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

Percentage Accurate: 72.4% → 99.8%

Time: 15.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y \cdot 3}{x \cdot 3 + -3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999995)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (+ 1.0 (log (/ (* y 3.0) (+ (* x 3.0) -3.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999995) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + log(((y * 3.0) / ((x * 3.0) + -3.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999995) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + Math.log(((y * 3.0) / ((x * 3.0) + -3.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999995:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + math.log(((y * 3.0) / ((x * 3.0) + -3.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999995)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + log(Float64(Float64(y * 3.0) / Float64(Float64(x * 3.0) + -3.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999995], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(N[(y * 3.0), $MachinePrecision] / N[(N[(x * 3.0), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.99999499999999997

   1. Initial program 99.8%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 99.9%

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

   if 0.99999499999999997 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.1%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 7.1%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 7.1%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{\frac{y + -1}{x - y}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{y + -1}\right), \left(x - y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y + -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      6. --lowering--.f64 7.8%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right)\right) \]

   6. Applied egg-rr 7.8%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{1}{\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

   8. Applied egg-rr 4.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(3 \cdot \frac{y}{\left(x + 2 \cdot x\right) - 3}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{3 \cdot y}{\left(x + 2 \cdot x\right) - 3}\right)\right)\right)\right) \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(3 \cdot y\right), \left(\left(x + 2 \cdot x\right) - 3\right)\right)\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\left(x + 2 \cdot x\right) - 3\right)\right)\right)\right)\right) \]

       4. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\left(x + 2 \cdot x\right) + \left(\mathsf{neg}\left(3\right)\right)\right)\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\left(x + 2 \cdot x\right) + -3\right)\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(x + 2 \cdot x\right), -3\right)\right)\right)\right)\right) \]

       7. distribute-rgt1-in N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(\left(2 + 1\right) \cdot x\right), -3\right)\right)\right)\right)\right) \]

       8. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot x\right), -3\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 99.0%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, x\right), -3\right)\right)\right)\right)\right) \]

   11. Simplified 99.0%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999995:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y \cdot 3}{x \cdot 3 + -3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double t_0 = 1.0 + log(((y + -1.0) / x));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + log((y * (-1.0 + (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(((y + (-1.0d0)) / x))
    if (x <= (-1.0d0)) then
        tmp = t_0
    else if (x <= 1.0d0) then
        tmp = 1.0d0 + log((y * ((-1.0d0) + (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(((y + -1.0) / x));
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 + Math.log((y * (-1.0 + (1.0 / y))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + log(((y + -1.0) / x));
	tmp = 0.0;
	if (x <= -1.0)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = 1.0 + log((y * (-1.0 + (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[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], t$95$0, If[LessEqual[x, 1.0], N[(1.0 + N[Log[N[(y * N[(-1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 74.4%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 74.4%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 74.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{\frac{y + -1}{x - y}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{y + -1}\right), \left(x - y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y + -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      6. --lowering--.f64 74.4%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right)\right) \]

   6. Applied egg-rr 74.4%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{1}{\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

   8. Applied egg-rr 27.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{y - 1}{x}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y - 1\right), x\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + -1\right), x\right)\right)\right)\right) \]

       4. +-lowering-+.f64 98.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right) \]

   11. Simplified 98.9%

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

   if -1 < x < 1

   1. Initial program 65.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 65.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 65.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{\frac{y + -1}{x - y}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{y + -1}\right), \left(x - y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y + -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      6. --lowering--.f64 66.3%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right)\right) \]

   6. Applied egg-rr 66.3%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{1}{\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

   8. Applied egg-rr 65.4%

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

   9. Taylor expanded in y around -inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot \left(y \cdot \left(-1 \cdot \frac{\left(-2 \cdot \frac{x}{\left(x + 2 \cdot x\right) - 3} + \left(-1 \cdot \frac{x - 1}{\left(x + 2 \cdot x\right) - 3} + 2 \cdot \frac{1}{\left(x + 2 \cdot x\right) - 3}\right)\right) - -3 \cdot \frac{\left(2 \cdot {x}^{2} + {x}^{2}\right) - \left(3 + 3 \cdot \left(\left(x + 2 \cdot x\right) - 3\right)\right)}{{\left(\left(x + 2 \cdot x\right) - 3\right)}^{2}}}{y} - 3 \cdot \frac{1}{\left(x + 2 \cdot x\right) - 3}\right)\right)\right)}\right)\right)\right) \]

   11. Simplified 99.8%

       \[\leadsto 1 - \left(-\log \color{blue}{\left(\left(-y\right) \cdot \left(\left(-\frac{\frac{x \cdot -2}{3 \cdot x + -3} + \left(\left(\frac{-1 \cdot \left(x + -1\right)}{3 \cdot x + -3} + \frac{2}{3 \cdot x + -3}\right) - -3 \cdot \frac{\left(3 \cdot \left(x \cdot x\right) - 3\right) - 3 \cdot \left(3 \cdot x + -3\right)}{\left(3 \cdot x + -3\right) \cdot \left(3 \cdot x + -3\right)}\right)}{y}\right) - \frac{3}{3 \cdot x + -3}\right)\right)}\right) \]

   12. Taylor expanded in x around 0

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

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\log \left(-1 \cdot \left(y \cdot \left(1 - \frac{1}{y}\right)\right)\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{log.f64}\left(\left(-1 \cdot \left(y \cdot \left(1 - \frac{1}{y}\right)\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(y \cdot \left(1 - \frac{1}{y}\right)\right)\right)\right)\right) \]

       4. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{log.f64}\left(\left(0 - y \cdot \left(1 - \frac{1}{y}\right)\right)\right)\right) \]

       5. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \left(y \cdot \left(1 - \frac{1}{y}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \left(1 - \frac{1}{y}\right)\right)\right)\right)\right) \]

       7. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \left(\frac{1}{y}\right)\right)\right)\right)\right)\right) \]

       8. /-lowering-/.f64 98.8%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{log.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(1, y\right)\right)\right)\right)\right)\right) \]

   14. Simplified 98.8%

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

4. Final simplification 98.9%

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

Alternative 3: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;y \leq -2.36 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (log (/ y x)))))
   (if (<= y -2.36e+27)
     t_0
     (if (<= y 2e+15) (- 1.0 (log1p (/ (- x y) (+ y -1.0)))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + log((y / x));
	double tmp;
	if (y <= -2.36e+27) {
		tmp = t_0;
	} else if (y <= 2e+15) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.log((y / x));
	double tmp;
	if (y <= -2.36e+27) {
		tmp = t_0;
	} else if (y <= 2e+15) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.log((y / x))
	tmp = 0
	if y <= -2.36e+27:
		tmp = t_0
	elif y <= 2e+15:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + log(Float64(y / x)))
	tmp = 0.0
	if (y <= -2.36e+27)
		tmp = t_0;
	elseif (y <= 2e+15)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.36e+27], t$95$0, If[LessEqual[y, 2e+15], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.36000000000000012e27 or 2e15 < y

   1. Initial program 24.5%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 24.6%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 24.6%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{\frac{y + -1}{x - y}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{y + -1}\right), \left(x - y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y + -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      6. --lowering--.f64 25.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right)\right) \]

   6. Applied egg-rr 25.2%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{1}{\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

   8. Applied egg-rr 5.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{y - 1}{x}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y - 1\right), x\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + -1\right), x\right)\right)\right)\right) \]

       4. +-lowering-+.f64 49.8%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right) \]

   11. Simplified 49.8%

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

   12. Taylor expanded in y around inf

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 49.8%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right)\right) \]

   14. Simplified 49.8%

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

   if -2.36000000000000012e27 < y < 2e15

   1. Initial program 97.1%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 97.2%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 97.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.36 \cdot 10^{+27}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+15}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \log \left(\frac{y + -1}{x}\right)\\ \mathbf{if}\;x \leq -0.116:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (log (/ (+ y -1.0) x)))))
   (if (<= x -0.116) t_0 (if (<= x 1.0) (- 1.0 (log1p (/ y (- 1.0 y)))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + log(((y + -1.0) / x));
	double tmp;
	if (x <= -0.116) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 - log1p((y / (1.0 - y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.log(((y + -1.0) / x));
	double tmp;
	if (x <= -0.116) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = 1.0 - Math.log1p((y / (1.0 - y)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.log(((y + -1.0) / x))
	tmp = 0
	if x <= -0.116:
		tmp = t_0
	elif x <= 1.0:
		tmp = 1.0 - math.log1p((y / (1.0 - y)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + log(Float64(Float64(y + -1.0) / x)))
	tmp = 0.0
	if (x <= -0.116)
		tmp = t_0;
	elseif (x <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(y / Float64(1.0 - y))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Log[N[(N[(y + -1.0), $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.116], t$95$0, If[LessEqual[x, 1.0], N[(1.0 - N[Log[1 + N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.116000000000000006 or 1 < x

   1. Initial program 74.4%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 74.4%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 74.4%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{\frac{y + -1}{x - y}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{y + -1}\right), \left(x - y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y + -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      6. --lowering--.f64 74.4%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right)\right) \]

   6. Applied egg-rr 74.4%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{1}{\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

   8. Applied egg-rr 27.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{y - 1}{x}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y - 1\right), x\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + -1\right), x\right)\right)\right)\right) \]

       4. +-lowering-+.f64 98.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right) \]

   11. Simplified 98.9%

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

   if -0.116000000000000006 < x < 1

   1. Initial program 65.9%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 65.9%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 65.9%

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

   5. Taylor expanded in x around 0

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{y}{y - 1}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{y}{y - 1}\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + -1 \cdot \frac{y}{y - 1}\right)\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(-1 \cdot \frac{y}{y - 1}\right)\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(-1 \cdot \frac{y}{y - 1}\right)\right)\right) \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{y}{y - 1}\right)\right)\right)\right) \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{y}{\mathsf{neg}\left(\left(y - 1\right)\right)}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\left(y - 1\right)\right)\right)\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\left(y + -1\right)\right)\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(\mathsf{neg}\left(\left(-1 + y\right)\right)\right)\right)\right)\right) \]

      12. distribute-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      14. rgt-mult-inverse N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(y \cdot \frac{1}{y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right)\right)\right) \]

      15. unsub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(y \cdot \frac{1}{y} - y\right)\right)\right)\right) \]

      16. rgt-mult-inverse N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \left(1 - y\right)\right)\right)\right) \]

      17. --lowering--.f64 65.1%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(1, y\right)\right)\right)\right) \]

   7. Simplified 65.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.116:\\ \;\;\;\;1 + \log \left(\frac{y + -1}{x}\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y + -1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;y \leq -1.5 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9600000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (log (/ y x)))))
   (if (<= y -1.5e+27)
     t_0
     (if (<= y 9600000000000.0) (- 1.0 (log1p (/ x (+ y -1.0)))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + log((y / x));
	double tmp;
	if (y <= -1.5e+27) {
		tmp = t_0;
	} else if (y <= 9600000000000.0) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.log((y / x));
	double tmp;
	if (y <= -1.5e+27) {
		tmp = t_0;
	} else if (y <= 9600000000000.0) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.log((y / x))
	tmp = 0
	if y <= -1.5e+27:
		tmp = t_0
	elif y <= 9600000000000.0:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + log(Float64(y / x)))
	tmp = 0.0
	if (y <= -1.5e+27)
		tmp = t_0;
	elseif (y <= 9600000000000.0)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.5e+27], t$95$0, If[LessEqual[y, 9600000000000.0], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.49999999999999988e27 or 9.6e12 < y

   1. Initial program 25.3%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 25.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 25.3%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{\frac{y + -1}{x - y}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{y + -1}\right), \left(x - y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y + -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      6. --lowering--.f64 26.0%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right)\right) \]

   6. Applied egg-rr 26.0%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{1}{\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

   8. Applied egg-rr 6.6%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{y - 1}{x}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y - 1\right), x\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + -1\right), x\right)\right)\right)\right) \]

       4. +-lowering-+.f64 50.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right) \]

   11. Simplified 50.3%

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

   12. Taylor expanded in y around inf

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 50.3%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right)\right) \]

   14. Simplified 50.3%

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

   if -1.49999999999999988e27 < y < 9.6e12

   1. Initial program 97.1%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 97.1%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 97.1%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(\frac{x}{y - 1}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(x, \left(y - 1\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(x, \left(y + -1\right)\right)\right)\right) \]

      4. +-lowering-+.f64 92.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   7. Simplified 92.2%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+27}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 9600000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \log \left(\frac{y}{x}\right)\\ \mathbf{if}\;y \leq -1.3 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (log (/ y x)))))
   (if (<= y -1.3e+18) t_0 (if (<= y 1.0) (- 1.0 (log1p (- 0.0 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + log((y / x));
	double tmp;
	if (y <= -1.3e+18) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + Math.log((y / x));
	double tmp;
	if (y <= -1.3e+18) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + math.log((y / x))
	tmp = 0
	if y <= -1.3e+18:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - math.log1p((0.0 - x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + log(Float64(y / x)))
	tmp = 0.0
	if (y <= -1.3e+18)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(0.0 - x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[Log[N[(y / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.3e+18], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e18 or 1 < y

   1. Initial program 26.5%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 26.5%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 26.5%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{\frac{y + -1}{x - y}}\right)\right)\right) \]

      2. associate-/r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{y + -1}\right), \left(x - y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(y + -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \left(x - y\right)\right)\right)\right) \]

      6. --lowering--.f64 27.2%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{+.f64}\left(y, -1\right)\right), \mathsf{\_.f64}\left(x, y\right)\right)\right)\right) \]

   6. Applied egg-rr 27.2%

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

      1. flip3-+ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(\frac{1}{\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}}\right)\right) \]

      3. log-rec N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{neg}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\log \left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{1 \cdot 1 + \left(\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right) - 1 \cdot \left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)\right)}{{1}^{3} + {\left(\frac{1}{y + -1} \cdot \left(x - y\right)\right)}^{3}}\right)\right)\right)\right) \]

   8. Applied egg-rr 6.4%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{y - 1}{x}\right)}\right)\right)\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y - 1\right), x\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right) \]

       3. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\left(y + -1\right), x\right)\right)\right)\right) \]

       4. +-lowering-+.f64 50.8%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right) \]

   11. Simplified 50.8%

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

   12. Taylor expanded in y around inf

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(\frac{y}{x}\right)}\right)\right)\right) \]
   13. Step-by-step derivation

       1. /-lowering-/.f64 50.4%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(y, x\right)\right)\right)\right) \]

   14. Simplified 50.4%

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

   if -1.3e18 < y < 1

   1. Initial program 97.7%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      3. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      4. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

      5. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

      8. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

      9. associate--r- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

      12. +-lowering-+.f64 97.7%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

   3. Simplified 97.7%

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

   5. Taylor expanded in x around -inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\left(-1 \cdot x\right) \cdot \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

      4. neg-sub0 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} + \left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right)\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right) + \frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      9. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(y - 1\right)\right)}\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y - 1\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y + -1\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 + y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      14. distribute-neg-in N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      16. rgt-mult-inverse N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot \frac{1}{y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      17. unsub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot \frac{1}{y} - y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      18. rgt-mult-inverse N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 - y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      19. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \left(x \cdot \left(y - 1\right)\right)\right)\right)\right)\right)\right) \]

      21. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \left(\left(y - 1\right) \cdot x\right)\right)\right)\right)\right)\right) \]

      22. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y - 1\right), x\right)\right)\right)\right)\right)\right) \]

      23. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right)\right)\right) \]

      24. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y + -1\right), x\right)\right)\right)\right)\right)\right) \]

      25. +-lowering-+.f64 97.7%

          \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right)\right)\right) \]

   7. Simplified 97.7%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 91.6%

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

   10. Simplified 91.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+18}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ 1 - \mathsf{log1p}\left(0 - x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 1.0 (log1p (- 0.0 x))))

C

double code(double x, double y) {
	return 1.0 - log1p((0.0 - x));
}

Java

public static double code(double x, double y) {
	return 1.0 - Math.log1p((0.0 - x));
}

Python

def code(x, y):
	return 1.0 - math.log1p((0.0 - x))

Julia

function code(x, y)
	return Float64(1.0 - log1p(Float64(0.0 - x)))
end

Wolfram

code[x_, y_] := N[(1.0 - N[Log[1 + N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

   9. associate--r- N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

   12. +-lowering-+.f64 69.1%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

3. Simplified 69.1%

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

5. Taylor expanded in x around -inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\left(-1 \cdot x\right) \cdot \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} + \left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right)\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right) + \frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(y - 1\right)\right)}\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y - 1\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y + -1\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 + y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   16. rgt-mult-inverse N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot \frac{1}{y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot \frac{1}{y} - y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   18. rgt-mult-inverse N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 - y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   19. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \left(x \cdot \left(y - 1\right)\right)\right)\right)\right)\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \left(\left(y - 1\right) \cdot x\right)\right)\right)\right)\right)\right) \]

   22. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y - 1\right), x\right)\right)\right)\right)\right)\right) \]

   23. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right)\right)\right) \]

   24. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y + -1\right), x\right)\right)\right)\right)\right)\right) \]

   25. +-lowering-+.f64 70.2%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right)\right)\right) \]

7. Simplified 70.2%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   2. neg-lowering-neg.f64 58.6%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

10. Simplified 58.6%

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

11. Final simplification 58.6%

    \[\leadsto 1 - \mathsf{log1p}\left(0 - x\right) \]
12. Add Preprocessing

Alternative 8: 42.9% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 69.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

   9. associate--r- N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

   12. +-lowering-+.f64 69.1%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

3. Simplified 69.1%

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

5. Taylor expanded in x around -inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\left(-1 \cdot x\right) \cdot \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} + \left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right)\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right) + \frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(y - 1\right)\right)}\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y - 1\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y + -1\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 + y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   16. rgt-mult-inverse N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot \frac{1}{y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot \frac{1}{y} - y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   18. rgt-mult-inverse N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 - y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   19. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \left(x \cdot \left(y - 1\right)\right)\right)\right)\right)\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \left(\left(y - 1\right) \cdot x\right)\right)\right)\right)\right)\right) \]

   22. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y - 1\right), x\right)\right)\right)\right)\right)\right) \]

   23. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right)\right)\right) \]

   24. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y + -1\right), x\right)\right)\right)\right)\right)\right) \]

   25. +-lowering-+.f64 70.2%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right)\right)\right) \]

7. Simplified 70.2%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   2. neg-lowering-neg.f64 58.6%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

10. Simplified 58.6%

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

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{1 + x} \]
12. Step-by-step derivation

    1. +-lowering-+.f64 42.3%

       \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{x}\right) \]

13. Simplified 42.3%

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

14. Final simplification 42.3%

    \[\leadsto x + 1 \]
15. Add Preprocessing

Alternative 9: 42.6% accurate, 111.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 69.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \color{blue}{\log \left(1 - \frac{x - y}{1 - y}\right)}\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \log \left(1 + \left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   3. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \left(\mathsf{log1p}\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   4. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(\frac{x - y}{1 - y}\right)\right)\right)\right) \]

   5. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\frac{x - y}{\mathsf{neg}\left(\left(1 - y\right)\right)}\right)\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\left(x - y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\mathsf{neg}\left(\left(1 - y\right)\right)\right)\right)\right)\right) \]

   8. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(0 - \left(1 - y\right)\right)\right)\right)\right) \]

   9. associate--r- N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\left(0 - 1\right) + y\right)\right)\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(-1 + y\right)\right)\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(y + -1\right)\right)\right)\right) \]

   12. +-lowering-+.f64 69.1%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{+.f64}\left(y, -1\right)\right)\right)\right) \]

3. Simplified 69.1%

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

5. Taylor expanded in x around -inf

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot \left(x \cdot \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\left(-1 \cdot x\right) \cdot \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(x\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   4. neg-sub0 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\left(0 - x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} - \frac{1}{y - 1}\right)\right)\right)\right) \]

   6. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\frac{y}{x \cdot \left(y - 1\right)} + \left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right)\right)\right)\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \left(\left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right) + \frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{1}{y - 1}\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   9. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\left(\frac{1}{\mathsf{neg}\left(\left(y - 1\right)\right)}\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y - 1\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(y + -1\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   13. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\mathsf{neg}\left(\left(-1 + y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   14. distribute-neg-in N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\left(\mathsf{neg}\left(-1\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   16. rgt-mult-inverse N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot \frac{1}{y} + \left(\mathsf{neg}\left(y\right)\right)\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   17. unsub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(y \cdot \frac{1}{y} - y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   18. rgt-mult-inverse N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(1 - y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   19. --lowering--.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \left(\frac{y}{x \cdot \left(y - 1\right)}\right)\right)\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \left(x \cdot \left(y - 1\right)\right)\right)\right)\right)\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \left(\left(y - 1\right) \cdot x\right)\right)\right)\right)\right)\right) \]

   22. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y - 1\right), x\right)\right)\right)\right)\right)\right) \]

   23. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y + \left(\mathsf{neg}\left(1\right)\right)\right), x\right)\right)\right)\right)\right)\right) \]

   24. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\left(y + -1\right), x\right)\right)\right)\right)\right)\right) \]

   25. +-lowering-+.f64 70.2%

       \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(1, y\right)\right), \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(\mathsf{+.f64}\left(y, -1\right), x\right)\right)\right)\right)\right)\right) \]

7. Simplified 70.2%

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

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
9. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

   2. neg-lowering-neg.f64 58.6%

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{neg.f64}\left(x\right)\right)\right) \]

10. Simplified 58.6%

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

11. Taylor expanded in x around 0

    \[\leadsto \color{blue}{1} \]
12. Step-by-step derivation

13. Simplified 42.0%

    \[\leadsto \color{blue}{1} \]
14. Add Preprocessing

Developer Target 1: 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 2024152 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8128475261947241/100000000) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 30094271212461764000000000) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))))))

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