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

Percentage Accurate: 72.0% → 99.9%

Time: 12.8s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 72.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{1 - y}, y - x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.998)
   (- 1.0 (log (fma (/ 1.0 (- 1.0 y)) (- y x) 1.0)))
   (- 1.0 (log (/ (+ (+ x -1.0) (/ (+ -1.0 (+ x (/ (+ x -1.0) y))) y)) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - log(fma((1.0 / (1.0 - y)), (y - x), 1.0));
	} else {
		tmp = 1.0 - log((((x + -1.0) + ((-1.0 + (x + ((x + -1.0) / y))) / y)) / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.998)
		tmp = Float64(1.0 - log(fma(Float64(1.0 / Float64(1.0 - y)), Float64(y - x), 1.0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(Float64(x + -1.0) + Float64(Float64(-1.0 + Float64(x + Float64(Float64(x + -1.0) / y))) / y)) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.998], N[(1.0 - N[Log[N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(N[(x + -1.0), $MachinePrecision] + N[(N[(-1.0 + N[(x + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998

   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-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. associate--r+ N/A

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

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 0.998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 10.2%

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

   3. Taylor expanded in y around -inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{1 - y}, y - x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{\left(x + -1\right) + \frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x - y}{y + -1}\\ \mathbf{if}\;t\_0 \leq 0.004:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(y + 1, y - x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (- x y) (+ y -1.0)))))
   (if (<= t_0 0.004)
     (- 1.0 (log (/ -1.0 y)))
     (if (<= t_0 2.0)
       (- 1.0 (log (fma (+ y 1.0) (- y x) 1.0)))
       (- 1.0 (log (/ x (+ y -1.0))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + ((x - y) / (y + -1.0));
	double tmp;
	if (t_0 <= 0.004) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (t_0 <= 2.0) {
		tmp = 1.0 - log(fma((y + 1.0), (y - x), 1.0));
	} else {
		tmp = 1.0 - log((x / (y + -1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(Float64(x - y) / Float64(y + -1.0)))
	tmp = 0.0
	if (t_0 <= 0.004)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 - log(fma(Float64(y + 1.0), Float64(y - x), 1.0)));
	else
		tmp = Float64(1.0 - log(Float64(x / Float64(y + -1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.004], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(1.0 - N[Log[N[(N[(y + 1.0), $MachinePrecision] * N[(y - x), $MachinePrecision] + 1.0), $MachinePrecision]], $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 (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 0.0040000000000000001

   1. Initial program 11.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 9.1

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

   5. Applied rewrites 9.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 67.8%

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

   if 0.0040000000000000001 < (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 2

   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-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. associate--r+ N/A

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

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 2 < (-.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 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. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 99.1

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

   5. Applied rewrites 99.1%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 0.004:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;1 + \frac{x - y}{y + -1} \leq 2:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(y + 1, y - x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{1 - y}\\ \mathbf{if}\;t\_0 \leq -5:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{elif}\;t\_0 \leq 0.005:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(y + 1, y - x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (- 1.0 y))))
   (if (<= t_0 -5.0)
     (- 1.0 (log (/ x (+ y -1.0))))
     (if (<= t_0 0.005)
       (- 1.0 (log (fma (+ y 1.0) (- y x) 1.0)))
       (- 1.0 (log (/ (+ x -1.0) y)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / (1.0 - y);
	double tmp;
	if (t_0 <= -5.0) {
		tmp = 1.0 - log((x / (y + -1.0)));
	} else if (t_0 <= 0.005) {
		tmp = 1.0 - log(fma((y + 1.0), (y - x), 1.0));
	} else {
		tmp = 1.0 - log(((x + -1.0) / y));
	}
	return tmp;
}

Julia

function code(x, y)
	t_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(y + -1.0))));
	elseif (t_0 <= 0.005)
		tmp = Float64(1.0 - log(fma(Float64(y + 1.0), Float64(y - x), 1.0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(x + -1.0) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5.0], N[(1.0 - N[Log[N[(x / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.005], N[(1.0 - N[Log[N[(N[(y + 1.0), $MachinePrecision] * N[(y - x), $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 3 regimes

2. if (/.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. 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. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -5 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.0050000000000000001

   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-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. associate--r+ N/A

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

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 99.2

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

   7. Applied rewrites 99.2%

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

   if 0.0050000000000000001 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 11.6%

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 96.7

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

   5. Applied rewrites 96.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq -5:\\ \;\;\;\;1 - \log \left(\frac{x}{y + -1}\right)\\ \mathbf{elif}\;\frac{x - y}{1 - y} \leq 0.005:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(y + 1, y - x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x + -1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{1 - y}, y - x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (/ (- x y) (- 1.0 y)) 0.998)
   (- 1.0 (log (fma (/ 1.0 (- 1.0 y)) (- y x) 1.0)))
   (- 1.0 (log (/ (+ -1.0 (+ x (/ (+ x -1.0) y))) y)))))

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.998) {
		tmp = 1.0 - log(fma((1.0 / (1.0 - y)), (y - x), 1.0));
	} else {
		tmp = 1.0 - log(((-1.0 + (x + ((x + -1.0) / y))) / y));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.998)
		tmp = Float64(1.0 - log(fma(Float64(1.0 / Float64(1.0 - y)), Float64(y - x), 1.0)));
	else
		tmp = Float64(1.0 - log(Float64(Float64(-1.0 + Float64(x + Float64(Float64(x + -1.0) / y))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.998], N[(1.0 - N[Log[N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(N[(-1.0 + N[(x + N[(N[(x + -1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.998

   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-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. associate--r+ N/A

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

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   if 0.998 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 10.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 99.3%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \leq 0.998:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{1 - y}, y - x, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{-1 + \left(x + \frac{x + -1}{y}\right)}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(1.0 - y)) <= 0.999998)
		tmp = Float64(1.0 - log(fma(Float64(1.0 / Float64(1.0 - y)), Float64(y - x), 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[(x - y), $MachinePrecision] / N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.999998], N[(1.0 - N[Log[N[(N[(1.0 / N[(1.0 - y), $MachinePrecision]), $MachinePrecision] * N[(y - x), $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)) < 0.999998000000000054

   1. Initial program 99.7%

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

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. associate--r+ N/A

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

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 0.999998000000000054 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.7%

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. 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. Final simplification 99.7%

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

Alternative 6: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (((x - y) / (1.0 - y)) <= 0.999998) {
		tmp = 1.0 - log((1.0 + ((x - y) / (y + -1.0))));
	} 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) :: tmp
    if (((x - y) / (1.0d0 - y)) <= 0.999998d0) then
        tmp = 1.0d0 - log((1.0d0 + ((x - y) / (y + (-1.0d0)))))
    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 tmp;
	if (((x - y) / (1.0 - y)) <= 0.999998) {
		tmp = 1.0 - Math.log((1.0 + ((x - y) / (y + -1.0))));
	} else {
		tmp = 1.0 - Math.log(((x + -1.0) / y));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x - y) / (1.0 - y)) <= 0.999998)
		tmp = 1.0 - log((1.0 + ((x - y) / (y + -1.0))));
	else
		tmp = 1.0 - log(((x + -1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y)) < 0.999998000000000054

   1. Initial program 99.7%

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

   if 0.999998000000000054 < (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))

   1. Initial program 7.7%

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

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. 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. Final simplification 99.7%

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

Alternative 7: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 0.004:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, y - x, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (+ 1.0 (/ (- x y) (+ y -1.0))) 0.004)
   (- 1.0 (log (/ -1.0 y)))
   (- 1.0 (log (fma 1.0 (- y x) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((1.0 + ((x - y) / (y + -1.0))) <= 0.004) {
		tmp = 1.0 - log((-1.0 / y));
	} else {
		tmp = 1.0 - log(fma(1.0, (y - x), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(1.0 + Float64(Float64(x - y) / Float64(y + -1.0))) <= 0.004)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	else
		tmp = Float64(1.0 - log(fma(1.0, Float64(y - x), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(1.0 + N[(N[(x - y), $MachinePrecision] / N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.004], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[N[(1.0 * N[(y - x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 #s(literal 1 binary64) (/.f64 (-.f64 x y) (-.f64 #s(literal 1 binary64) y))) < 0.0040000000000000001

   1. Initial program 11.6%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 9.1

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

   5. Applied rewrites 9.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 67.8%

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

   if 0.0040000000000000001 < (-.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. 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-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. associate--r+ N/A

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

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 84.6%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 + \frac{x - y}{y + -1} \leq 0.004:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, y - x, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.0) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - log(fma((y + 1.0), (y - x), 1.0));
	} 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 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - log(fma(Float64(y + 1.0), Float64(y - x), 1.0)));
	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[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[Log[N[(N[(y + 1.0), $MachinePrecision] * N[(y - x), $MachinePrecision] + 1.0), $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 31.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 8.1

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

   5. Applied rewrites 8.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 62.8%

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

   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--.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-rgt-neg-in N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

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

      11. lift--.f64 N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. associate--r+ N/A

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

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 1 < y

   1. Initial program 68.9%

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

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

      7. distribute-neg-in N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 96.5

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

   5. Applied rewrites 96.5%

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

   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 94.3%

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

4. Final simplification 89.8%

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

Alternative 9: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -22.5:\\ \;\;\;\;1 - \log \left(\frac{-1}{y}\right)\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;1 - \left(y + \mathsf{log1p}\left(-x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -22.5)
   (- 1.0 (log (/ -1.0 y)))
   (if (<= y 1.0) (- 1.0 (+ y (log1p (- x)))) (- 1.0 (log (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -22.5) {
		tmp = 1.0 - log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + log1p(-x));
	} else {
		tmp = 1.0 - log((x / y));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -22.5) {
		tmp = 1.0 - Math.log((-1.0 / y));
	} else if (y <= 1.0) {
		tmp = 1.0 - (y + Math.log1p(-x));
	} else {
		tmp = 1.0 - Math.log((x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -22.5:
		tmp = 1.0 - math.log((-1.0 / y))
	elif y <= 1.0:
		tmp = 1.0 - (y + math.log1p(-x))
	else:
		tmp = 1.0 - math.log((x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -22.5)
		tmp = Float64(1.0 - log(Float64(-1.0 / y)));
	elseif (y <= 1.0)
		tmp = Float64(1.0 - Float64(y + log1p(Float64(-x))));
	else
		tmp = Float64(1.0 - log(Float64(x / y)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -22.5], N[(1.0 - N[Log[N[(-1.0 / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.0], N[(1.0 - N[(y + N[Log[1 + (-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 < -22.5

   1. Initial program 31.1%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 8.1

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

   5. Applied rewrites 8.1%

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

   6. Taylor expanded in y around -inf

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

   8. Applied rewrites 62.8%

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

   if -22.5 < 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. +-commutative N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. mul-1-neg N/A

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

      9. div-sub N/A

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

      10. sub-neg N/A

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

      11. mul-1-neg N/A

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

      12. *-inverses N/A

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

      13. *-rgt-identity N/A

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

      14. lower-+.f64 N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 98.9%

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

   if 1 < y

   1. Initial program 68.9%

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

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

      7. distribute-neg-in N/A

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

      8. neg-mul-1 N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 96.5

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

   5. Applied rewrites 96.5%

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

   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 94.3%

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

Alternative 10: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-35}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(y \cdot \left(y + 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-35) (- 1.0 (log1p (- x))) (- 1.0 (log1p (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-35) {
		tmp = 1.0 - log1p(-x);
	} else {
		tmp = 1.0 - log1p((y * (y + 1.0)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-35) {
		tmp = 1.0 - Math.log1p(-x);
	} else {
		tmp = 1.0 - Math.log1p((y * (y + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.3e-35:
		tmp = 1.0 - math.log1p(-x)
	else:
		tmp = 1.0 - math.log1p((y * (y + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.3e-35)
		tmp = Float64(1.0 - log1p(Float64(-x)));
	else
		tmp = Float64(1.0 - log1p(Float64(y * Float64(y + 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.3e-35], N[(1.0 - N[Log[1 + (-x)], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Log[1 + N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.3000000000000002e-35

   1. Initial program 80.2%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 74.0

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

   5. Applied rewrites 74.0%

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

   if 4.3000000000000002e-35 < y

   1. Initial program 75.4%

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
   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 22.4

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

   5. Applied rewrites 22.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 30.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-35}:\\ \;\;\;\;1 - \mathsf{log1p}\left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \mathsf{log1p}\left(y \cdot \left(y + 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.1% 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 79.6%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 66.6

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

5. Applied rewrites 66.6%

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

Alternative 12: 43.5% 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 79.6%

   \[1 - \log \left(1 - \frac{x - y}{1 - y}\right) \]
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 66.6

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

5. Applied rewrites 66.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 47.0%

   \[\leadsto 1 - \left(-x\right) \]
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 2024233 
(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))))))