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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 7

Speedup: 0.1×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := N[(y * N[(x + -0.5), $MachinePrecision] + 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. Step-by-step derivation

   1. cancel-sign-sub-inv 100.0%

      \[\leadsto \color{blue}{\left(x \cdot \left(y - 1\right) + \left(-y\right) \cdot 0.5\right)} + 0.918938533204673 \]

   2. +-commutative 100.0%

      \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + x \cdot \left(y - 1\right)\right)} + 0.918938533204673 \]

   3. sub-neg 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + x \cdot \color{blue}{\left(y + \left(-1\right)\right)}\right) + 0.918938533204673 \]

   4. distribute-rgt-in 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + \color{blue}{\left(y \cdot x + \left(-1\right) \cdot x\right)}\right) + 0.918938533204673 \]

   5. metadata-eval 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(y \cdot x + \color{blue}{-1} \cdot x\right)\right) + 0.918938533204673 \]

   6. neg-mul-1 100.0%

      \[\leadsto \left(\left(-y\right) \cdot 0.5 + \left(y \cdot x + \color{blue}{\left(-x\right)}\right)\right) + 0.918938533204673 \]

   7. associate-+r+ 100.0%

      \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + y \cdot x\right) + \left(-x\right)\right)} + 0.918938533204673 \]

   8. unsub-neg 100.0%

      \[\leadsto \color{blue}{\left(\left(\left(-y\right) \cdot 0.5 + y \cdot x\right) - x\right)} + 0.918938533204673 \]

   9. associate-+l- 100.0%

      \[\leadsto \color{blue}{\left(\left(-y\right) \cdot 0.5 + y \cdot x\right) - \left(x - 0.918938533204673\right)} \]

   10. distribute-lft-neg-out 100.0%

       \[\leadsto \left(\color{blue}{\left(-y \cdot 0.5\right)} + y \cdot x\right) - \left(x - 0.918938533204673\right) \]

   11. distribute-rgt-neg-in 100.0%

       \[\leadsto \left(\color{blue}{y \cdot \left(-0.5\right)} + y \cdot x\right) - \left(x - 0.918938533204673\right) \]

   12. distribute-lft-out 100.0%

       \[\leadsto \color{blue}{y \cdot \left(\left(-0.5\right) + x\right)} - \left(x - 0.918938533204673\right) \]

   13. fma-neg 100.0%

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

   14. +-commutative 100.0%

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

   15. metadata-eval 100.0%

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

   16. neg-sub0 100.0%

       \[\leadsto \mathsf{fma}\left(y, x + -0.5, \color{blue}{0 - \left(x - 0.918938533204673\right)}\right) \]

   17. associate-+l- 100.0%

       \[\leadsto \mathsf{fma}\left(y, x + -0.5, \color{blue}{\left(0 - x\right) + 0.918938533204673}\right) \]

   18. neg-sub0 100.0%

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

   19. +-commutative 100.0%

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

   20. unsub-neg 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 73.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -310 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e+49)
   (* y -0.5)
   (if (or (<= y -310.0) (not (<= y 1.0))) (* y x) (- 0.918938533204673 x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e+49) {
		tmp = y * -0.5;
	} else if ((y <= -310.0) || !(y <= 1.0)) {
		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 <= (-1.1d+49)) then
        tmp = y * (-0.5d0)
    else if ((y <= (-310.0d0)) .or. (.not. (y <= 1.0d0))) 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 <= -1.1e+49) {
		tmp = y * -0.5;
	} else if ((y <= -310.0) || !(y <= 1.0)) {
		tmp = y * x;
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e+49:
		tmp = y * -0.5
	elif (y <= -310.0) or not (y <= 1.0):
		tmp = y * x
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e+49)
		tmp = Float64(y * -0.5);
	elseif ((y <= -310.0) || !(y <= 1.0))
		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 <= -1.1e+49)
		tmp = y * -0.5;
	elseif ((y <= -310.0) || ~((y <= 1.0)))
		tmp = y * x;
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e+49], N[(y * -0.5), $MachinePrecision], If[Or[LessEqual[y, -310.0], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1e49

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 100.0%

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

   6. Taylor expanded in x around 0 59.5%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]

   if -1.1e49 < y < -310 or 1 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 96.7%

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

   6. Taylor expanded in x around inf 63.0%

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

      1. *-commutative 63.0%

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

   8. Simplified 63.0%

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

   if -310 < y < 1

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.7%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. neg-mul-1 98.7%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+49}:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq -310 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2800000000000 \lor \neg \left(y \leq 900000\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2800000000000.0) (not (<= y 900000.0)))
   (* y (- x 0.5))
   (+ 0.918938533204673 (* x (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2800000000000.0) || !(y <= 900000.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2800000000000.0d0)) .or. (.not. (y <= 900000.0d0))) then
        tmp = y * (x - 0.5d0)
    else
        tmp = 0.918938533204673d0 + (x * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2800000000000.0) || !(y <= 900000.0)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 + (x * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2800000000000.0) or not (y <= 900000.0):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 + (x * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2800000000000.0) || !(y <= 900000.0))
		tmp = Float64(y * Float64(x - 0.5));
	else
		tmp = Float64(0.918938533204673 + Float64(x * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2800000000000.0) || ~((y <= 900000.0)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 + (x * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2800000000000.0], N[Not[LessEqual[y, 900000.0]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 + N[(x * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e12 or 9e5 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 99.1%

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

   if -2.8e12 < y < 9e5

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
   6. Step-by-step derivation

      1. associate--l+ 100.0%

         \[\leadsto \color{blue}{0.918938533204673 + \left(x \cdot \left(y - 1\right) - 0.5 \cdot y\right)} \]

      2. sub-neg 100.0%

         \[\leadsto 0.918938533204673 + \left(x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - 0.5 \cdot y\right) \]

      3. metadata-eval 100.0%

         \[\leadsto 0.918938533204673 + \left(x \cdot \left(y + \color{blue}{-1}\right) - 0.5 \cdot y\right) \]

      4. distribute-rgt-in 100.0%

         \[\leadsto 0.918938533204673 + \left(\color{blue}{\left(y \cdot x + -1 \cdot x\right)} - 0.5 \cdot y\right) \]

      5. neg-mul-1 100.0%

         \[\leadsto 0.918938533204673 + \left(\left(y \cdot x + \color{blue}{\left(-x\right)}\right) - 0.5 \cdot y\right) \]

      6. *-commutative 100.0%

         \[\leadsto 0.918938533204673 + \left(\left(\color{blue}{x \cdot y} + \left(-x\right)\right) - 0.5 \cdot y\right) \]

      7. +-commutative 100.0%

         \[\leadsto 0.918938533204673 + \left(\color{blue}{\left(\left(-x\right) + x \cdot y\right)} - 0.5 \cdot y\right) \]

      8. associate--l+ 100.0%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(\left(-x\right) + \left(x \cdot y - 0.5 \cdot y\right)\right)} \]

      9. cancel-sign-sub-inv 100.0%

         \[\leadsto 0.918938533204673 + \left(\left(-x\right) + \color{blue}{\left(x \cdot y + \left(-0.5\right) \cdot y\right)}\right) \]

      10. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(\left(-x\right) + \left(x \cdot y + \color{blue}{-0.5} \cdot y\right)\right) \]

      11. distribute-rgt-in 100.0%

          \[\leadsto 0.918938533204673 + \left(\left(-x\right) + \color{blue}{y \cdot \left(x + -0.5\right)}\right) \]

      12. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x + \color{blue}{\left(-0.5\right)}\right)\right) \]

      13. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(\left(-x\right) + y \cdot \color{blue}{\left(x - 0.5\right)}\right) \]

      14. +-commutative 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

      15. unsub-neg 100.0%

          \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

      16. sub-neg 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

      17. metadata-eval 100.0%

          \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

   8. Taylor expanded in x around inf 99.6%

      \[\leadsto 0.918938533204673 + \color{blue}{x \cdot \left(y - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2800000000000 \lor \neg \left(y \leq 900000\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 + x \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.35 \lor \neg \left(y \leq 1.05\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -1.35) (not (<= y 1.05)))
   (* y (- x 0.5))
   (- 0.918938533204673 x)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -1.35) || !(y <= 1.05)) {
		tmp = y * (x - 0.5);
	} 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 <= (-1.35d0)) .or. (.not. (y <= 1.05d0))) then
        tmp = y * (x - 0.5d0)
    else
        tmp = 0.918938533204673d0 - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -1.35) || !(y <= 1.05)) {
		tmp = y * (x - 0.5);
	} else {
		tmp = 0.918938533204673 - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -1.35) or not (y <= 1.05):
		tmp = y * (x - 0.5)
	else:
		tmp = 0.918938533204673 - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -1.35) || !(y <= 1.05))
		tmp = Float64(y * Float64(x - 0.5));
	else
		tmp = Float64(0.918938533204673 - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -1.35) || ~((y <= 1.05)))
		tmp = y * (x - 0.5);
	else
		tmp = 0.918938533204673 - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -1.35], N[Not[LessEqual[y, 1.05]], $MachinePrecision]], N[(y * N[(x - 0.5), $MachinePrecision]), $MachinePrecision], N[(0.918938533204673 - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3500000000000001 or 1.05000000000000004 < y

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 98.1%

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

   if -1.3500000000000001 < y < 1.05000000000000004

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 98.7%

      \[\leadsto \color{blue}{0.918938533204673 + -1 \cdot x} \]
   6. Step-by-step derivation

      1. neg-mul-1 98.7%

         \[\leadsto 0.918938533204673 + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \lor \neg \left(y \leq 1.05\right):\\ \;\;\;\;y \cdot \left(x - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;0.918938533204673 - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+19} \lor \neg \left(x \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.15e+19) (not (<= x 2e-5))) (* y x) (* y -0.5)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.15e+19) || !(x <= 2e-5)) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.15d+19)) .or. (.not. (x <= 2d-5))) then
        tmp = y * x
    else
        tmp = y * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.15e+19) || !(x <= 2e-5)) {
		tmp = y * x;
	} else {
		tmp = y * -0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.15e+19) or not (x <= 2e-5):
		tmp = y * x
	else:
		tmp = y * -0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.15e+19) || !(x <= 2e-5))
		tmp = Float64(y * x);
	else
		tmp = Float64(y * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.15e+19) || ~((x <= 2e-5)))
		tmp = y * x;
	else
		tmp = y * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.15e+19], N[Not[LessEqual[x, 2e-5]], $MachinePrecision]], N[(y * x), $MachinePrecision], N[(y * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.15e19 or 2.00000000000000016e-5 < x

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 55.1%

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

   6. Taylor expanded in x around inf 55.0%

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

      1. *-commutative 55.0%

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

   8. Simplified 55.0%

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

   if -1.15e19 < x < 2.00000000000000016e-5

   1. Initial program 100.0%

      \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
   2. Step-by-step derivation

      1. associate-+l- 100.0%

         \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

      2. sub-neg 100.0%

         \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

      3. metadata-eval 100.0%

         \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 50.1%

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

   6. Taylor expanded in x around 0 49.5%

      \[\leadsto \color{blue}{-0.5 \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+19} \lor \neg \left(x \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. sub-neg 100.0%

      \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. metadata-eval 100.0%

      \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(0.918938533204673 + x \cdot \left(y - 1\right)\right) - 0.5 \cdot y} \]
6. Step-by-step derivation

   1. associate--l+ 100.0%

      \[\leadsto \color{blue}{0.918938533204673 + \left(x \cdot \left(y - 1\right) - 0.5 \cdot y\right)} \]

   2. sub-neg 100.0%

      \[\leadsto 0.918938533204673 + \left(x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - 0.5 \cdot y\right) \]

   3. metadata-eval 100.0%

      \[\leadsto 0.918938533204673 + \left(x \cdot \left(y + \color{blue}{-1}\right) - 0.5 \cdot y\right) \]

   4. distribute-rgt-in 100.0%

      \[\leadsto 0.918938533204673 + \left(\color{blue}{\left(y \cdot x + -1 \cdot x\right)} - 0.5 \cdot y\right) \]

   5. neg-mul-1 100.0%

      \[\leadsto 0.918938533204673 + \left(\left(y \cdot x + \color{blue}{\left(-x\right)}\right) - 0.5 \cdot y\right) \]

   6. *-commutative 100.0%

      \[\leadsto 0.918938533204673 + \left(\left(\color{blue}{x \cdot y} + \left(-x\right)\right) - 0.5 \cdot y\right) \]

   7. +-commutative 100.0%

      \[\leadsto 0.918938533204673 + \left(\color{blue}{\left(\left(-x\right) + x \cdot y\right)} - 0.5 \cdot y\right) \]

   8. associate--l+ 100.0%

      \[\leadsto 0.918938533204673 + \color{blue}{\left(\left(-x\right) + \left(x \cdot y - 0.5 \cdot y\right)\right)} \]

   9. cancel-sign-sub-inv 100.0%

      \[\leadsto 0.918938533204673 + \left(\left(-x\right) + \color{blue}{\left(x \cdot y + \left(-0.5\right) \cdot y\right)}\right) \]

   10. metadata-eval 100.0%

       \[\leadsto 0.918938533204673 + \left(\left(-x\right) + \left(x \cdot y + \color{blue}{-0.5} \cdot y\right)\right) \]

   11. distribute-rgt-in 100.0%

       \[\leadsto 0.918938533204673 + \left(\left(-x\right) + \color{blue}{y \cdot \left(x + -0.5\right)}\right) \]

   12. metadata-eval 100.0%

       \[\leadsto 0.918938533204673 + \left(\left(-x\right) + y \cdot \left(x + \color{blue}{\left(-0.5\right)}\right)\right) \]

   13. sub-neg 100.0%

       \[\leadsto 0.918938533204673 + \left(\left(-x\right) + y \cdot \color{blue}{\left(x - 0.5\right)}\right) \]

   14. +-commutative 100.0%

       \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) + \left(-x\right)\right)} \]

   15. unsub-neg 100.0%

       \[\leadsto 0.918938533204673 + \color{blue}{\left(y \cdot \left(x - 0.5\right) - x\right)} \]

   16. sub-neg 100.0%

       \[\leadsto 0.918938533204673 + \left(y \cdot \color{blue}{\left(x + \left(-0.5\right)\right)} - x\right) \]

   17. metadata-eval 100.0%

       \[\leadsto 0.918938533204673 + \left(y \cdot \left(x + \color{blue}{-0.5}\right) - x\right) \]

7. Simplified 100.0%

   \[\leadsto \color{blue}{0.918938533204673 + \left(y \cdot \left(x + -0.5\right) - x\right)} \]

8. Final simplification 100.0%

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

Alternative 7: 27.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ y \cdot -0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y -0.5))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
2. Step-by-step derivation

   1. associate-+l- 100.0%

      \[\leadsto \color{blue}{x \cdot \left(y - 1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]

   2. sub-neg 100.0%

      \[\leadsto x \cdot \color{blue}{\left(y + \left(-1\right)\right)} - \left(y \cdot 0.5 - 0.918938533204673\right) \]

   3. metadata-eval 100.0%

      \[\leadsto x \cdot \left(y + \color{blue}{-1}\right) - \left(y \cdot 0.5 - 0.918938533204673\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \left(y + -1\right) - \left(y \cdot 0.5 - 0.918938533204673\right)} \]
4. Add Preprocessing

5. Taylor expanded in y around inf 52.7%

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

6. Taylor expanded in x around 0 25.4%

   \[\leadsto \color{blue}{-0.5 \cdot y} \]

7. Final simplification 25.4%

   \[\leadsto y \cdot -0.5 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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))