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

Percentage Accurate: 72.3% → 98.7%

Time: 15.1s

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: 72.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -225000000000:\\ \;\;\;\;\left(1 - \log \left(1 - x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+146}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log y\right) - \log \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -225000000000.0)
   (- (- 1.0 (log (- 1.0 x))) (log (/ -1.0 y)))
   (if (<= y 4.8e+146)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- (+ 1.0 (log y)) (log (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -225000000000.0) {
		tmp = (1.0 - log((1.0 - x))) - log((-1.0 / y));
	} else if (y <= 4.8e+146) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 + log(y)) - log((x + -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -225000000000.0) {
		tmp = (1.0 - Math.log((1.0 - x))) - Math.log((-1.0 / y));
	} else if (y <= 4.8e+146) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = (1.0 + Math.log(y)) - Math.log((x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -225000000000.0:
		tmp = (1.0 - math.log((1.0 - x))) - math.log((-1.0 / y))
	elif y <= 4.8e+146:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = (1.0 + math.log(y)) - math.log((x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -225000000000.0)
		tmp = Float64(Float64(1.0 - log(Float64(1.0 - x))) - log(Float64(-1.0 / y)));
	elseif (y <= 4.8e+146)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(Float64(1.0 + log(y)) - log(Float64(x + -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -225000000000.0], N[(N[(1.0 - N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+146], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Log[y], $MachinePrecision]), $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.25e11

   1. Initial program 24.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 24.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in y around -inf

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

   7. Simplified 99.5%

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

   if -2.25e11 < y < 4.8000000000000004e146

   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

   if 4.8000000000000004e146 < y

   1. Initial program 21.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 21.1%

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

   3. Simplified 21.1%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -225000000000:\\ \;\;\;\;\left(1 - \log \left(1 - x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+146}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log y\right) - \log \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+19}:\\ \;\;\;\;\left(1 - \log \left(0 - x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+146}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(1 - \frac{y}{x}\right) \cdot \frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log y\right) - \log \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.8e+19)
   (- (- 1.0 (log (- 0.0 x))) (log (/ -1.0 y)))
   (if (<= y 4.8e+146)
     (- 1.0 (log1p (* (- 1.0 (/ y x)) (/ x (+ y -1.0)))))
     (- (+ 1.0 (log y)) (log (+ x -1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+19) {
		tmp = (1.0 - log((0.0 - x))) - log((-1.0 / y));
	} else if (y <= 4.8e+146) {
		tmp = 1.0 - log1p(((1.0 - (y / x)) * (x / (y + -1.0))));
	} else {
		tmp = (1.0 + log(y)) - log((x + -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.8e+19) {
		tmp = (1.0 - Math.log((0.0 - x))) - Math.log((-1.0 / y));
	} else if (y <= 4.8e+146) {
		tmp = 1.0 - Math.log1p(((1.0 - (y / x)) * (x / (y + -1.0))));
	} else {
		tmp = (1.0 + Math.log(y)) - Math.log((x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.8e+19:
		tmp = (1.0 - math.log((0.0 - x))) - math.log((-1.0 / y))
	elif y <= 4.8e+146:
		tmp = 1.0 - math.log1p(((1.0 - (y / x)) * (x / (y + -1.0))))
	else:
		tmp = (1.0 + math.log(y)) - math.log((x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.8e+19)
		tmp = Float64(Float64(1.0 - log(Float64(0.0 - x))) - log(Float64(-1.0 / y)));
	elseif (y <= 4.8e+146)
		tmp = Float64(1.0 - log1p(Float64(Float64(1.0 - Float64(y / x)) * Float64(x / Float64(y + -1.0)))));
	else
		tmp = Float64(Float64(1.0 + log(y)) - log(Float64(x + -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.8e+19], N[(N[(1.0 - N[Log[N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+146], N[(1.0 - N[Log[1 + N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Log[y], $MachinePrecision]), $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.8e19

   1. Initial program 25.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 25.1%

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

   3. Simplified 25.1%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 26.6%

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

   6. Applied egg-rr 26.6%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 26.6%

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

   10. Taylor expanded in y around -inf

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

       1. associate--r+ N/A

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

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

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

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

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

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

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

       5. mul-1-neg N/A

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

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

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

       7. metadata-eval N/A

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

       8. distribute-neg-frac N/A

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

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

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

       10. distribute-neg-frac N/A

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

       11. metadata-eval N/A

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

       12. /-lowering-/.f64 39.0%

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

   12. Simplified 39.0%

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

   if -3.8e19 < y < 4.8000000000000004e146

   1. Initial program 99.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

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

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

      8. sub-neg N/A

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

      9. metadata-eval N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. unsub-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 99.4%

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

   7. Simplified 99.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 4.8000000000000004e146 < y

   1. Initial program 21.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 21.1%

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

   3. Simplified 21.1%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 99.1%

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

   7. Simplified 99.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+19}:\\ \;\;\;\;\left(1 - \log \left(0 - x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+146}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\left(1 - \frac{y}{x}\right) \cdot \frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log y\right) - \log \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+48}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log y\right) - \log \left(x + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.2e+48)
   (- 1.0 (log1p (/ x (+ y -1.0))))
   (- (+ 1.0 (log y)) (log (+ x -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+48) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = (1.0 + log(y)) - log((x + -1.0));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e+48) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = (1.0 + Math.log(y)) - Math.log((x + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.2e+48:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = (1.0 + math.log(y)) - math.log((x + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.2e+48)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = Float64(Float64(1.0 + log(y)) - log(Float64(x + -1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.2e+48], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Log[y], $MachinePrecision]), $MachinePrecision] - N[Log[N[(x + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.2000000000000001e48

   1. Initial program 72.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 72.0%

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

   3. Simplified 72.0%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 74.0%

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

   7. Simplified 74.0%

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

   if 1.2000000000000001e48 < y

   1. Initial program 60.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 60.5%

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

   3. Simplified 60.5%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{+48}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \log y\right) - \log \left(x + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.4% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.0)
   (- 1.0 (log1p (/ x y)))
   (if (<= y 1.0) (- 1.0 (log1p (- 0.0 x))) (- 1.0 (log1p (+ -1.0 (/ x y)))))))

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   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. 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 34.1%

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

   7. Simplified 34.1%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 33.8%

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

   10. Simplified 33.8%

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

   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. Step-by-step derivation

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. clear-num N/A

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

      4. flip-- N/A

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

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

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

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

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

      7. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
   8. 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 98.0%

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

   9. Simplified 98.0%

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

   if 1 < y

   1. Initial program 68.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 68.4%

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

   3. Simplified 68.4%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 68.4%

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

   6. Applied egg-rr 68.4%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 67.3%

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

   10. Taylor expanded in y around inf

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

       1. sub-neg N/A

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

       2. metadata-eval N/A

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

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

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

       4. /-lowering-/.f64 67.3%

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

   12. Simplified 67.3%

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

4. Final simplification 72.4%

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

Alternative 5: 72.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-7}:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log1p (/ x y)))))
   (if (<= y -1.0) t_0 (if (<= y 6.7e-7) (- 1.0 (log1p (- 0.0 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log1p((x / y));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 6.7e-7) {
		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.log1p((x / y));
	double tmp;
	if (y <= -1.0) {
		tmp = t_0;
	} else if (y <= 6.7e-7) {
		tmp = 1.0 - Math.log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log1p((x / y))
	tmp = 0
	if y <= -1.0:
		tmp = t_0
	elif y <= 6.7e-7:
		tmp = 1.0 - math.log1p((0.0 - x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log1p(Float64(x / y)))
	tmp = 0.0
	if (y <= -1.0)
		tmp = t_0;
	elseif (y <= 6.7e-7)
		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[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.0], t$95$0, If[LessEqual[y, 6.7e-7], 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 6.70000000000000044e-7 < y

   1. Initial program 36.7%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 36.7%

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

   3. Simplified 36.7%

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

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(\frac{x}{y - 1}\right)}\right)\right) \]
   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 41.2%

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

   7. Simplified 41.2%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 40.7%

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

   10. Simplified 40.7%

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

   if -1 < y < 6.70000000000000044e-7

   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. Step-by-step derivation

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. clear-num N/A

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

      4. flip-- N/A

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

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

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

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

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

      7. --lowering--.f64 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
   8. 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 98.7%

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

   9. Simplified 98.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 6.7 \cdot 10^{-7}:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.1%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 71.1%

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

3. Simplified 71.1%

   \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
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 72.8%

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

7. Simplified 72.8%

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

Alternative 7: 62.8% 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 71.1%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 71.1%

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

3. Simplified 71.1%

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

   1. flip-- N/A

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

   2. clear-num N/A

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

   3. clear-num N/A

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

   4. flip-- N/A

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

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

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

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

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

   7. --lowering--.f64 71.1%

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

6. Applied egg-rr 71.1%

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

7. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\color{blue}{\left(-1 \cdot x\right)}\right)\right) \]
8. 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 58.8%

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

9. Simplified 58.8%

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

10. Final simplification 58.8%

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

Alternative 8: 45.0% 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 71.1%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 71.1%

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

3. Simplified 71.1%

   \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
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 72.8%

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

7. Simplified 72.8%

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

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. +-lowering-+.f64 41.6%

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

10. Simplified 41.6%

    \[\leadsto \color{blue}{1 - \frac{x}{y + -1}} \]
11. Add Preprocessing

Alternative 9: 43.3% 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 71.1%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 71.1%

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

3. Simplified 71.1%

   \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
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 72.8%

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

7. Simplified 72.8%

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

8. Taylor expanded in x around 0

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

10. Simplified 39.8%

    \[\leadsto \color{blue}{1} \]
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 2024161 
(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))))))