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

Percentage Accurate: 100.0% → 100.0%

Time: 7.4s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 2: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+166}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{+42}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -50:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+229}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.42e+166)
   (- x)
   (if (<= x -5.1e+42)
     (* x y)
     (if (<= x -50.0)
       (- 0.918938533204673 x)
       (if (<= x 2.95e-5)
         (fma -0.5 y 0.918938533204673)
         (if (<= x 1.15e+229) (- 0.918938533204673 x) (* x y)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.42e+166) {
		tmp = -x;
	} else if (x <= -5.1e+42) {
		tmp = x * y;
	} else if (x <= -50.0) {
		tmp = 0.918938533204673 - x;
	} else if (x <= 2.95e-5) {
		tmp = fma(-0.5, y, 0.918938533204673);
	} else if (x <= 1.15e+229) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.42e+166)
		tmp = Float64(-x);
	elseif (x <= -5.1e+42)
		tmp = Float64(x * y);
	elseif (x <= -50.0)
		tmp = Float64(0.918938533204673 - x);
	elseif (x <= 2.95e-5)
		tmp = fma(-0.5, y, 0.918938533204673);
	elseif (x <= 1.15e+229)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.42e+166], (-x), If[LessEqual[x, -5.1e+42], N[(x * y), $MachinePrecision], If[LessEqual[x, -50.0], N[(0.918938533204673 - x), $MachinePrecision], If[LessEqual[x, 2.95e-5], N[(-0.5 * y + 0.918938533204673), $MachinePrecision], If[LessEqual[x, 1.15e+229], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.41999999999999995e166

   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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot y + x \cdot -1} \]

      4. *-commutative N/A

         \[\leadsto x \cdot y + \color{blue}{-1 \cdot x} \]

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.3%

      \[\leadsto -x \]

   if -1.41999999999999995e166 < x < -5.0999999999999999e42 or 1.15e229 < 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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot y + x \cdot -1} \]

      4. *-commutative N/A

         \[\leadsto x \cdot y + \color{blue}{-1 \cdot x} \]

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 69.8%

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

   if -5.0999999999999999e42 < x < -50 or 2.9499999999999999e-5 < x < 1.15e229

   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 65.5

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

   5. Applied rewrites 65.5%

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

   if -50 < x < 2.9499999999999999e-5

   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 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.42 \cdot 10^{+166}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq -5.1 \cdot 10^{+42}:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;x \leq -50:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{elif}\;x \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y, 0.918938533204673\right)\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{+229}:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(x + -0.5\right)\\ \mathbf{if}\;y \leq -1.3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.72:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (+ x -0.5))))
   (if (<= y -1.3) t_0 (if (<= y 1.72) (- 0.918938533204673 x) t_0))))

C

double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.3) {
		tmp = t_0;
	} else if (y <= 1.72) {
		tmp = 0.918938533204673 - x;
	} 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 = y * (x + (-0.5d0))
    if (y <= (-1.3d0)) then
        tmp = t_0
    else if (y <= 1.72d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (x + -0.5);
	double tmp;
	if (y <= -1.3) {
		tmp = t_0;
	} else if (y <= 1.72) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (x + -0.5)
	tmp = 0
	if y <= -1.3:
		tmp = t_0
	elif y <= 1.72:
		tmp = 0.918938533204673 - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(x + -0.5))
	tmp = 0.0
	if (y <= -1.3)
		tmp = t_0;
	elseif (y <= 1.72)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (x + -0.5);
	tmp = 0.0;
	if (y <= -1.3)
		tmp = t_0;
	elseif (y <= 1.72)
		tmp = 0.918938533204673 - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(x + -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.3], t$95$0, If[LessEqual[y, 1.72], N[(0.918938533204673 - x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000004 or 1.71999999999999997 < 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 y around inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 97.6

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

   5. Applied rewrites 97.6%

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

   if -1.30000000000000004 < y < 1.71999999999999997

   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 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 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \mathbf{elif}\;y \leq 1.72:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x + -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -43:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.35:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -43.0) (* y -0.5) (if (<= y 1.35) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -43.0) {
		tmp = y * -0.5;
	} else if (y <= 1.35) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-43.0d0)) then
        tmp = y * (-0.5d0)
    else if (y <= 1.35d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -43.0) {
		tmp = y * -0.5;
	} else if (y <= 1.35) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -43.0:
		tmp = y * -0.5
	elif y <= 1.35:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -43.0)
		tmp = Float64(y * -0.5);
	elseif (y <= 1.35)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -43.0)
		tmp = y * -0.5;
	elseif (y <= 1.35)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -43.0], N[(y * -0.5), $MachinePrecision], If[LessEqual[y, 1.35], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -43

   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 inf

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-neg-in N/A

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

      5. +-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-neg-in N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower-+.f64 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 60.9%

      \[\leadsto y \cdot -0.5 \]

   if -43 < y < 1.3500000000000001

   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%

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

   if 1.3500000000000001 < 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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot y + x \cdot -1} \]

      4. *-commutative N/A

         \[\leadsto x \cdot y + \color{blue}{-1 \cdot x} \]

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 51.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -43:\\ \;\;\;\;y \cdot -0.5\\ \mathbf{elif}\;y \leq 1.35:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -112000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.35:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -112000000000.0)
   (* x y)
   (if (<= y 1.35) (- 0.918938533204673 x) (* x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -112000000000.0) {
		tmp = x * y;
	} else if (y <= 1.35) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-112000000000.0d0)) then
        tmp = x * y
    else if (y <= 1.35d0) then
        tmp = 0.918938533204673d0 - x
    else
        tmp = x * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -112000000000.0) {
		tmp = x * y;
	} else if (y <= 1.35) {
		tmp = 0.918938533204673 - x;
	} else {
		tmp = x * y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -112000000000.0:
		tmp = x * y
	elif y <= 1.35:
		tmp = 0.918938533204673 - x
	else:
		tmp = x * y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -112000000000.0)
		tmp = Float64(x * y);
	elseif (y <= 1.35)
		tmp = Float64(0.918938533204673 - x);
	else
		tmp = Float64(x * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -112000000000.0)
		tmp = x * y;
	elseif (y <= 1.35)
		tmp = 0.918938533204673 - x;
	else
		tmp = x * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -112000000000.0], N[(x * y), $MachinePrecision], If[LessEqual[y, 1.35], N[(0.918938533204673 - x), $MachinePrecision], N[(x * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.12e11 or 1.3500000000000001 < 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 inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot y + x \cdot -1} \]

      4. *-commutative N/A

         \[\leadsto x \cdot y + \color{blue}{-1 \cdot x} \]

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 44.4%

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

   if -1.12e11 < y < 1.3500000000000001

   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 95.7

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

   5. Applied rewrites 95.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -112000000000:\\ \;\;\;\;x \cdot y\\ \mathbf{elif}\;y \leq 1.35:\\ \;\;\;\;0.918938533204673 - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.92:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \leq 0.92:\\ \;\;\;\;0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.92) (- x) (if (<= x 0.92) 0.918938533204673 (- x))))

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.92d0)) then
        tmp = -x
    else if (x <= 0.92d0) then
        tmp = 0.918938533204673d0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

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

Python

def code(x, y):
	tmp = 0
	if x <= -0.92:
		tmp = -x
	elif x <= 0.92:
		tmp = 0.918938533204673
	else:
		tmp = -x
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -0.92], (-x), If[LessEqual[x, 0.92], 0.918938533204673, (-x)]]

Derivation

1. Split input into 2 regimes

2. if x < -0.92000000000000004 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 x around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot y + x \cdot -1} \]

      4. *-commutative N/A

         \[\leadsto x \cdot y + \color{blue}{-1 \cdot x} \]

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.4%

      \[\leadsto -x \]

   if -0.92000000000000004 < 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 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.0%

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

Alternative 7: 49.5% 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 51.1

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

5. Applied rewrites 51.1%

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

Alternative 8: 25.5% 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 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 27.1%

   \[\leadsto 0.918938533204673 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))