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

Percentage Accurate: 72.3% → 99.6%

Time: 13.5s

Alternatives: 10

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: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -13600:\\ \;\;\;\;1 + \left(\frac{-1 + \frac{-0.5}{y}}{y} - \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -13600.0)
   (+ 1.0 (- (/ (+ -1.0 (/ -0.5 y)) y) (+ (log (- 1.0 x)) (log (/ -1.0 y)))))
   (if (<= y 5e+14)
     (- 1.0 (log1p (/ (- x y) (+ y -1.0))))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -13600.0) {
		tmp = 1.0 + (((-1.0 + (-0.5 / y)) / y) - (log((1.0 - x)) + log((-1.0 / y))));
	} else if (y <= 5e+14) {
		tmp = 1.0 - log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -13600.0) {
		tmp = 1.0 + (((-1.0 + (-0.5 / y)) / y) - (Math.log((1.0 - x)) + Math.log((-1.0 / y))));
	} else if (y <= 5e+14) {
		tmp = 1.0 - Math.log1p(((x - y) / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -13600.0:
		tmp = 1.0 + (((-1.0 + (-0.5 / y)) / y) - (math.log((1.0 - x)) + math.log((-1.0 / y))))
	elif y <= 5e+14:
		tmp = 1.0 - math.log1p(((x - y) / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -13600.0)
		tmp = Float64(1.0 + Float64(Float64(Float64(-1.0 + Float64(-0.5 / y)) / y) - Float64(log(Float64(1.0 - x)) + log(Float64(-1.0 / y)))));
	elseif (y <= 5e+14)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -13600.0], N[(1.0 + N[(N[(N[(-1.0 + N[(-0.5 / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] - N[(N[Log[N[(1.0 - x), $MachinePrecision]], $MachinePrecision] + N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+14], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -13600

   1. Initial program 20.8%

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

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

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

      2. sub-neg N/A

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

      3. log1p-define N/A

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

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

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

      5. distribute-neg-frac2 N/A

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

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

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

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

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 20.8%

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

   3. Simplified 20.8%

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

   5. Taylor expanded in y around -inf

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

   7. Simplified 99.5%

      \[\leadsto 1 - \color{blue}{\left(\left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right) - \frac{-1 + \frac{-0.5}{y}}{y}\right)} \]

   if -13600 < y < 5e14

   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 5e14 < y

   1. Initial program 36.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 36.3%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 36.3%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-+l- N/A

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

      7. metadata-eval N/A

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

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

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

      9. /-lowering-/.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -13600:\\ \;\;\;\;1 + \left(\frac{-1 + \frac{-0.5}{y}}{y} - \left(\log \left(1 - x\right) + \log \left(\frac{-1}{y}\right)\right)\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;\left(1 - \log \left(1 - x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\frac{1 - \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}{\left(-1 - y\right) \cdot \left(1 - \left(y \cdot y\right) \cdot \left(-1 - y \cdot y\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e+18)
   (- (- 1.0 (log (- 1.0 x))) (log (/ -1.0 y)))
   (if (<= y 5e+14)
     (-
      1.0
      (log1p
       (/
        (- x y)
        (/
         (- 1.0 (* (* y y) (* (* y y) (* y y))))
         (* (- -1.0 y) (- 1.0 (* (* y y) (- -1.0 (* y y)))))))))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+18) {
		tmp = (1.0 - log((1.0 - x))) - log((-1.0 / y));
	} else if (y <= 5e+14) {
		tmp = 1.0 - log1p(((x - y) / ((1.0 - ((y * y) * ((y * y) * (y * y)))) / ((-1.0 - y) * (1.0 - ((y * y) * (-1.0 - (y * y))))))));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+18) {
		tmp = (1.0 - Math.log((1.0 - x))) - Math.log((-1.0 / y));
	} else if (y <= 5e+14) {
		tmp = 1.0 - Math.log1p(((x - y) / ((1.0 - ((y * y) * ((y * y) * (y * y)))) / ((-1.0 - y) * (1.0 - ((y * y) * (-1.0 - (y * y))))))));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e+18:
		tmp = (1.0 - math.log((1.0 - x))) - math.log((-1.0 / y))
	elif y <= 5e+14:
		tmp = 1.0 - math.log1p(((x - y) / ((1.0 - ((y * y) * ((y * y) * (y * y)))) / ((-1.0 - y) * (1.0 - ((y * y) * (-1.0 - (y * y))))))))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e+18)
		tmp = Float64(Float64(1.0 - log(Float64(1.0 - x))) - log(Float64(-1.0 / y)));
	elseif (y <= 5e+14)
		tmp = Float64(1.0 - log1p(Float64(Float64(x - y) / Float64(Float64(1.0 - Float64(Float64(y * y) * Float64(Float64(y * y) * Float64(y * y)))) / Float64(Float64(-1.0 - y) * Float64(1.0 - Float64(Float64(y * y) * Float64(-1.0 - Float64(y * y)))))))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e+18], 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, 5e+14], N[(1.0 - N[Log[1 + N[(N[(x - y), $MachinePrecision] / N[(N[(1.0 - N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(-1.0 - y), $MachinePrecision] * N[(1.0 - N[(N[(y * y), $MachinePrecision] * N[(-1.0 - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2e18

   1. Initial program 15.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 15.6%

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

   3. Simplified 15.6%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 -3.2e18 < y < 5e14

   1. Initial program 99.9%

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

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

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

      2. sub-neg N/A

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

      3. log1p-define N/A

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

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

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

      5. distribute-neg-frac2 N/A

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

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

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

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

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

      1. +-commutative N/A

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

      2. flip-+ N/A

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

      3. div-inv N/A

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

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

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

      5. metadata-eval N/A

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

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

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

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

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

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

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

      9. --lowering--.f64 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. un-div-inv N/A

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

      2. flip3-- N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{\frac{{1}^{3} - {\left(y \cdot y\right)}^{3}}{1 \cdot 1 + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1 \cdot \left(y \cdot y\right)\right)}}{-1 - y}\right)\right)\right)\right) \]

      3. associate-/l/ N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \left(\frac{{1}^{3} - {\left(y \cdot y\right)}^{3}}{\left(-1 - y\right) \cdot \left(1 \cdot 1 + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1 \cdot \left(y \cdot y\right)\right)\right)}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(\left({1}^{3} - {\left(y \cdot y\right)}^{3}\right), \left(\left(-1 - y\right) \cdot \left(1 \cdot 1 + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1 \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(\left(1 - {\left(y \cdot y\right)}^{3}\right), \left(\left(-1 - y\right) \cdot \left(1 \cdot 1 + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1 \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), \mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \left({\left(y \cdot y\right)}^{3}\right)\right), \left(\left(-1 - y\right) \cdot \left(1 \cdot 1 + \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right) + 1 \cdot \left(y \cdot y\right)\right)\right)\right)\right)\right)\right)\right) \]

      7. cube-mult N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. metadata-eval N/A

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

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

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

   8. Applied egg-rr 99.9%

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

   if 5e14 < y

   1. Initial program 36.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 36.3%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 36.3%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-+l- N/A

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

      7. metadata-eval N/A

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

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

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

      9. /-lowering-/.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;\left(1 - \log \left(1 - x\right)\right) - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x - y}{\frac{1 - \left(y \cdot y\right) \cdot \left(\left(y \cdot y\right) \cdot \left(y \cdot y\right)\right)}{\left(-1 - y\right) \cdot \left(1 - \left(y \cdot y\right) \cdot \left(-1 - y \cdot y\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(x \cdot \left(\frac{1}{y + -1} + \frac{\frac{y}{1 - y}}{x}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (/ (- x y) (+ y -1.0))) 5e-7)
   (- 1.0 (log (/ x y)))
   (- 1.0 (log1p (* x (+ (/ 1.0 (+ y -1.0)) (/ (/ y (- 1.0 y)) x)))))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + ((x - y) / (y + -1.0))) <= 5e-7) {
		tmp = 1.0 - log((x / y));
	} else {
		tmp = 1.0 - log1p((x * ((1.0 / (y + -1.0)) + ((y / (1.0 - y)) / x))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 + ((x - y) / (y + -1.0))) <= 5e-7) {
		tmp = 1.0 - Math.log((x / y));
	} else {
		tmp = 1.0 - Math.log1p((x * ((1.0 / (y + -1.0)) + ((y / (1.0 - y)) / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 + ((x - y) / (y + -1.0))) <= 5e-7:
		tmp = 1.0 - math.log((x / y))
	else:
		tmp = 1.0 - math.log1p((x * ((1.0 / (y + -1.0)) + ((y / (1.0 - y)) / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + Float64(Float64(x - y) / Float64(y + -1.0))) <= 5e-7)
		tmp = Float64(1.0 - log(Float64(x / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(x * Float64(Float64(1.0 / Float64(y + -1.0)) + Float64(Float64(y / Float64(1.0 - y)) / x)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-7], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(x * N[(N[(1.0 / N[(y + -1.0), $MachinePrecision]), $MachinePrecision] + N[(N[(y / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 4.99999999999999977e-7

   1. Initial program 5.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 5.4%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-+l- N/A

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

      7. metadata-eval N/A

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

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

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

      9. /-lowering-/.f64 32.5%

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

   9. Applied egg-rr 32.5%

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

   if 4.99999999999999977e-7 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.9%

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

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

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

      2. sub-neg N/A

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

      3. log1p-define N/A

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

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

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

      5. distribute-neg-frac2 N/A

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

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

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

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

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf

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

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

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

      2. +-commutative N/A

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

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

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

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

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

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

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

      8. associate-*r/ N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. associate-*r/ N/A

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

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

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

      13. mul-1-neg N/A

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

      14. distribute-neg-frac2 N/A

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

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

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

      16. sub-neg N/A

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

      17. metadata-eval N/A

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

      18. +-commutative N/A

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

      19. distribute-neg-in N/A

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

      20. metadata-eval N/A

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

      21. rgt-mult-inverse N/A

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

      22. unsub-neg N/A

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

      23. rgt-mult-inverse N/A

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

      24. --lowering--.f64 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 79.6%

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

Alternative 4: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{y + -1}\\ \mathbf{if}\;1 + t\_0 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (+ y -1.0))))
   (if (<= (+ 1.0 t_0) 5e-7) (- 1.0 (log (/ x y))) (- 1.0 (log1p t_0)))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (y + -1.0);
	double tmp;
	if ((1.0 + t_0) <= 5e-7) {
		tmp = 1.0 - log((x / y));
	} else {
		tmp = 1.0 - log1p(t_0);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / (y + -1.0);
	double tmp;
	if ((1.0 + t_0) <= 5e-7) {
		tmp = 1.0 - Math.log((x / y));
	} else {
		tmp = 1.0 - Math.log1p(t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / (y + -1.0)
	tmp = 0
	if (1.0 + t_0) <= 5e-7:
		tmp = 1.0 - math.log((x / y))
	else:
		tmp = 1.0 - math.log1p(t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(y + -1.0))
	tmp = 0.0
	if (Float64(1.0 + t_0) <= 5e-7)
		tmp = Float64(1.0 - log(Float64(x / y)));
	else
		tmp = Float64(1.0 - log1p(t_0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(1.0 + t$95$0), $MachinePrecision], 5e-7], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 4.99999999999999977e-7

   1. Initial program 5.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 5.4%

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

   3. Simplified 5.4%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 5.4%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-+l- N/A

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

      7. metadata-eval N/A

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

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

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

      9. /-lowering-/.f64 32.5%

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

   9. Applied egg-rr 32.5%

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

   if 4.99999999999999977e-7 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))

   1. Initial program 99.9%

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

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

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

      2. sub-neg N/A

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

      3. log1p-define N/A

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

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

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

      5. distribute-neg-frac2 N/A

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

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

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

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

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 79.6%

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

Alternative 5: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -280000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (/ x y)))))
   (if (<= y -280000.0) t_0 (if (<= y 1.0) (- 1.0 (log1p (- 0.0 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((x / y));
	double tmp;
	if (y <= -280000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log((x / y));
	double tmp;
	if (y <= -280000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log((x / y))
	tmp = 0
	if y <= -280000.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - math.log1p((0.0 - x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(x / y)))
	tmp = 0.0
	if (y <= -280000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(0.0 - x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -280000.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e5 or 1 < y

   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

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

   7. Simplified 23.8%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-+l- N/A

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

      7. metadata-eval N/A

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

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

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

      9. /-lowering-/.f64 45.3%

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

   9. Applied egg-rr 45.3%

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

   if -2.8e5 < y < 1

   1. Initial program 99.9%

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

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

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

      2. sub-neg N/A

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

      3. log1p-define N/A

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

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

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

      5. distribute-neg-frac2 N/A

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

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

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

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

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 98.1%

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

   7. Simplified 98.1%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -280000:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \mathsf{log1p}\left(\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -240000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(0 - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log1p (/ x y)))))
   (if (<= y -240000.0) t_0 (if (<= y 1.0) (- 1.0 (log1p (- 0.0 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log1p((x / y));
	double tmp;
	if (y <= -240000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = 1.0 - Math.log1p((x / y));
	double tmp;
	if (y <= -240000.0) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p((0.0 - x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 - math.log1p((x / y))
	tmp = 0
	if y <= -240000.0:
		tmp = t_0
	elif y <= 1.0:
		tmp = 1.0 - math.log1p((0.0 - x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 - log1p(Float64(x / y)))
	tmp = 0.0
	if (y <= -240000.0)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(0.0 - x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[1 + N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -240000.0], t$95$0, If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + N[(0.0 - x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e5 or 1 < y

   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 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 31.1%

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

   7. Simplified 31.1%

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

   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 30.0%

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

   10. Simplified 30.0%

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

   if -2.4e5 < y < 1

   1. Initial program 99.9%

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

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

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

      2. sub-neg N/A

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

      3. log1p-define N/A

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

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

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

      5. distribute-neg-frac2 N/A

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

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

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

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

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

      8. neg-sub0 N/A

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

      9. associate--r- N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. +-lowering-+.f64 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 98.1%

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

   7. Simplified 98.1%

      \[\leadsto 1 - \mathsf{log1p}\left(\color{blue}{0 - x}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e+14) (- 1.0 (log1p (/ x (+ y -1.0)))) (- 1.0 (log (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+14) {
		tmp = 1.0 - log1p((x / (y + -1.0)));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+14) {
		tmp = 1.0 - Math.log1p((x / (y + -1.0)));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+14:
		tmp = 1.0 - math.log1p((x / (y + -1.0)))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+14)
		tmp = Float64(1.0 - log1p(Float64(x / Float64(y + -1.0))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+14], N[(1.0 - N[Log[1 + N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5e14

   1. Initial program 74.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 74.3%

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

   3. Simplified 74.3%

      \[\leadsto \color{blue}{1 - \mathsf{log1p}\left(\frac{x - y}{y + -1}\right)} \]
   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 75.7%

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

   7. Simplified 75.7%

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

   if 5e14 < y

   1. Initial program 36.3%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 36.3%

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

   3. Simplified 36.3%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 36.3%

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

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

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

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

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

      3. +-commutative N/A

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

      6. associate-+l- N/A

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

      7. metadata-eval N/A

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

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

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

      9. /-lowering-/.f64 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+14}:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{x}{y + -1}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.7% 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 71.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 71.5%

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

3. Simplified 71.5%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. neg-sub0 N/A

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

   3. --lowering--.f64 64.9%

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

7. Simplified 64.9%

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

Alternative 9: 45.3% accurate, 15.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 71.5%

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

3. Simplified 71.5%

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

5. Taylor expanded in x around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-lowering-+.f64 73.2%

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

7. Simplified 73.2%

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

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 44.2%

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

10. Simplified 44.2%

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

11. Final simplification 44.2%

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

Alternative 10: 43.7% 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 71.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 71.5%

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

3. Simplified 71.5%

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

5. Taylor expanded in x around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. +-lowering-+.f64 73.2%

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

7. Simplified 73.2%

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

8. Taylor expanded in x around 0

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

10. Simplified 42.8%

    \[\leadsto \color{blue}{1} \]
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 2024160 
(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))))))