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

Percentage Accurate: 73.0% → 100.0%

Time: 10.4s

Alternatives: 11

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: 73.0% 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: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 6:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(-1 + \frac{\left(x - \frac{\left(1 - \frac{x - 1}{y}\right) - x}{y}\right) - 1}{y}\right) + x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 6.0)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (-
    1.0
    (log
     (/
      (+ (+ -1.0 (/ (- (- x (/ (- (- 1.0 (/ (- x 1.0) y)) x) y)) 1.0) y)) x)
      y)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 6.0) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log((((-1.0 + (((x - (((1.0 - ((x - 1.0) / y)) - x) / y)) - 1.0) / y)) + x) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 6.0) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log((((-1.0 + (((x - (((1.0 - ((x - 1.0) / y)) - x) / y)) - 1.0) / y)) + x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 6.0:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log((((-1.0 + (((x - (((1.0 - ((x - 1.0) / y)) - x) / y)) - 1.0) / y)) + x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 6.0)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(-1.0 + Float64(Float64(Float64(x - Float64(Float64(Float64(1.0 - Float64(Float64(x - 1.0) / y)) - x) / y)) - 1.0) / y)) + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 6.0], 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[(N[(-1.0 + N[(N[(N[(x - N[(N[(N[(1.0 - N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 6

   1. Initial program 99.9%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 6 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 8.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 6:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(-1 + \frac{\left(x - \frac{\left(1 - \frac{x - 1}{y}\right) - x}{y}\right) - 1}{y}\right) + x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y)))))))
   (if (<= t_0 -5.0)
     (- 1.0 (log (/ x (+ -1.0 y))))
     (if (<= t_0 5.0)
       (- 1.0 (log1p (fma (- 1.0 x) y (- x))))
       (- 1.0 (log (/ (+ -1.0 x) y)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 - log((1.0 - ((x - y) / (1.0 - y))));
	double tmp;
	if (t_0 <= -5.0) {
		tmp = 1.0 - log((x / (-1.0 + y)));
	} else if (t_0 <= 5.0) {
		tmp = 1.0 - log1p(fma((1.0 - x), y, -x));
	} else {
		tmp = 1.0 - log(((-1.0 + x) / y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))))
	tmp = 0.0
	if (t_0 <= -5.0)
		tmp = Float64(1.0 - log(Float64(x / Float64(-1.0 + y))));
	elseif (t_0 <= 5.0)
		tmp = Float64(1.0 - log1p(fma(Float64(1.0 - x), y, Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[(1.0 - N[Log[N[(x / N[(-1.0 + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5.0], N[(1.0 - N[Log[1 + N[(N[(1.0 - x), $MachinePrecision] * y + (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < -5

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -5 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 5

   1. Initial program 99.9%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 5 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 9.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

4. Final simplification 99.2%

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

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 9.52:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(-1 + \frac{x - 1}{y}\right) + x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 9.52)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (/ (+ (+ -1.0 (/ (- x 1.0) y)) x) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 9.52) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log((((-1.0 + ((x - 1.0) / y)) + x) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 9.52) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log((((-1.0 + ((x - 1.0) / y)) + x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 9.52:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log((((-1.0 + ((x - 1.0) / y)) + x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 9.52)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(-1.0 + Float64(Float64(x - 1.0) / y)) + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 9.52], 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[(N[(-1.0 + N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 9.5199999999999996

   1. Initial program 99.9%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 9.5199999999999996 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 6.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. associate--l+ N/A

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

      4. +-commutative N/A

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

      5. *-lft-identity N/A

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

      6. metadata-eval N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. +-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-/.f64 N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 9.52:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(-1 + \frac{x - 1}{y}\right) + x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - \log \left(1 - \frac{x - y}{1 - y}\right) \leq 9.52:\\ \;\;\;\;1 - \mathsf{log1p}\left(\frac{y - x}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))) 9.52)
   (- 1.0 (log1p (/ (- y x) (- 1.0 y))))
   (- 1.0 (log (/ (+ -1.0 x) y)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 - log((1.0 - ((x - y) / (1.0 - y))))) <= 9.52) {
		tmp = 1.0 - log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - log(((-1.0 + x) / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((1.0 - Math.log((1.0 - ((x - y) / (1.0 - y))))) <= 9.52) {
		tmp = 1.0 - Math.log1p(((y - x) / (1.0 - y)));
	} else {
		tmp = 1.0 - Math.log(((-1.0 + x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (1.0 - math.log((1.0 - ((x - y) / (1.0 - y))))) <= 9.52:
		tmp = 1.0 - math.log1p(((y - x) / (1.0 - y)))
	else:
		tmp = 1.0 - math.log(((-1.0 + x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 - log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y))))) <= 9.52)
		tmp = Float64(1.0 - log1p(Float64(Float64(y - x) / Float64(1.0 - y))));
	else
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + x) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 - N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 9.52], 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[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))) < 9.5199999999999996

   1. Initial program 99.9%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 9.5199999999999996 < (-.f64 #s(literal 1 binary64) (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))))

   1. Initial program 6.9%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 99.3

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

   5. Applied rewrites 99.3%

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

Alternative 5: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(1 - \frac{x - y}{1 - y}\right) \leq -4:\\ \;\;\;\;1 - \left(-\log \left(-y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (log (- 1.0 (/ (- x y) (- 1.0 y)))) -4.0)
   (- 1.0 (- (log (- y))))
   (- 1.0 (log1p (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (log((1.0 - ((x - y) / (1.0 - y)))) <= -4.0) {
		tmp = 1.0 - -log(-y);
	} else {
		tmp = 1.0 - log1p(-x);
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.log((1.0 - ((x - y) / (1.0 - y)))) <= -4.0) {
		tmp = 1.0 - -Math.log(-y);
	} else {
		tmp = 1.0 - Math.log1p(-x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.log((1.0 - ((x - y) / (1.0 - y)))) <= -4.0:
		tmp = 1.0 - -math.log(-y)
	else:
		tmp = 1.0 - math.log1p(-x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (log(Float64(1.0 - Float64(Float64(x - y) / Float64(1.0 - y)))) <= -4.0)
		tmp = Float64(1.0 - Float64(-log(Float64(-y))));
	else
		tmp = Float64(1.0 - log1p(Float64(-x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[Log[N[(1.0 - N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -4.0], N[(1.0 - (-N[Log[(-y)], $MachinePrecision])), $MachinePrecision], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)))) < -4

   1. Initial program 9.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 7.3

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

   5. Applied rewrites 7.3%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 69.1%

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

   10. Applied rewrites 69.1%

       \[\leadsto 1 - \left(-\log \left(-y\right)\right) \]

   if -4 < (log.f64 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))))

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower-log1p.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 86.1

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

   5. Applied rewrites 86.1%

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

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.84:\\ \;\;\;\;1 - \log \left(\frac{-1 + x}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(\mathsf{fma}\left(1 - x, y, -x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.84)
   (- 1.0 (log (/ (+ -1.0 x) y)))
   (if (<= y 1.0)
     (- 1.0 (log1p (fma (- 1.0 x) y (- x))))
     (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.84) {
		tmp = 1.0 - log(((-1.0 + x) / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(fma((1.0 - x), y, -x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.84)
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + x) / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(fma(Float64(1.0 - x), y, Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.84], N[(1.0 - N[Log[N[(N[(-1.0 + x), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + N[(N[(1.0 - x), $MachinePrecision] * y + (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.839999999999999969

   1. Initial program 23.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac N/A

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

      3. lower-/.f64 N/A

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

      4. distribute-neg-in N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. lower-+.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if -0.839999999999999969 < y < 1

   1. Initial program 100.0%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 1 < y

   1. Initial program 54.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.2%

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

Alternative 7: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if y < -1

   1. Initial program 23.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 71.2%

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

   10. Applied rewrites 71.2%

       \[\leadsto 1 - \left(-\log \left(-y\right)\right) \]

   if -1 < y < 1

   1. Initial program 100.0%

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

      1. lift-log.f64 N/A

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

      2. lift--.f64 N/A

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

      3. *-lft-identity N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. distribute-lft-neg-in N/A

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

      6. *-lft-identity N/A

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

      7. lower-log1p.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-frac-neg N/A

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

      10. lower-/.f64 N/A

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

      11. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. lower-fma.f64 N/A

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

      7. fp-cancel-sign-sub-inv N/A

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

      8. metadata-eval N/A

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

      9. *-lft-identity N/A

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

      10. lower--.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 1 < y

   1. Initial program 54.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.2%

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

Alternative 8: 89.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -1.0)
		tmp = 1.0 - -log(-y);
	elseif (y <= 1.0)
		tmp = 1.0 - log(((y + 1.0) * (1.0 - x)));
	else
		tmp = 1.0 - log((x / y));
	end
	tmp_2 = tmp;
end

Wolfram

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

   1. Initial program 23.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 71.2%

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

   10. Applied rewrites 71.2%

       \[\leadsto 1 - \left(-\log \left(-y\right)\right) \]

   if -1 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. *-lft-identity N/A

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

      4. metadata-eval N/A

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

      5. fp-cancel-sign-sub-inv N/A

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

      6. +-commutative N/A

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

      7. distribute-rgt1-in N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. fp-cancel-sign-sub-inv N/A

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

      11. metadata-eval N/A

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

      12. *-lft-identity N/A

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

      13. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if 1 < y

   1. Initial program 54.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.2%

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

Alternative 9: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8:\\ \;\;\;\;1 - \left(-\log \left(-y\right)\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.0)
   (- 1.0 (- (log (- y))))
   (if (<= y 1.0) (- 1.0 (log1p (- x))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.0) {
		tmp = 1.0 - -log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.0) {
		tmp = 1.0 - -Math.log(-y);
	} else if (y <= 1.0) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.0:
		tmp = 1.0 - -math.log(-y)
	elif y <= 1.0:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.0)
		tmp = Float64(1.0 - Float64(-log(Float64(-y))));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.0], N[(1.0 - (-N[Log[(-y)], $MachinePrecision])), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8

   1. Initial program 23.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-log1p.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 71.2%

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

   10. Applied rewrites 71.2%

       \[\leadsto 1 - \left(-\log \left(-y\right)\right) \]

   if -8 < y < 1

   1. Initial program 100.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-lft-identity N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. lower-log1p.f64 N/A

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

      5. mul-1-neg N/A

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

      6. lower-neg.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if 1 < y

   1. Initial program 54.8%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower--.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 98.2%

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

Alternative 10: 63.4% accurate, 1.2× 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 73.7%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. lower-log1p.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 64.2

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

5. Applied rewrites 64.2%

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

Alternative 11: 43.7% accurate, 20.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.7%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
2. Add Preprocessing

3. Taylor expanded in y around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

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

   3. fp-cancel-sign-sub-inv N/A

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

   4. lower-log1p.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 64.2

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

5. Applied rewrites 64.2%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 46.8%

   \[\leadsto 1 - \left(-x\right) \]
9. 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 2024337 
(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))))))