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

Percentage Accurate: 72.6% → 99.9%

Time: 8.9s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.6% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(0 - \left(1 - y\right)\right)}}^{-1}, x - y, 1\right)\right) \]

      13. lift--.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 - y\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      14. sub-neg N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      16. associate--r+ N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - 1\right)}}^{-1}, x - y, 1\right)\right) \]

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left({\left(y - 1\right)}^{-1}, x - y, 1\right)\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

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

   1. Initial program 5.2%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r/ N/A

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], 0.999], N[(1.0 - N[Log[N[(N[(-1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision] + 1.0), $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 #s(literal 1 binary64) y)) < 0.998999999999999999

   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--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(0 - \left(1 - y\right)\right)}}^{-1}, x - y, 1\right)\right) \]

      13. lift--.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 - y\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      14. sub-neg N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      16. associate--r+ N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - 1\right)}}^{-1}, x - y, 1\right)\right) \]

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left({\left(y - 1\right)}^{-1}, x - y, 1\right)\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. unpow-1 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. metadata-eval N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lower-/.f64 99.9

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

      12. sub-neg N/A

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

      13. lift--.f64 99.9

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

   6. Applied rewrites 99.9%

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

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

   1. Initial program 5.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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-frac-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y - x}{y - 1}\\ \mathbf{if}\;t\_0 \leq 0.999:\\ \;\;\;\;1 - \log \left(1 - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- y x) (- y 1.0))))
   (if (<= t_0 0.999)
     (- 1.0 (log (- 1.0 t_0)))
     (- 1.0 (log (/ (- x 1.0) y))))))

C

double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double tmp;
	if (t_0 <= 0.999) {
		tmp = 1.0 - log((1.0 - t_0));
	} else {
		tmp = 1.0 - log(((x - 1.0) / y));
	}
	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 = (y - x) / (y - 1.0d0)
    if (t_0 <= 0.999d0) then
        tmp = 1.0d0 - log((1.0d0 - t_0))
    else
        tmp = 1.0d0 - log(((x - 1.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y - x) / (y - 1.0);
	double tmp;
	if (t_0 <= 0.999) {
		tmp = 1.0 - Math.log((1.0 - t_0));
	} else {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y - x) / (y - 1.0)
	tmp = 0
	if t_0 <= 0.999:
		tmp = 1.0 - math.log((1.0 - t_0))
	else:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y - x) / Float64(y - 1.0))
	tmp = 0.0
	if (t_0 <= 0.999)
		tmp = Float64(1.0 - log(Float64(1.0 - t_0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y - x) / (y - 1.0);
	tmp = 0.0;
	if (t_0 <= 0.999)
		tmp = 1.0 - log((1.0 - t_0));
	else
		tmp = 1.0 - log(((x - 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.999], N[(1.0 - N[Log[N[(1.0 - t$95$0), $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 #s(literal 1 binary64) y)) < 0.998999999999999999

   1. Initial program 99.9%

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

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

   1. Initial program 5.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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-frac-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.9%

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

Alternative 4: 80.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.002:\\ \;\;\;\;1 - \mathsf{log1p}\left(-1 \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- y x) (- y 1.0)) 0.002)
   (- 1.0 (log1p (* -1.0 (- x y))))
   (- 1.0 (log (/ -1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (((y - x) / (y - 1.0)) <= 0.002) {
		tmp = 1.0 - log1p((-1.0 * (x - y)));
	} else {
		tmp = 1.0 - log((-1.0 / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (((y - x) / (y - 1.0)) <= 0.002) {
		tmp = 1.0 - Math.log1p((-1.0 * (x - y)));
	} else {
		tmp = 1.0 - Math.log((-1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((y - x) / (y - 1.0)) <= 0.002:
		tmp = 1.0 - math.log1p((-1.0 * (x - y)))
	else:
		tmp = 1.0 - math.log((-1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(y - x) / Float64(y - 1.0)) <= 0.002)
		tmp = Float64(1.0 - log1p(Float64(-1.0 * Float64(x - y))));
	else
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(y - x), $MachinePrecision] / N[(y - 1.0), $MachinePrecision]), $MachinePrecision], 0.002], N[(1.0 - N[Log[1 + N[(-1.0 * N[(x - y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 2e-3

   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--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

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

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

      8. distribute-frac-neg2 N/A

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

      9. lower-fma.f64 N/A

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

      10. inv-pow N/A

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

      11. lower-pow.f64 N/A

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

      12. neg-sub0 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(0 - \left(1 - y\right)\right)}}^{-1}, x - y, 1\right)\right) \]

      13. lift--.f64 N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 - y\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      14. sub-neg N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      15. +-commutative N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + 1\right)}\right)}^{-1}, x - y, 1\right)\right) \]

      16. associate--r+ N/A

          \[\leadsto 1 - \log \left(\mathsf{fma}\left({\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - 1\right)}}^{-1}, x - y, 1\right)\right) \]

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 87.5%

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

      1. lift-log.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-log1p.f64 N/A

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

      5. lower-*.f64 87.5

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

   9. Applied rewrites 87.5%

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

   if 2e-3 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 6.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 5.1

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

   5. Applied rewrites 5.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 66.9%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y - x}{y - 1} \leq 0.002:\\ \;\;\;\;1 - \mathsf{log1p}\left(-1 \cdot \left(x - y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.75:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{elif}\;y \leq 0.051:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.75)
   (- 1.0 (log (/ (- x 1.0) y)))
   (if (<= y 0.051)
     (- 1.0 (+ (log1p (- x)) y))
     (- 1.0 (log (/ x (- y 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.75) {
		tmp = 1.0 - log(((x - 1.0) / y));
	} else if (y <= 0.051) {
		tmp = 1.0 - (log1p(-x) + y);
	} else {
		tmp = 1.0 - log((x / (y - 1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -1.75) {
		tmp = 1.0 - Math.log(((x - 1.0) / y));
	} else if (y <= 0.051) {
		tmp = 1.0 - (Math.log1p(-x) + y);
	} else {
		tmp = 1.0 - Math.log((x / (y - 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.75:
		tmp = 1.0 - math.log(((x - 1.0) / y))
	elif y <= 0.051:
		tmp = 1.0 - (math.log1p(-x) + y)
	else:
		tmp = 1.0 - math.log((x / (y - 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.75)
		tmp = Float64(1.0 - log(Float64(Float64(x - 1.0) / y)));
	elseif (y <= 0.051)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + y));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y - 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.75], N[(1.0 - N[Log[N[(N[(x - 1.0), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.051], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.75

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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-frac-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. remove-double-neg N/A

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

      7. sub-neg N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if -1.75 < y < 0.0509999999999999967

   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

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. div-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   if 0.0509999999999999967 < y

   1. Initial program 53.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.75:\\ \;\;\;\;1 - \log \left(\frac{x - 1}{y}\right)\\ \mathbf{elif}\;y \leq 0.051:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.051:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -18.5)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 0.051)
     (- 1.0 (+ (log1p (- x)) y))
     (- 1.0 (log (/ x (- y 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -18.5) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 0.051) {
		tmp = 1.0 - (log1p(-x) + y);
	} else {
		tmp = 1.0 - log((x / (y - 1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -18.5) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 0.051) {
		tmp = 1.0 - (Math.log1p(-x) + y);
	} else {
		tmp = 1.0 - Math.log((x / (y - 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -18.5:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 0.051:
		tmp = 1.0 - (math.log1p(-x) + y)
	else:
		tmp = 1.0 - math.log((x / (y - 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -18.5)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 0.051)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + y));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y - 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -18.5], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.051], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / N[(y - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -18.5

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 68.1%

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

   if -18.5 < y < 0.0509999999999999967

   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

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. div-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   if 0.0509999999999999967 < y

   1. Initial program 53.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 0.051:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y - 1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(\mathsf{log1p}\left(-x\right) + y\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -18.5)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ (log1p (- x)) y)) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -18.5) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (log1p(-x) + y);
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -18.5) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (Math.log1p(-x) + y);
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -18.5:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (math.log1p(-x) + y)
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -18.5)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(log1p(Float64(-x)) + y));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -18.5], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(N[Log[1 + (-x)], $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -18.5

   1. Initial program 22.0%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 4.7

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

   5. Applied rewrites 4.7%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 68.1%

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

   if -18.5 < 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

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. mul-1-neg N/A

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

      10. div-sub N/A

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

      11. sub-neg N/A

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

      12. mul-1-neg N/A

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

      13. *-inverses N/A

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

      14. *-rgt-identity N/A

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

      15. lower-+.f64 N/A

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

   5. Applied rewrites 99.5%

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

   if 1 < y

   1. Initial program 53.5%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. neg-mul-1 N/A

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

      6. distribute-neg-in N/A

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

      7. metadata-eval N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. lower-+.f64 98.7

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

   5. Applied rewrites 98.7%

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

   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 96.8%

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

Alternative 8: 62.7% 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 74.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 65.4

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

5. Applied rewrites 65.4%

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

Alternative 9: 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 74.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 65.4

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

5. Applied rewrites 65.4%

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

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

Alternative 10: 42.5% accurate, 31.0× speedup

Math

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

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 - 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 74.4%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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. sub-neg N/A

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

   2. mul-1-neg N/A

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

   3. lower-log1p.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 65.4

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

5. Applied rewrites 65.4%

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

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. Step-by-step derivation

10. Applied rewrites 46.7%

    \[\leadsto 1 - \frac{0 - x \cdot x}{0 + \color{blue}{x}} \]
11. Step-by-step derivation

12. Applied rewrites 45.9%

    \[\leadsto 1 - x \]
13. 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 2024332 
(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))))))