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

Percentage Accurate: 100.0% → 100.0%

Time: 1.5s

Alternatives: 7

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

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, 1.0× 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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 55.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - 1\right)\\ t_1 := t\_0 - y \cdot 0.5\\ \mathbf{if}\;t\_1 \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 0.0004:\\ \;\;\;\;\left(\left(y - 1\right) + 0.918938533204673\right) + 0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- y 1.0))) (t_1 (- t_0 (* y 0.5))))
   (if (<= t_1 -1000.0)
     t_0
     (if (<= t_1 0.0004)
       (+ (+ (- y 1.0) 0.918938533204673) 0.918938533204673)
       t_0))))

C

double code(double x, double y) {
	double t_0 = x * (y - 1.0);
	double t_1 = t_0 - (y * 0.5);
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.0004) {
		tmp = ((y - 1.0) + 0.918938533204673) + 0.918938533204673;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = x * (y - 1.0d0)
    t_1 = t_0 - (y * 0.5d0)
    if (t_1 <= (-1000.0d0)) then
        tmp = t_0
    else if (t_1 <= 0.0004d0) then
        tmp = ((y - 1.0d0) + 0.918938533204673d0) + 0.918938533204673d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y - 1.0);
	double t_1 = t_0 - (y * 0.5);
	double tmp;
	if (t_1 <= -1000.0) {
		tmp = t_0;
	} else if (t_1 <= 0.0004) {
		tmp = ((y - 1.0) + 0.918938533204673) + 0.918938533204673;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y - 1.0)
	t_1 = t_0 - (y * 0.5)
	tmp = 0
	if t_1 <= -1000.0:
		tmp = t_0
	elif t_1 <= 0.0004:
		tmp = ((y - 1.0) + 0.918938533204673) + 0.918938533204673
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y - 1.0))
	t_1 = Float64(t_0 - Float64(y * 0.5))
	tmp = 0.0
	if (t_1 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 0.0004)
		tmp = Float64(Float64(Float64(y - 1.0) + 0.918938533204673) + 0.918938533204673);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y - 1.0);
	t_1 = t_0 - (y * 0.5);
	tmp = 0.0;
	if (t_1 <= -1000.0)
		tmp = t_0;
	elseif (t_1 <= 0.0004)
		tmp = ((y - 1.0) + 0.918938533204673) + 0.918938533204673;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1000.0], t$95$0, If[LessEqual[t$95$1, 0.0004], N[(N[(N[(y - 1.0), $MachinePrecision] + 0.918938533204673), $MachinePrecision] + 0.918938533204673), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) < -1e3 or 4.00000000000000019e-4 < (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64)))

   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 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 65.1%

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

   if -1e3 < (-.f64 (*.f64 x (-.f64 y #s(literal 1 binary64))) (*.f64 y #s(literal 1/2 binary64))) < 4.00000000000000019e-4

   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

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 18.8%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 22.9%

       \[\leadsto \left(\left(y - 1\right) + 0.918938533204673\right) + 0.918938533204673 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(y - 1\right)\\ t_1 := t\_0 - y \cdot 0.5\\ \mathbf{if}\;y \leq -1.82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0 + 0.918938533204673\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* x (- y 1.0))) (t_1 (- t_0 (* y 0.5))))
   (if (<= y -1.82) t_1 (if (<= y 4.5e-6) (+ t_0 0.918938533204673) t_1))))

C

double code(double x, double y) {
	double t_0 = x * (y - 1.0);
	double t_1 = t_0 - (y * 0.5);
	double tmp;
	if (y <= -1.82) {
		tmp = t_1;
	} else if (y <= 4.5e-6) {
		tmp = t_0 + 0.918938533204673;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = x * (y - 1.0d0)
    t_1 = t_0 - (y * 0.5d0)
    if (y <= (-1.82d0)) then
        tmp = t_1
    else if (y <= 4.5d-6) then
        tmp = t_0 + 0.918938533204673d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x * (y - 1.0);
	double t_1 = t_0 - (y * 0.5);
	double tmp;
	if (y <= -1.82) {
		tmp = t_1;
	} else if (y <= 4.5e-6) {
		tmp = t_0 + 0.918938533204673;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x * (y - 1.0)
	t_1 = t_0 - (y * 0.5)
	tmp = 0
	if y <= -1.82:
		tmp = t_1
	elif y <= 4.5e-6:
		tmp = t_0 + 0.918938533204673
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x * Float64(y - 1.0))
	t_1 = Float64(t_0 - Float64(y * 0.5))
	tmp = 0.0
	if (y <= -1.82)
		tmp = t_1;
	elseif (y <= 4.5e-6)
		tmp = Float64(t_0 + 0.918938533204673);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x * (y - 1.0);
	t_1 = t_0 - (y * 0.5);
	tmp = 0.0;
	if (y <= -1.82)
		tmp = t_1;
	elseif (y <= 4.5e-6)
		tmp = t_0 + 0.918938533204673;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x * N[(y - 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(y * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.82], t$95$1, If[LessEqual[y, 4.5e-6], N[(t$95$0 + 0.918938533204673), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.82000000000000006 or 4.50000000000000011e-6 < 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 98.8%

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

   if -1.82000000000000006 < y < 4.50000000000000011e-6

   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

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

   5. Applied rewrites 99.4%

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

Alternative 4: 75.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Applied rewrites 73.6%

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

Alternative 5: 51.0% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (- y 1.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(y - 1.0), $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 75.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 50.2%

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

Alternative 6: 2.6% accurate, 3.3× 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. 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 75.9%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 2.5%

   \[\leadsto y \cdot \color{blue}{0.5} \]
9. Add Preprocessing

Alternative 7: 2.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return y - 1.0;
}

Python

def code(x, y):
	return y - 1.0

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(y - 1.0), $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 75.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 2.1%

   \[\leadsto y - \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  :pre (TRUE)
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))