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

Percentage Accurate: 72.0% → 99.9%

Time: 20.0s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.0% 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: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.998999999999999999

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. log1p-def 99.9%

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

      3. distribute-neg-frac 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. neg-sub0 99.9%

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

      7. +-commutative 99.9%

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

      8. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   if 0.998999999999999999 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 7.7%

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

      1. sub-neg 7.7%

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

      2. log1p-def 7.7%

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

      3. distribute-neg-frac 7.7%

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

      4. neg-sub0 7.7%

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

      5. associate-+l- 7.7%

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

      6. neg-sub0 7.7%

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

      7. +-commutative 7.7%

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

      8. sub-neg 7.7%

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

   3. Simplified 7.7%

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

      1. div-inv 9.6%

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

      2. *-commutative 9.6%

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

   5. Applied egg-rr 9.6%

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

   7. Applied egg-rr 7.7%

      \[\leadsto 1 - \color{blue}{\log \left(\frac{1}{1 + \frac{\frac{y - x}{1 - y} - 1}{\frac{1 - y}{y - x}}} \cdot \left(1 + {\left(\frac{y - x}{1 - y}\right)}^{3}\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 7.7%

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

      2. div-inv 7.7%

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

      3. metadata-eval 7.7%

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

      4. div-sub 7.7%

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

      5. div-inv 7.7%

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

      6. clear-num 7.7%

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

      7. clear-num 7.7%

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

      8. metadata-eval 7.7%

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

      9. *-rgt-identity 7.7%

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

      10. *-commutative 7.7%

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

   9. Applied egg-rr 11.8%

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

   10. Taylor expanded in y around inf 99.7%

       \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + \frac{x}{{y}^{2}}\right) - \left(\frac{1}{y} + \frac{1}{{y}^{2}}\right)\right)} \]
   11. Step-by-step derivation

       1. associate--l+ 99.7%

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

       2. unpow2 99.7%

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

       3. unpow2 99.7%

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

   12. Simplified 99.7%

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

4. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.999998)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (/ (+ x -1.0) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999998) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999998) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x - y) / (1.0 - y)) <= 0.999998:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((x + -1.0) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999998)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999998], N[(1.0 - N[Log[1 + N[(N[(y - x), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 1 y)) < 0.999998000000000054

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. log1p-def 99.8%

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

      3. distribute-neg-frac 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate-+l- 99.8%

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

      6. neg-sub0 99.8%

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

      7. +-commutative 99.8%

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

      8. sub-neg 99.8%

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

   3. Simplified 99.8%

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

   if 0.999998000000000054 < (/.f64 (-.f64 x y) (-.f64 1 y))

   1. Initial program 6.6%

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

      1. sub-neg 6.6%

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

      2. log1p-def 6.6%

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

      3. distribute-neg-frac 6.6%

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

      4. neg-sub0 6.6%

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

      5. associate-+l- 6.6%

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

      6. neg-sub0 6.6%

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

      7. +-commutative 6.6%

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

      8. sub-neg 6.6%

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

   3. Simplified 6.6%

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

      1. div-inv 8.5%

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

      2. *-commutative 8.5%

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

   5. Applied egg-rr 8.5%

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

   7. Applied egg-rr 6.6%

      \[\leadsto 1 - \color{blue}{\log \left(\frac{1}{1 + \frac{\frac{y - x}{1 - y} - 1}{\frac{1 - y}{y - x}}} \cdot \left(1 + {\left(\frac{y - x}{1 - y}\right)}^{3}\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 6.6%

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

      2. div-inv 6.6%

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

      3. metadata-eval 6.6%

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

      4. div-sub 6.6%

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

      5. div-inv 6.6%

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

      6. clear-num 6.6%

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

      7. clear-num 6.6%

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

      8. metadata-eval 6.6%

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

      9. *-rgt-identity 6.6%

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

      10. *-commutative 6.6%

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

   9. Applied egg-rr 10.8%

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

   10. Taylor expanded in y around inf 99.5%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.999998:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.65) (not (<= y 1.0)))
   (- 1.0 (log (/ (+ x -1.0) y)))
   (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.65) || !(y <= 1.0)) {
		tmp = 1.0 - log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.65) || !(y <= 1.0)) {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.65) or not (y <= 1.0):
		tmp = 1.0 - math.log(((x + -1.0) / y))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.65) || !(y <= 1.0))
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.65], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(1.0 - N[Log[N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6499999999999999 or 1 < y

   1. Initial program 35.4%

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

      1. sub-neg 35.4%

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

      2. log1p-def 35.4%

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

      3. distribute-neg-frac 35.4%

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

      4. neg-sub0 35.4%

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

      5. associate-+l- 35.4%

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

      6. neg-sub0 35.4%

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

      7. +-commutative 35.4%

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

      8. sub-neg 35.4%

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

   3. Simplified 35.4%

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

      1. div-inv 36.7%

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

      2. *-commutative 36.7%

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

   5. Applied egg-rr 36.7%

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

   7. Applied egg-rr 26.2%

      \[\leadsto 1 - \color{blue}{\log \left(\frac{1}{1 + \frac{\frac{y - x}{1 - y} - 1}{\frac{1 - y}{y - x}}} \cdot \left(1 + {\left(\frac{y - x}{1 - y}\right)}^{3}\right)\right)} \]
   8. Step-by-step derivation

      1. *-commutative 26.2%

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

      2. div-inv 26.2%

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

      3. metadata-eval 26.2%

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

      4. div-sub 26.2%

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

      5. div-inv 26.2%

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

      6. clear-num 26.2%

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

      7. clear-num 26.2%

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

      8. metadata-eval 26.2%

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

      9. *-rgt-identity 26.2%

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

      10. *-commutative 26.2%

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

   9. Applied egg-rr 38.3%

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

   10. Taylor expanded in y around inf 97.8%

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

   if -1.6499999999999999 < y < 1

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. log1p-def 100.0%

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

      3. distribute-neg-frac 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 97.8%

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

      1. +-commutative 97.8%

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

      2. div-sub 97.8%

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

      3. mul-1-neg 97.8%

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

      4. sub-neg 97.8%

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

      5. *-inverses 97.8%

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

      6. *-rgt-identity 97.8%

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

      7. log1p-def 97.8%

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

      8. mul-1-neg 97.8%

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

   6. Simplified 97.8%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 4: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -35:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -35.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (+ y (log1p (- x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -35.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + log1p(-x));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -35.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - (y + Math.log1p(-x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -35.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - (y + math.log1p(-x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -35.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -35.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(y + N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -35

   1. Initial program 24.1%

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

      1. sub-neg 24.1%

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

      2. log1p-def 24.1%

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

      3. distribute-neg-frac 24.1%

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

      4. neg-sub0 24.1%

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

      5. associate-+l- 24.1%

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

      6. neg-sub0 24.1%

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

      7. +-commutative 24.1%

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

      8. sub-neg 24.1%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in y around -inf 96.6%

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

      1. associate--r+ 96.6%

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

      2. mul-1-neg 96.6%

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

      3. neg-sub0 96.6%

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

      4. associate-+l- 96.6%

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

      5. neg-sub0 96.6%

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

      6. mul-1-neg 96.6%

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

      7. +-commutative 96.6%

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

      8. log1p-def 96.6%

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

      9. mul-1-neg 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in x around 0 64.2%

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

   if -35 < y

   1. Initial program 95.6%

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

      1. sub-neg 95.6%

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

      2. log1p-def 95.6%

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

      3. distribute-neg-frac 95.6%

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

      4. neg-sub0 95.6%

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

      5. associate-+l- 95.6%

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

      6. neg-sub0 95.6%

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

      7. +-commutative 95.6%

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

      8. sub-neg 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in y around 0 84.6%

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

      1. +-commutative 84.6%

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

      2. div-sub 84.6%

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

      3. mul-1-neg 84.6%

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

      4. sub-neg 84.6%

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

      5. *-inverses 84.6%

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

      6. *-rgt-identity 84.6%

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

      7. log1p-def 84.6%

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

      8. mul-1-neg 84.6%

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

   6. Simplified 84.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -35:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \end{array} \]

Alternative 5: 80.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -160:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -160.0) (- 1.0 (log (/ -1.0 y))) (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -160.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -160.0) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -160.0:
		tmp = 1.0 - math.log((-1.0 / y))
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -160.0)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -160.0], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -160

   1. Initial program 24.1%

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

      1. sub-neg 24.1%

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

      2. log1p-def 24.1%

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

      3. distribute-neg-frac 24.1%

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

      4. neg-sub0 24.1%

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

      5. associate-+l- 24.1%

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

      6. neg-sub0 24.1%

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

      7. +-commutative 24.1%

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

      8. sub-neg 24.1%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in y around -inf 96.6%

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

      1. associate--r+ 96.6%

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

      2. mul-1-neg 96.6%

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

      3. neg-sub0 96.6%

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

      4. associate-+l- 96.6%

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

      5. neg-sub0 96.6%

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

      6. mul-1-neg 96.6%

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

      7. +-commutative 96.6%

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

      8. log1p-def 96.6%

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

      9. mul-1-neg 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in x around 0 64.2%

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

   if -160 < y

   1. Initial program 95.6%

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

      1. sub-neg 95.6%

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

      2. log1p-def 95.6%

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

      3. distribute-neg-frac 95.6%

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

      4. neg-sub0 95.6%

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

      5. associate-+l- 95.6%

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

      6. neg-sub0 95.6%

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

      7. +-commutative 95.6%

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

      8. sub-neg 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in y around 0 84.2%

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

      1. log1p-def 84.2%

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

      2. mul-1-neg 84.2%

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

   6. Simplified 84.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -160:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \]

Alternative 6: 63.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. sub-neg 75.8%

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

   2. log1p-def 75.8%

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

   3. distribute-neg-frac 75.8%

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

   4. neg-sub0 75.8%

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

   5. associate-+l- 75.8%

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

   6. neg-sub0 75.8%

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

   7. +-commutative 75.8%

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

   8. sub-neg 75.8%

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

3. Simplified 75.8%

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

4. Taylor expanded in y around 0 64.3%

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

   1. log1p-def 64.3%

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

   2. mul-1-neg 64.3%

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

6. Simplified 64.3%

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

7. Final simplification 64.3%

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

Alternative 7: 43.5% accurate, 37.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.8%

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

   1. sub-neg 75.8%

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

   2. log1p-def 75.8%

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

   3. distribute-neg-frac 75.8%

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

   4. neg-sub0 75.8%

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

   5. associate-+l- 75.8%

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

   6. neg-sub0 75.8%

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

   7. +-commutative 75.8%

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

   8. sub-neg 75.8%

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

3. Simplified 75.8%

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

4. Taylor expanded in y around 0 64.3%

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

   1. log1p-def 64.3%

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

   2. mul-1-neg 64.3%

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

6. Simplified 64.3%

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

7. Taylor expanded in x around 0 42.6%

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

8. Final simplification 42.6%

   \[\leadsto x + 1 \]

Alternative 8: 43.2% 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 75.8%

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

   1. sub-neg 75.8%

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

   2. log1p-def 75.8%

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

   3. distribute-neg-frac 75.8%

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

   4. neg-sub0 75.8%

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

   5. associate-+l- 75.8%

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

   6. neg-sub0 75.8%

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

   7. +-commutative 75.8%

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

   8. sub-neg 75.8%

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

3. Simplified 75.8%

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

4. Taylor expanded in y around 0 64.3%

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

   1. log1p-def 64.3%

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

   2. mul-1-neg 64.3%

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

6. Simplified 64.3%

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

7. Taylor expanded in x around 0 42.5%

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

8. Final simplification 42.5%

   \[\leadsto 1 \]

Developer target: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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