Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 12

Speedup: 1.5×

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))

C

double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x * (y - 1.0d0)) - (y * 0.5d0)) + 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}

Python

def code(x, y):
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673

Julia

function code(x, y)
	return Float64(Float64(Float64(x * Float64(y - 1.0)) - Float64(y * 0.5)) + 0.918938533204673)
end

MATLAB

function tmp = code(x, y)
	tmp = ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
end

Wolfram

code[x_, y_] := N[(N[(N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision] - N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + 0.918938533204673), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (fma (- x 0.5) y (- 0.918938533204673 x)))

C

double code(double x, double y) {
	return fma((x - 0.5), y, (0.918938533204673 - x));
}

Julia

function code(x, y)
	return fma(Float64(x - 0.5), y, Float64(0.918938533204673 - x))
end

Wolfram

code[x_, y_] := N[(N[(x - 0.5), $MachinePrecision] * y + N[(0.918938533204673 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]
6. Add Preprocessing

Alternative 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.7 \cdot 10^{+90}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -14.2:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.32:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+68} \lor \neg \left(y \leq 4.7 \cdot 10^{+141} \lor \neg \left(y \leq 2.95 \cdot 10^{+262}\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -6.7e+90)
   (* -0.5 y)
   (if (<= y -14.2)
     (* y x)
     (if (<= y 1.32)
       (- 0.918938533204673 x)
       (if (or (<= y 3.1e+68)
               (not (or (<= y 4.7e+141) (not (<= y 2.95e+262)))))
         (* y x)
         (* -0.5 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -6.7e+90) {
		tmp = -0.5 * y;
	} else if (y <= -14.2) {
		tmp = y * x;
	} else if (y <= 1.32) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 3.1e+68) || !((y <= 4.7e+141) || !(y <= 2.95e+262))) {
		tmp = y * x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-6.7d+90)) then
        tmp = (-0.5d0) * y
    else if (y <= (-14.2d0)) then
        tmp = y * x
    else if (y <= 1.32d0) then
        tmp = 0.918938533204673d0 - x
    else if ((y <= 3.1d+68) .or. (.not. (y <= 4.7d+141) .or. (.not. (y <= 2.95d+262)))) then
        tmp = y * x
    else
        tmp = (-0.5d0) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -6.7e+90) {
		tmp = -0.5 * y;
	} else if (y <= -14.2) {
		tmp = y * x;
	} else if (y <= 1.32) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 3.1e+68) || !((y <= 4.7e+141) || !(y <= 2.95e+262))) {
		tmp = y * x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -6.7e+90:
		tmp = -0.5 * y
	elif y <= -14.2:
		tmp = y * x
	elif y <= 1.32:
		tmp = 0.918938533204673 - x
	elif (y <= 3.1e+68) or not ((y <= 4.7e+141) or not (y <= 2.95e+262)):
		tmp = y * x
	else:
		tmp = -0.5 * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -6.7e+90)
		tmp = Float64(-0.5 * y);
	elseif (y <= -14.2)
		tmp = Float64(y * x);
	elseif (y <= 1.32)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 3.1e+68) || !((y <= 4.7e+141) || !(y <= 2.95e+262)))
		tmp = Float64(y * x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -6.7e+90)
		tmp = -0.5 * y;
	elseif (y <= -14.2)
		tmp = y * x;
	elseif (y <= 1.32)
		tmp = 0.918938533204673 - x;
	elseif ((y <= 3.1e+68) || ~(((y <= 4.7e+141) || ~((y <= 2.95e+262)))))
		tmp = y * x;
	else
		tmp = -0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -6.7e+90], N[(-0.5 * y), $MachinePrecision], If[LessEqual[y, -14.2], N[(y * x), $MachinePrecision], If[LessEqual[y, 1.32], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 3.1e+68], N[Not[Or[LessEqual[y, 4.7e+141], N[Not[LessEqual[y, 2.95e+262]], $MachinePrecision]]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.7000000000000003e90 or 3.0999999999999998e68 < y < 4.69999999999999979e141 or 2.94999999999999994e262 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \frac{1}{2}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(0 - x\right)} + \frac{1}{2}\right)\right)\right) \]

       6. associate-+l- N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(0 - \left(x - \frac{1}{2}\right)\right)}\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right)}\right)\right) \]

       8. remove-double-neg N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 100.0

           \[\leadsto \color{blue}{\left(x - 0.5\right)} \cdot y \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{-1}{2} \cdot y \]
   13. Step-by-step derivation

   14. Applied rewrites 71.1%

       \[\leadsto -0.5 \cdot y \]

   if -6.7000000000000003e90 < y < -14.199999999999999 or 1.32000000000000006 < y < 3.0999999999999998e68 or 4.69999999999999979e141 < y < 2.94999999999999994e262

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.4%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \frac{1}{2}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(0 - x\right)} + \frac{1}{2}\right)\right)\right) \]

       6. associate-+l- N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(0 - \left(x - \frac{1}{2}\right)\right)}\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right)}\right)\right) \]

       8. remove-double-neg N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 95.4

           \[\leadsto \color{blue}{\left(x - 0.5\right)} \cdot y \]

   11. Applied rewrites 95.4%

       \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot y} \]

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 70.3%

       \[\leadsto y \cdot \color{blue}{x} \]

   if -14.199999999999999 < y < 1.32000000000000006

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 97.7

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.7 \cdot 10^{+90}:\\ \;\;\;\;-0.5 \cdot y\\ \mathbf{elif}\;y \leq -14.2:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 1.32:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+68} \lor \neg \left(y \leq 4.7 \cdot 10^{+141} \lor \neg \left(y \leq 2.95 \cdot 10^{+262}\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 1.32:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+68} \lor \neg \left(y \leq 4.7 \cdot 10^{+141} \lor \neg \left(y \leq 2.95 \cdot 10^{+262}\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.2e-15)
   (fma -0.5 y 0.918938533204673)
   (if (<= y 1.32)
     (- 0.918938533204673 x)
     (if (or (<= y 3.1e+68) (not (or (<= y 4.7e+141) (not (<= y 2.95e+262)))))
       (* y x)
       (* -0.5 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.2e-15) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (y <= 1.32) {
		tmp = 0.918938533204673 - x;
	} else if ((y <= 3.1e+68) || !((y <= 4.7e+141) || !(y <= 2.95e+262))) {
		tmp = y * x;
	} else {
		tmp = -0.5 * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.2e-15)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (y <= 1.32)
		tmp = Float64(0.918938533204673 - x);
	elseif ((y <= 3.1e+68) || !((y <= 4.7e+141) || !(y <= 2.95e+262)))
		tmp = Float64(y * x);
	else
		tmp = Float64(-0.5 * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.2e-15], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[y, 1.32], N[(0.918938533204673 - x), $MachinePrecision], If[Or[LessEqual[y, 3.1e+68], N[Not[Or[LessEqual[y, 4.7e+141], N[Not[LessEqual[y, 2.95e+262]], $MachinePrecision]]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(-0.5 * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.19999999999999986e-15

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) + \frac{918938533204673}{1000000000000000}} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} + \frac{918938533204673}{1000000000000000} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y + \frac{918938533204673}{1000000000000000} \]

      5. lower-fma.f64 58.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]

   5. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]

   if -2.19999999999999986e-15 < y < 1.32000000000000006

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 98.5

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 98.5%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 1.32000000000000006 < y < 3.0999999999999998e68 or 4.69999999999999979e141 < y < 2.94999999999999994e262

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \frac{1}{2}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(0 - x\right)} + \frac{1}{2}\right)\right)\right) \]

       6. associate-+l- N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(0 - \left(x - \frac{1}{2}\right)\right)}\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right)}\right)\right) \]

       8. remove-double-neg N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 99.2

           \[\leadsto \color{blue}{\left(x - 0.5\right)} \cdot y \]

   11. Applied rewrites 99.2%

       \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot y} \]

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 74.7%

       \[\leadsto y \cdot \color{blue}{x} \]

   if 3.0999999999999998e68 < y < 4.69999999999999979e141 or 2.94999999999999994e262 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \frac{1}{2}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(0 - x\right)} + \frac{1}{2}\right)\right)\right) \]

       6. associate-+l- N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(0 - \left(x - \frac{1}{2}\right)\right)}\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right)}\right)\right) \]

       8. remove-double-neg N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 100.0

           \[\leadsto \color{blue}{\left(x - 0.5\right)} \cdot y \]

   11. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot y} \]

   12. Taylor expanded in x around 0

       \[\leadsto \frac{-1}{2} \cdot y \]
   13. Step-by-step derivation

   14. Applied rewrites 77.2%

       \[\leadsto -0.5 \cdot y \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;y \leq 1.32:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+68} \lor \neg \left(y \leq 4.7 \cdot 10^{+141} \lor \neg \left(y \leq 2.95 \cdot 10^{+262}\right)\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-15} \lor \neg \left(y \leq 4.2 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.2e-15) (not (<= y 4.2e-18)))
   (fma (- x 0.5) y 0.918938533204673)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.2e-15) || !(y <= 4.2e-18)) {
		tmp = fma((x - 0.5), y, 0.918938533204673);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.2e-15) || !(y <= 4.2e-18))
		tmp = fma(Float64(x - 0.5), y, 0.918938533204673);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.2e-15], N[Not[LessEqual[y, 4.2e-18]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y + 0.918938533204673), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.19999999999999986e-15 or 4.19999999999999999e-18 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 97.6%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   if -2.19999999999999986e-15 < y < 4.19999999999999999e-18

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 100.0

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-15} \lor \neg \left(y \leq 4.2 \cdot 10^{-18}\right):\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.041:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, -x\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-18}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -0.041)
   (fma (- x 0.5) y (- x))
   (if (<= y 4.2e-18)
     (- 0.918938533204673 x)
     (fma (- x 0.5) y 0.918938533204673))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -0.041) {
		tmp = fma((x - 0.5), y, -x);
	} else if (y <= 4.2e-18) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = fma((x - 0.5), y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -0.041)
		tmp = fma(Float64(x - 0.5), y, Float64(-x));
	elseif (y <= 4.2e-18)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = fma(Float64(x - 0.5), y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -0.041], N[(N[(x - 0.5), $MachinePrecision] * y + (-x)), $MachinePrecision], If[LessEqual[y, 4.2e-18], N[(0.918938533204673 - x), $MachinePrecision], N[(N[(x - 0.5), $MachinePrecision] * y + 0.918938533204673), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -0.0410000000000000017

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, -1 \cdot x\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, -x\right) \]

   if -0.0410000000000000017 < y < 4.19999999999999999e-18

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 99.1

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   if 4.19999999999999999e-18 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.6%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.78\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -0.72) (not (<= x 0.78)))
   (* (- y 1.0) x)
   (fma -0.5 y 0.918938533204673)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -0.72) || !(x <= 0.78)) {
		tmp = (y - 1.0) * x;
	} else {
		tmp = fma(-0.5, y, 0.918938533204673);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -0.72) || !(x <= 0.78))
		tmp = Float64(Float64(y - 1.0) * x);
	else
		tmp = fma(-0.5, y, 0.918938533204673);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -0.72], N[Not[LessEqual[x, 0.78]], $MachinePrecision]], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], N[(-0.5 * y + 0.918938533204673), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.71999999999999997 or 0.78000000000000003 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 49.2%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 97.6

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

   11. Applied rewrites 97.6%

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

   if -0.71999999999999997 < x < 0.78000000000000003

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) + \frac{918938533204673}{1000000000000000}} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} + \frac{918938533204673}{1000000000000000} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y + \frac{918938533204673}{1000000000000000} \]

      5. lower-fma.f64 98.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.72 \lor \neg \left(x \leq 0.78\right):\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.52 \lor \neg \left(y \leq 1.55\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.52) (not (<= y 1.55)))
   (* (- x 0.5) y)
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.52) || !(y <= 1.55)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-1.52d0)) .or. (.not. (y <= 1.55d0))) then
        tmp = (x - 0.5d0) * y
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.52) || !(y <= 1.55)) {
		tmp = (x - 0.5) * y;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.52) or not (y <= 1.55):
		tmp = (x - 0.5) * y
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.52) || !(y <= 1.55))
		tmp = Float64(Float64(x - 0.5) * y);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.52) || ~((y <= 1.55)))
		tmp = (x - 0.5) * y;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.52], N[Not[LessEqual[y, 1.55]], $MachinePrecision]], N[(N[(x - 0.5), $MachinePrecision] * y), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.52 or 1.55000000000000004 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \frac{1}{2}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(0 - x\right)} + \frac{1}{2}\right)\right)\right) \]

       6. associate-+l- N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(0 - \left(x - \frac{1}{2}\right)\right)}\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right)}\right)\right) \]

       8. remove-double-neg N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 97.7

           \[\leadsto \color{blue}{\left(x - 0.5\right)} \cdot y \]

   11. Applied rewrites 97.7%

       \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot y} \]

   if -1.52 < y < 1.55000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 97.7

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.52 \lor \neg \left(y \leq 1.55\right):\\ \;\;\;\;\left(x - 0.5\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.72:\\ \;\;\;\;\left(y - 1\right) \cdot x\\ \mathbf{elif}\;x \leq 0.78:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.72)
   (* (- y 1.0) x)
   (if (<= x 0.78) (fma -0.5 y 0.918938533204673) (fma y x (- x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.72) {
		tmp = (y - 1.0) * x;
	} else if (x <= 0.78) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else {
		tmp = fma(y, x, -x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.72)
		tmp = Float64(Float64(y - 1.0) * x);
	elseif (x <= 0.78)
		tmp = fma(-0.5, y, 0.918938533204673);
	else
		tmp = fma(y, x, Float64(-x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.72], N[(N[(y - 1.0), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 0.78], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], N[(y * x + (-x)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.71999999999999997

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 50.4%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 97.1

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

   11. Applied rewrites 97.1%

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

   if -0.71999999999999997 < x < 0.78000000000000003

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - \frac{1}{2} \cdot y} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + \left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot y\right)\right) + \frac{918938533204673}{1000000000000000}} \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot y} + \frac{918938533204673}{1000000000000000} \]

      4. metadata-eval N/A

         \[\leadsto \color{blue}{\frac{-1}{2}} \cdot y + \frac{918938533204673}{1000000000000000} \]

      5. lower-fma.f64 98.7

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)} \]

   if 0.78000000000000003 < x

   1. Initial program 99.9%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 47.8%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower--.f64 98.2

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

   11. Applied rewrites 98.2%

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

   13. Applied rewrites 98.2%

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

Alternative 9: 73.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -14.2 \lor \neg \left(y \leq 1.32\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -14.2) (not (<= y 1.32))) (* y x) (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -14.2) || !(y <= 1.32)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-14.2d0)) .or. (.not. (y <= 1.32d0))) then
        tmp = y * x
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -14.2) || !(y <= 1.32)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -14.2) or not (y <= 1.32):
		tmp = y * x
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -14.2) || !(y <= 1.32))
		tmp = Float64(y * x);
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -14.2) || ~((y <= 1.32)))
		tmp = y * x;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -14.2], N[Not[LessEqual[y, 1.32]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -14.199999999999999 or 1.32000000000000006 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\frac{918938533204673}{1000000000000000} + x \cdot \left(y - 1\right)\right) - \frac{1}{2} \cdot y} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - 0.5, y, 0.918938533204673 - x\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - \frac{1}{2}, y, \frac{918938533204673}{1000000000000000}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 98.2%

      \[\leadsto \mathsf{fma}\left(x - 0.5, y, 0.918938533204673\right) \]

   9. Taylor expanded in y around -inf

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

       1. mul-1-neg N/A

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

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

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

       3. +-commutative N/A

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

       4. mul-1-neg N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(x\right)\right)} + \frac{1}{2}\right)\right)\right) \]

       5. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\left(0 - x\right)} + \frac{1}{2}\right)\right)\right) \]

       6. associate-+l- N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(0 - \left(x - \frac{1}{2}\right)\right)}\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto y \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(x - \frac{1}{2}\right)\right)\right)}\right)\right) \]

       8. remove-double-neg N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 N/A

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

       11. lower--.f64 97.7

           \[\leadsto \color{blue}{\left(x - 0.5\right)} \cdot y \]

   11. Applied rewrites 97.7%

       \[\leadsto \color{blue}{\left(x - 0.5\right) \cdot y} \]

   12. Taylor expanded in x around inf

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

   14. Applied rewrites 49.9%

       \[\leadsto y \cdot \color{blue}{x} \]

   if -14.199999999999999 < y < 1.32000000000000006

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 97.7

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 97.7%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -14.2 \lor \neg \left(y \leq 1.32\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.8) (not (<= x 0.92))) (- x) 0.918938533204673))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.8) || !(x <= 0.92)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.8d0)) .or. (.not. (x <= 0.92d0))) then
        tmp = -x
    else
        tmp = 0.918938533204673d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.8) || !(x <= 0.92)) {
		tmp = -x;
	} else {
		tmp = 0.918938533204673;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.8) or not (x <= 0.92):
		tmp = -x
	else:
		tmp = 0.918938533204673
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.8) || !(x <= 0.92))
		tmp = Float64(-x);
	else
		tmp = 0.918938533204673;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.8) || ~((x <= 0.92)))
		tmp = -x;
	else
		tmp = 0.918938533204673;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.8], N[Not[LessEqual[x, 0.92]], $MachinePrecision]], (-x), 0.918938533204673]

Derivation

1. Split input into 2 regimes

2. if x < -1.80000000000000004 or 0.92000000000000004 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 52.7

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 52.7%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 51.2%

      \[\leadsto -x \]

   if -1.80000000000000004 < x < 0.92000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

      3. lower--.f64 53.4

         \[\leadsto \color{blue}{0.918938533204673 - x} \]

   5. Applied rewrites 53.4%

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{918938533204673}{1000000000000000} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.5%

      \[\leadsto 0.918938533204673 \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \lor \neg \left(x \leq 0.92\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 - x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- 0.918938533204673 x))

C

double code(double x, double y) {
	return 0.918938533204673 - x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0 - x
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673 - x;
}

Python

def code(x, y):
	return 0.918938533204673 - x

Julia

function code(x, y)
	return Float64(0.918938533204673 - x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673 - x;
end

Wolfram

code[x_, y_] := N[(0.918938533204673 - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

   3. lower--.f64 53.0

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

5. Applied rewrites 53.0%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]
6. Add Preprocessing

Alternative 12: 26.6% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ 0.918938533204673 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.918938533204673)

C

double code(double x, double y) {
	return 0.918938533204673;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.918938533204673d0
end function

Java

public static double code(double x, double y) {
	return 0.918938533204673;
}

Python

def code(x, y):
	return 0.918938533204673

Julia

function code(x, y)
	return 0.918938533204673
end

MATLAB

function tmp = code(x, y)
	tmp = 0.918938533204673;
end

Wolfram

code[x_, y_] := 0.918938533204673

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} + -1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{918938533204673}{1000000000000000} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\frac{918938533204673}{1000000000000000} - x} \]

   3. lower--.f64 53.0

      \[\leadsto \color{blue}{0.918938533204673 - x} \]

5. Applied rewrites 53.0%

   \[\leadsto \color{blue}{0.918938533204673 - x} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{918938533204673}{1000000000000000} \]
7. Step-by-step derivation

8. Applied rewrites 28.1%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))