg-frac2 24.2%

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

      4. neg-sub0 24.2%

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

      5. associate--r- 24.2%

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

      6. metadata-eval 24.2%

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

      7. +-commutative 24.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in y around -inf 98.8%

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

      1. sub-neg 98.8%

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

      2. metadata-eval 98.8%

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

      3. distribute-lft-in 98.8%

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

      4. metadata-eval 98.8%

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

      5. +-commutative 98.8%

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

      6. log1p-define 98.8%

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

      7. mul-1-neg 98.8%

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

   7. Simplified 98.8%

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

   if -6.9e9 < y < 2.59999999999999973e61

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-define 100.0%

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

      3. distribute-neg-frac2 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate--r- 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   if 2.59999999999999973e61 < y

   1. Initial program 36.4%

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

      1. sub-neg 36.4%

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

      2. log1p-define 36.4%

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

      3. distribute-neg-frac2 36.4%

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

      4. neg-sub0 36.4%

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

      5. associate--r- 36.4%

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

      6. metadata-eval 36.4%

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

      7. +-commutative 36.4%

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

   3. Simplified 36.4%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. log-rec 99.2%

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

      2. unsub-neg 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

   7. Simplified 99.2%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6900000000:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + \log \left(\frac{-1}{y}\right)\right)\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+61}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\log y - \log \left(x + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ t_1 := 1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{if}\;y \leq -9.4 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -180000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-18}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))) (t_1 (- 1.0 (log1p (/ x (+ y -1.0))))))
   (if (<= y -9.4e+39)
     t_0
     (if (<= y -1.3e+19)
       t_1
       (if (<= y -180000000.0)
         t_0
         (if (<= y -6.4e-18) (- 1.0 (log1p (/ y (- 1.0 y)))) t_1))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((-1.0 / y));
	double t_1 = 1.0 - log1p((x / (y + -1.0)));
	double tmp;
	if (y <= -9.4e+39) {
		tmp = t_0;
	} else if (y <= -1.3e+19) {
		tmp = t_1;
	} else if (y <= -180000000.0) {
		tmp = t_0;
	} else if (y <= -6.4e-18) {
		tmp = 1.0 - log1p((y / (1.0 - y)));
	} 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.log1p((x / (y + -1.0)));
	double tmp;
	if (y <= -9.4e+39) {
		tmp = t_0;
	} else if (y <= -1.3e+19) {
		tmp = t_1;
	} else if (y <= -180000000.0) {
		tmp = t_0;
	} else if (y <= -6.4e-18) {
		tmp = 1.0 - Math.log1p((y / (1.0 - y)));
	} 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.log1p((x / (y + -1.0)))
	tmp = 0
	if y <= -9.4e+39:
		tmp = t_0
	elif y <= -1.3e+19:
		tmp = t_1
	elif y <= -180000000.0:
		tmp = t_0
	elif y <= -6.4e-18:
		tmp = 1.0 - math.log1p((y / (1.0 - y)))
	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 - log1p(Float64(x / Float64(y + -1.0))))
	tmp = 0.0
	if (y <= -9.4e+39)
		tmp = t_0;
	elseif (y <= -1.3e+19)
		tmp = t_1;
	elseif (y <= -180000000.0)
		tmp = t_0;
	elseif (y <= -6.4e-18)
		tmp = Float64(1.0 - log1p(Float64(y / Float64(1.0 - y))));
	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[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -9.4e+39], t$95$0, If[LessEqual[y, -1.3e+19], t$95$1, If[LessEqual[y, -180000000.0], t$95$0, If[LessEqual[y, -6.4e-18], N[(1.0 - N[Log[1 + N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.3999999999999998e39 or -1.3e19 < y < -1.8e8

   1. Initial program 18.4%

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

      1. sub-neg 18.4%

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

      2. log1p-define 18.4%

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

      3. distribute-neg-frac2 18.4%

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

      4. neg-sub0 18.4%

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

      5. associate--r- 18.4%

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

      6. metadata-eval 18.4%

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

      7. +-commutative 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in x around 0 6.0%

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

      1. sub-neg 6.0%

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

      2. metadata-eval 6.0%

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

      3. neg-mul-1 6.0%

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

      4. distribute-neg-frac 6.0%

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

   7. Simplified 6.0%

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

   8. Taylor expanded in y around -inf 70.5%

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

   if -9.3999999999999998e39 < y < -1.3e19 or -6.3999999999999998e-18 < y

   1. Initial program 91.4%

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

      1. sub-neg 91.4%

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

      2. log1p-define 91.4%

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

      3. distribute-neg-frac2 91.4%

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

      4. neg-sub0 91.4%

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

      5. associate--r- 91.4%

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

      6. metadata-eval 91.4%

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

      7. +-commutative 91.4%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in x around inf 90.6%

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

   if -1.8e8 < y < -6.3999999999999998e-18

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. neg-sub0 99.7%

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

      5. associate--r- 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 83.3%

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

      1. sub-neg 83.3%

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

      2. mul-1-neg 83.3%

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

      3. log1p-define 83.3%

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

      4. mul-1-neg 83.3%

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

      5. sub-neg 83.3%

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

      6. metadata-eval 83.3%

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

      7. distribute-neg-frac2 83.3%

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

      8. +-commutative 83.3%

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

      9. distribute-neg-in 83.3%

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

      10. metadata-eval 83.3%

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

      11. unsub-neg 83.3%

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

   9. Simplified 83.3%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.4 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+19}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{elif}\;y \leq -180000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-18}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{+19}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-1}{y} \cdot \left(y - x\right)\right)\\ \mathbf{elif}\;y \leq -180000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-18}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ -1.0 y)))))
   (if (<= y -3.8e+39)
     t_0
     (if (<= y -1.42e+19)
       (- 1.0 (log1p (* (/ -1.0 y) (- y x))))
       (if (<= y -180000000.0)
         t_0
         (if (<= y -6.4e-18)
           (- 1.0 (log1p (/ y (- 1.0 y))))
           (- 1.0 (log1p (/ x (+ y -1.0))))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((-1.0 / y));
	double tmp;
	if (y <= -3.8e+39) {
		tmp = t_0;
	} else if (y <= -1.42e+19) {
		tmp = 1.0 - log1p(((-1.0 / y) * (y - x)));
	} else if (y <= -180000000.0) {
		tmp = t_0;
	} else if (y <= -6.4e-18) {
		tmp = 1.0 - log1p((y / (1.0 - y)));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((-1.0 / y));
	double tmp;
	if (y <= -3.8e+39) {
		tmp = t_0;
	} else if (y <= -1.42e+19) {
		tmp = 1.0 - Math.log1p(((-1.0 / y) * (y - x)));
	} else if (y <= -180000000.0) {
		tmp = t_0;
	} else if (y <= -6.4e-18) {
		tmp = 1.0 - Math.log1p((y / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((-1.0 / y))
	tmp = 0
	if y <= -3.8e+39:
		tmp = t_0
	elif y <= -1.42e+19:
		tmp = 1.0 - math.log1p(((-1.0 / y) * (y - x)))
	elif y <= -180000000.0:
		tmp = t_0
	elif y <= -6.4e-18:
		tmp = 1.0 - math.log1p((y / (1.0 - y)))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(-1.0 / y)))
	tmp = 0.0
	if (y <= -3.8e+39)
		tmp = t_0;
	elseif (y <= -1.42e+19)
		tmp = Float64(1.0 - log1p(Float64(Float64(-1.0 / y) * Float64(y - x))));
	elseif (y <= -180000000.0)
		tmp = t_0;
	elseif (y <= -6.4e-18)
		tmp = Float64(1.0 - log1p(Float64(y / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e+39], t$95$0, If[LessEqual[y, -1.42e+19], N[(1.0 - N[Log[1 + N[(N[(-1.0 / y), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -180000000.0], t$95$0, If[LessEqual[y, -6.4e-18], N[(1.0 - N[Log[1 + N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.7999999999999998e39 or -1.42e19 < y < -1.8e8

   1. Initial program 18.4%

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

      1. sub-neg 18.4%

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

      2. log1p-define 18.4%

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

      3. distribute-neg-frac2 18.4%

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

      4. neg-sub0 18.4%

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

      5. associate--r- 18.4%

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

      6. metadata-eval 18.4%

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

      7. +-commutative 18.4%

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

   3. Simplified 18.4%

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

   5. Taylor expanded in x around 0 6.0%

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

      1. sub-neg 6.0%

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

      2. metadata-eval 6.0%

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

      3. neg-mul-1 6.0%

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

      4. distribute-neg-frac 6.0%

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

   7. Simplified 6.0%

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

   8. Taylor expanded in y around -inf 70.5%

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

   if -3.7999999999999998e39 < y < -1.42e19

   1. Initial program 83.9%

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

      1. sub-neg 83.9%

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

      2. log1p-define 83.9%

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

      3. distribute-neg-frac2 83.9%

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

      4. neg-sub0 83.9%

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

      5. associate--r- 83.9%

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

      6. metadata-eval 83.9%

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

      7. +-commutative 83.9%

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

   3. Simplified 83.9%

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

      1. clear-num 83.9%

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

      2. associate-/r/ 86.4%

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

   6. Applied egg-rr 86.4%

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

   7. Taylor expanded in y around inf 86.4%

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

   if -1.8e8 < y < -6.3999999999999998e-18

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. neg-sub0 99.7%

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

      5. associate--r- 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0 83.3%

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

      1. sub-neg 83.3%

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

      2. mul-1-neg 83.3%

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

      3. log1p-define 83.3%

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

      4. mul-1-neg 83.3%

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

      5. sub-neg 83.3%

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

      6. metadata-eval 83.3%

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

      7. distribute-neg-frac2 83.3%

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

      8. +-commutative 83.3%

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

      9. distribute-neg-in 83.3%

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

      10. metadata-eval 83.3%

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

      11. unsub-neg 83.3%

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

   9. Simplified 83.3%

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

   if -6.3999999999999998e-18 < y

   1. Initial program 91.6%

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

      1. sub-neg 91.6%

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

      2. log1p-define 91.7%

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

      3. distribute-neg-frac2 91.7%

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

      4. neg-sub0 91.7%

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

      5. associate--r- 91.7%

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

      6. metadata-eval 91.7%

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

      7. +-commutative 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in x around inf 90.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -1.42 \cdot 10^{+19}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{-1}{y} \cdot \left(y - x\right)\right)\\ \mathbf{elif}\;y \leq -180000000:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-18}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 700000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(x - y\right) \cdot \frac{-1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{\frac{1 - x}{y}}{1 - x} + \log \left(\frac{1 + x}{y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e+39)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 700000000.0)
     (- 1.0 (log1p (* (- x y) (/ -1.0 (- 1.0 y)))))
     (- 1.0 (+ (/ (/ (- 1.0 x) y) (- 1.0 x)) (log (/ (+ 1.0 x) y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e+39) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 700000000.0) {
		tmp = 1.0 - log1p(((x - y) * (-1.0 / (1.0 - y))));
	} else {
		tmp = 1.0 - ((((1.0 - x) / y) / (1.0 - x)) + log(((1.0 + x) / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4e+39) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 700000000.0) {
		tmp = 1.0 - Math.log1p(((x - y) * (-1.0 / (1.0 - y))));
	} else {
		tmp = 1.0 - ((((1.0 - x) / y) / (1.0 - x)) + Math.log(((1.0 + x) / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e+39:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 700000000.0:
		tmp = 1.0 - math.log1p(((x - y) * (-1.0 / (1.0 - y))))
	else:
		tmp = 1.0 - ((((1.0 - x) / y) / (1.0 - x)) + math.log(((1.0 + x) / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e+39)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 700000000.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) * Float64(-1.0 / Float64(1.0 - y)))));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(Float64(1.0 - x) / y) / Float64(1.0 - x)) + log(Float64(Float64(1.0 + x) / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e+39], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 700000000.0], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] * N[(-1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(N[(1.0 - x), $MachinePrecision] / y), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.99999999999999976e39

   1. Initial program 16.3%

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

      1. sub-neg 16.3%

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

      2. log1p-define 16.3%

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

      3. distribute-neg-frac2 16.3%

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

      4. neg-sub0 16.3%

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

      5. associate--r- 16.3%

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

      6. metadata-eval 16.3%

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

      7. +-commutative 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in x around 0 2.8%

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

      1. sub-neg 2.8%

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

      2. metadata-eval 2.8%

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

      3. neg-mul-1 2.8%

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

      4. distribute-neg-frac 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in y around -inf 68.8%

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

   if -3.99999999999999976e39 < y < 7e8

   1. Initial program 97.7%

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

      1. sub-neg 97.7%

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

      2. log1p-define 97.7%

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

      3. distribute-neg-frac2 97.7%

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

      4. neg-sub0 97.7%

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

      5. associate--r- 97.7%

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

      6. metadata-eval 97.7%

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

      7. +-commutative 97.7%

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

   3. Simplified 97.7%

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

      1. clear-num 97.7%

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

      2. associate-/r/ 97.8%

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

   6. Applied egg-rr 97.8%

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

   if 7e8 < y

   1. Initial program 47.4%

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

      1. sub-neg 47.4%

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

      2. log1p-define 47.4%

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

      3. distribute-neg-frac2 47.4%

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

      4. neg-sub0 47.4%

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

      5. associate--r- 47.4%

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

      6. metadata-eval 47.4%

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

      7. +-commutative 47.4%

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

   3. Simplified 47.4%

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

   5. Taylor expanded in y around -inf 0.0%

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

   7. Simplified 0.0%

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

      1. log1p-undefine 0.0%

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

      2. sub-neg 0.0%

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

      3. metadata-eval 0.0%

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

      4. add-sqr-sqrt 0.0%

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

      5. sqrt-unprod 0.0%

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

      6. sqr-neg 0.0%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 0.0%

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

      9. frac-2neg 0.0%

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

      10. log-prod 0.0%

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

      11. div-inv 0.0%

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

      12. sub-neg 0.0%

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

      13. add-sqr-sqrt 0.0%

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

      14. sqrt-unprod 60.6%

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

      15. sqr-neg 60.6%

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

      16. sqrt-unprod 97.7%

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

      17. add-sqr-sqrt 97.7%

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

   9. Applied egg-rr 97.7%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 700000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(x - y\right) \cdot \frac{-1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{\frac{1 - x}{y}}{1 - x} + \log \left(\frac{1 + x}{y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (/ (- x y) (+ y -1.0))) 1e-10)
   (+ 1.0 (- (/ -1.0 y) (log (/ -1.0 y))))
   (- 1.0 (log1p (* x (/ (- 1.0 (/ y x)) (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + ((x - y) / (y + -1.0))) <= 1e-10) {
		tmp = 1.0 + ((-1.0 / y) - log((-1.0 / y)));
	} else {
		tmp = 1.0 - log1p((x * ((1.0 - (y / x)) / (y + -1.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 + ((x - y) / (y + -1.0))) <= 1e-10) {
		tmp = 1.0 + ((-1.0 / y) - Math.log((-1.0 / y)));
	} else {
		tmp = 1.0 - Math.log1p((x * ((1.0 - (y / x)) / (y + -1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 + ((x - y) / (y + -1.0))) <= 1e-10:
		tmp = 1.0 + ((-1.0 / y) - math.log((-1.0 / y)))
	else:
		tmp = 1.0 - math.log1p((x * ((1.0 - (y / x)) / (y + -1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + Float64(Float64(x - y) / Float64(y + -1.0))) <= 1e-10)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - log(Float64(-1.0 / y))));
	else
		tmp = Float64(1.0 - log1p(Float64(x * Float64(Float64(1.0 - Float64(y / x)) / Float64(y + -1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-10], N[(1.0 + N[(N[(-1.0 / y), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x * N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 1.00000000000000004e-10

   1. Initial program 6.5%

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

      1. sub-neg 6.5%

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

      2. log1p-define 6.5%

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

      3. distribute-neg-frac2 6.5%

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

      4. neg-sub0 6.5%

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

      5. associate--r- 6.5%

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

      6. metadata-eval 6.5%

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

      7. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in x around 0 5.9%

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

      1. sub-neg 5.9%

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

      2. metadata-eval 5.9%

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

      3. neg-mul-1 5.9%

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

      4. distribute-neg-frac 5.9%

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

   7. Simplified 5.9%

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

   8. Taylor expanded in y around -inf 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 1.00000000000000004e-10 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. neg-sub0 99.7%

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

      5. associate--r- 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

      4. mul-1-neg 99.4%

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

      5. unsub-neg 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

      8. associate-/r* 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around inf 99.4%

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

      1. +-commutative 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

      4. +-commutative 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

      8. +-commutative 99.4%

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

      9. cancel-sign-sub-inv 99.4%

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

      10. *-lft-identity 99.4%

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

      11. associate-/r* 99.7%

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

      12. div-sub 99.7%

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

   10. Simplified 99.7%

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

4. Final simplification 89.7%

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

Alternative 10: 91.3% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (+ y -1.0))))
   (if (<= (+ 1.0 t_0) 1e-10)
     (+ 1.0 (- (/ -1.0 y) (log (/ -1.0 y))))
     (- 1.0 (log1p t_0)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (y + -1.0);
	double tmp;
	if ((1.0 + t_0) <= 1e-10) {
		tmp = 1.0 + ((-1.0 / y) - log((-1.0 / y)));
	} else {
		tmp = 1.0 - log1p(t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (y + -1.0);
	double tmp;
	if ((1.0 + t_0) <= 1e-10) {
		tmp = 1.0 + ((-1.0 / y) - Math.log((-1.0 / y)));
	} else {
		tmp = 1.0 - Math.log1p(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (y + -1.0)
	tmp = 0
	if (1.0 + t_0) <= 1e-10:
		tmp = 1.0 + ((-1.0 / y) - math.log((-1.0 / y)))
	else:
		tmp = 1.0 - math.log1p(t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(y + -1.0))
	tmp = 0.0
	if (Float64(1.0 + t_0) <= 1e-10)
		tmp = Float64(1.0 + Float64(Float64(-1.0 / y) - log(Float64(-1.0 / y))));
	else
		tmp = Float64(1.0 - log1p(t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 + t$95$0), $MachinePrecision], 1e-10], N[(1.0 + N[(N[(-1.0 / y), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 1.00000000000000004e-10

   1. Initial program 6.5%

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

      1. sub-neg 6.5%

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

      2. log1p-define 6.5%

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

      3. distribute-neg-frac2 6.5%

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

      4. neg-sub0 6.5%

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

      5. associate--r- 6.5%

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

      6. metadata-eval 6.5%

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

      7. +-commutative 6.5%

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

   3. Simplified 6.5%

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

   5. Taylor expanded in x around 0 5.9%

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

      1. sub-neg 5.9%

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

      2. metadata-eval 5.9%

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

      3. neg-mul-1 5.9%

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

      4. distribute-neg-frac 5.9%

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

   7. Simplified 5.9%

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

   8. Taylor expanded in y around -inf 63.2%

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

      1. +-commutative 63.2%

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

   10. Simplified 63.2%

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

   if 1.00000000000000004e-10 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. log1p-define 99.7%

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

      3. distribute-neg-frac2 99.7%

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

      4. neg-sub0 99.7%

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

      5. associate--r- 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

   3. Simplified 99.7%

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

4. Final simplification 89.7%

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

Alternative 11: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+39)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+39) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+39) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+39:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+39)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+39], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7999999999999998e39

   1. Initial program 16.3%

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

      1. sub-neg 16.3%

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

      2. log1p-define 16.3%

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

      3. distribute-neg-frac2 16.3%

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

      4. neg-sub0 16.3%

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

      5. associate--r- 16.3%

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

      6. metadata-eval 16.3%

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

      7. +-commutative 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in x around 0 2.8%

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

      1. sub-neg 2.8%

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

      2. metadata-eval 2.8%

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

      3. neg-mul-1 2.8%

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

      4. distribute-neg-frac 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in y around -inf 68.8%

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

   if -3.7999999999999998e39 < y

   1. Initial program 90.4%

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

      1. sub-neg 90.4%

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

      2. log1p-define 90.4%

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

      3. distribute-neg-frac2 90.4%

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

      4. neg-sub0 90.4%

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

      5. associate--r- 90.4%

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

      6. metadata-eval 90.4%

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

      7. +-commutative 90.4%

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

   3. Simplified 90.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.7e+39)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log1p (/ x (+ y -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+39) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.7e+39) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.7e+39:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.7e+39)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.7e+39], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6999999999999999e39

   1. Initial program 16.3%

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

      1. sub-neg 16.3%

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

      2. log1p-define 16.3%

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

      3. distribute-neg-frac2 16.3%

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

      4. neg-sub0 16.3%

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

      5. associate--r- 16.3%

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

      6. metadata-eval 16.3%

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

      7. +-commutative 16.3%

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

   3. Simplified 16.3%

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

   5. Taylor expanded in x around 0 2.8%

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

      1. sub-neg 2.8%

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

      2. metadata-eval 2.8%

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

      3. neg-mul-1 2.8%

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

      4. distribute-neg-frac 2.8%

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

   7. Simplified 2.8%

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

   8. Taylor expanded in y around -inf 68.8%

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

   if -4.6999999999999999e39 < y

   1. Initial program 90.4%

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

      1. sub-neg 90.4%

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

      2. log1p-define 90.4%

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

      3. distribute-neg-frac2 90.4%

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

      4. neg-sub0 90.4%

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

      5. associate--r- 90.4%

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

      6. metadata-eval 90.4%

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

      7. +-commutative 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in x around inf 87.9%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+39}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 80.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.39999999999999991

   1. Initial program 26.4%

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

      1. sub-neg 26.4%

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

      2. log1p-define 26.4%

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

      3. distribute-neg-frac2 26.4%

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

      4. neg-sub0 26.4%

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

      5. associate--r- 26.4%

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

      6. metadata-eval 26.4%

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

      7. +-commutative 26.4%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in x around 0 8.2%

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

      1. sub-neg 8.2%

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

      2. metadata-eval 8.2%

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

      3. neg-mul-1 8.2%

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

      4. distribute-neg-frac 8.2%

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

   7. Simplified 8.2%

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

   8. Taylor expanded in y around -inf 64.5%

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

   if -2.39999999999999991 < y

   1. Initial program 91.8%

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

      1. sub-neg 91.8%

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

      2. log1p-define 91.8%

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

      3. distribute-neg-frac2 91.8%

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

      4. neg-sub0 91.8%

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

      5. associate--r- 91.8%

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

      6. metadata-eval 91.8%

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

      7. +-commutative 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around 0 83.0%

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

      1. +-commutative 83.0%

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

      2. div-sub 83.0%

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

      3. fma-define 83.0%

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

      4. mul-1-neg 83.0%

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

      5. sub-neg 83.0%

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

      6. *-inverses 83.0%

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

      7. +-commutative 83.0%

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

      8. metadata-eval 83.0%

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

      9. distribute-lft-in 83.0%

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

      10. metadata-eval 83.0%

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

      11. sub-neg 83.0%

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

      12. fma-undefine 83.0%

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

      13. *-rgt-identity 83.0%

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

      14. sub-neg 83.0%

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

      15. metadata-eval 83.0%

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

      16. distribute-lft-in 83.0%

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

      17. metadata-eval 83.0%

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

      18. +-commutative 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4) {
		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 <= -2.4) {
		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 <= -2.4:
		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 <= -2.4)
		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, -2.4], 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 < -2.39999999999999991

   1. Initial program 26.4%

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

      1. sub-neg 26.4%

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

      2. log1p-define 26.4%

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

      3. distribute-neg-frac2 26.4%

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

      4. neg-sub0 26.4%

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

      5. associate--r- 26.4%

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

      6. metadata-eval 26.4%

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

      7. +-commutative 26.4%

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

   3. Simplified 26.4%

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

   5. Taylor expanded in x around 0 8.2%

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

      1. sub-neg 8.2%

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

      2. metadata-eval 8.2%

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

      3. neg-mul-1 8.2%

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

      4. distribute-neg-frac 8.2%

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

   7. Simplified 8.2%

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

   8. Taylor expanded in y around -inf 64.5%

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

   if -2.39999999999999991 < y

   1. Initial program 91.8%

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

      1. sub-neg 91.8%

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

      2. log1p-define 91.8%

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

      3. distribute-neg-frac2 91.8%

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

      4. neg-sub0 91.8%

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

      5. associate--r- 91.8%

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

      6. metadata-eval 91.8%

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

      7. +-commutative 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in y around 0 82.4%

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

      1. log1p-define 82.4%

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

      2. mul-1-neg 82.4%

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

   7. Simplified 82.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. sub-neg 74.2%

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

   2. log1p-define 74.2%

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

   3. distribute-neg-frac2 74.2%

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

   4. neg-sub0 74.2%

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

   5. associate--r- 74.2%

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

   6. metadata-eval 74.2%

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

   7. +-commutative 74.2%

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

3. Simplified 74.2%

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

5. Taylor expanded in y around 0 63.7%

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

   1. log1p-define 63.7%

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

   2. mul-1-neg 63.7%

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

7. Simplified 63.7%

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

8. Final simplification 63.7%

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

Alternative 16: 41.4% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.2%

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

   1. sub-neg 74.2%

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

   2. log1p-define 74.2%

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

   3. distribute-neg-frac2 74.2%

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

   4. neg-sub0 74.2%

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

   5. associate--r- 74.2%

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

   6. metadata-eval 74.2%

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

   7. +-commutative 74.2%

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

3. Simplified 74.2%

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

5. Taylor expanded in x around 0 39.8%

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

   1. sub-neg 39.8%

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

   2. metadata-eval 39.8%

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

   3. neg-mul-1 39.8%

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

   4. distribute-neg-frac 39.8%

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

7. Simplified 39.8%

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

8. Taylor expanded in y around 0 38.6%

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

9. Final simplification 38.6%

   \[\leadsto 1 - y \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024084 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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