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

Percentage Accurate: 71.5% → 99.8%

Time: 11.9s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 71.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\frac{1}{y + -1} + \frac{\frac{y}{1 - y}}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999)
   (- 1.0 (log1p (* x (+ (/ 1.0 (+ y -1.0)) (/ (/ y (- 1.0 y)) x)))))
   (+ 1.0 (log (/ y (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999) {
		tmp = 1.0 - log1p((x * ((1.0 / (y + -1.0)) + ((y / (1.0 - y)) / x))));
	} else {
		tmp = 1.0 + log((y / (x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999) {
		tmp = 1.0 - Math.log1p((x * ((1.0 / (y + -1.0)) + ((y / (1.0 - y)) / x))));
	} else {
		tmp = 1.0 + Math.log((y / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999:
		tmp = 1.0 - math.log1p((x * ((1.0 / (y + -1.0)) + ((y / (1.0 - y)) / x))))
	else:
		tmp = 1.0 + math.log((y / (x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999)
		tmp = Float64(1.0 - log1p(Float64(x * Float64(Float64(1.0 / Float64(y + -1.0)) + Float64(Float64(y / Float64(1.0 - y)) / x)))));
	else
		tmp = Float64(1.0 + log(Float64(y / Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999], N[(1.0 - N[Log[1 + N[(x * N[(N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 + N[Log[N[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998999999999999999

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

      12. /-lowering-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. distribute-neg-in N/A

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

      20. metadata-eval N/A

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

      21. rgt-mult-inverse N/A

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

      22. unsub-neg N/A

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

      23. rgt-mult-inverse N/A

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

      24. --lowering--.f64 100.0%

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

   7. Simplified 100.0%

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

   if 0.998999999999999999 < (/.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. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 20.3%

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

   7. Simplified 20.3%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

      4. diff-log N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\frac{1}{y + -1} + \frac{\frac{y}{1 - y}}{x}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999)
   (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
   (+ 1.0 (log (/ y (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + log((y / (x + -1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 + Math.log((y / (x + -1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 + math.log((y / (x + -1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 + log(Float64(y / Float64(x + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999], 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[(y / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998999999999999999

   1. Initial program 99.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 100.0%

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

   3. Simplified 100.0%

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

   if 0.998999999999999999 < (/.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. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 20.3%

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

   7. Simplified 20.3%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

      4. diff-log N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 99.8%

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

   9. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \log \left(\frac{y}{x + -1}\right)\\ \mathbf{if}\;y \leq -45:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1020000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (log (/ y (+ x -1.0))))))
   (if (<= y -45.0)
     t_0
     (if (<= y 1020000000000.0) (- 1.0 (log1p (/ x (+ y -1.0)))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + log((y / (x + -1.0)));
	double tmp;
	if (y <= -45.0) {
		tmp = t_0;
	} else if (y <= 1020000000000.0) {
		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 + -1.0)));
	double tmp;
	if (y <= -45.0) {
		tmp = t_0;
	} else if (y <= 1020000000000.0) {
		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 + -1.0)))
	tmp = 0
	if y <= -45.0:
		tmp = t_0
	elif y <= 1020000000000.0:
		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 / Float64(x + -1.0))))
	tmp = 0.0
	if (y <= -45.0)
		tmp = t_0;
	elseif (y <= 1020000000000.0)
		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 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -45.0], t$95$0, If[LessEqual[y, 1020000000000.0], 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 < -45 or 1.02e12 < y

   1. Initial program 30.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 30.0%

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

   3. Simplified 30.0%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 26.5%

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

   7. Simplified 26.5%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

      4. diff-log N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 99.2%

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

   9. Applied egg-rr 99.2%

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

   if -45 < y < 1.02e12

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -45:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \mathbf{elif}\;y \leq 1020000000000:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(\frac{y}{x + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (log (/ y (+ x -1.0))))))
   (if (<= y -1.0) 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 + -1.0)));
	double tmp;
	if (y <= -1.0) {
		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 + -1.0)));
	double tmp;
	if (y <= -1.0) {
		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 + -1.0)))
	tmp = 0
	if y <= -1.0:
		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 / Float64(x + -1.0))))
	tmp = 0.0
	if (y <= -1.0)
		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 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], 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 or 1 < y

   1. Initial program 30.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 30.7%

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

   3. Simplified 30.7%

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

   5. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. sub-neg N/A

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

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

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

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

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

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

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

      10. sub-neg N/A

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

      11. metadata-eval N/A

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

      12. +-commutative N/A

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

      13. +-lowering-+.f64 26.2%

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

   7. Simplified 26.2%

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

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

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

      4. diff-log N/A

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

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

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

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

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 98.5%

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

   9. Applied egg-rr 98.5%

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

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
   6. 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-sub0 N/A

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

      3. --lowering--.f64 97.8%

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

   7. Simplified 97.8%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
   8. Step-by-step derivation

      1. sub0-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 97.8%

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

   9. Applied egg-rr 97.8%

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

4. Final simplification 98.1%

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

Alternative 5: 79.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -10.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ x y)))))
   (if (<= y -10.5) 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((x / y));
	double tmp;
	if (y <= -10.5) {
		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((x / y));
	double tmp;
	if (y <= -10.5) {
		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((x / y))
	tmp = 0
	if y <= -10.5:
		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(x / y)))
	tmp = 0.0
	if (y <= -10.5)
		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[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -10.5], 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 < -10.5 or 1 < y

   1. Initial program 30.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 30.7%

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

   3. Simplified 30.7%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 28.6%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-+l- N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. --lowering--.f64 N/A

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

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

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

      11. metadata-eval 56.0%

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

   9. Applied egg-rr 56.0%

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

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

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

       2. --rgt-identity N/A

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

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

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

       4. /-lowering-/.f64 56.0%

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

   11. Applied egg-rr 56.0%

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

   if -10.5 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
   6. 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-sub0 N/A

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

      3. --lowering--.f64 97.8%

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

   7. Simplified 97.8%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
   8. Step-by-step derivation

      1. sub0-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 97.8%

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

   9. Applied egg-rr 97.8%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -10.5:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.2% 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 73.5%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 73.5%

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

3. Simplified 73.5%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
6. 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-sub0 N/A

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

   3. --lowering--.f64 63.9%

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

7. Simplified 63.9%

   \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
8. Step-by-step derivation

   1. sub0-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 63.9%

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

9. Applied egg-rr 63.9%

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

10. Final simplification 63.9%

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

Alternative 7: 44.4% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.5%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 73.5%

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

3. Simplified 73.5%

   \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
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 73.3%

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

7. Simplified 73.3%

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

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + -1 \cdot \frac{x}{y - 1}} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

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

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

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

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. +-commutative N/A

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

   8. +-lowering-+.f64 45.3%

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

10. Simplified 45.3%

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

11. Final simplification 45.3%

    \[\leadsto 1 - \frac{x}{y + -1} \]
12. Add Preprocessing

Alternative 8: 42.8% 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 73.5%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 73.5%

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

3. Simplified 73.5%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
6. 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-sub0 N/A

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

   3. --lowering--.f64 63.9%

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

7. Simplified 63.9%

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

8. Taylor expanded in x around 0

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

10. Simplified 44.1%

    \[\leadsto \color{blue}{1} \]
11. 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 2024158 
(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))))))