Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 2.9s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = ((log((x + y)) + log(z)) - t) + ((a - 0.5d0) * log(t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return ((Math.log((x + y)) + Math.log(z)) - t) + ((a - 0.5) * Math.log(t));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + ((a - 0.5) * math.log(t))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 65.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{+17}:\\ \;\;\;\;\left(a - 0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.05e+17) (* (- a 0.5) (log t)) (- (- a 0.5) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.05e+17) {
		tmp = (a - 0.5) * log(t);
	} else {
		tmp = (a - 0.5) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.05d+17) then
        tmp = (a - 0.5d0) * log(t)
    else
        tmp = (a - 0.5d0) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.05e+17) {
		tmp = (a - 0.5) * Math.log(t);
	} else {
		tmp = (a - 0.5) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.05e+17:
		tmp = (a - 0.5) * math.log(t)
	else:
		tmp = (a - 0.5) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.05e+17)
		tmp = Float64(Float64(a - 0.5) * log(t));
	else
		tmp = Float64(Float64(a - 0.5) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.05e+17)
		tmp = (a - 0.5) * log(t);
	else
		tmp = (a - 0.5) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.05e+17], N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.05e17

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \left(x \cdot \left(x \cdot \left(\frac{1}{3} \cdot \frac{x}{{y}^{3}} - \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \frac{1}{y}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right) - t} \]

   5. Applied rewrites 10.0%

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

   6. Taylor expanded in x around 0

      \[\leadsto \log \left(x + y\right) \]

   8. Applied rewrites 7.5%

      \[\leadsto \log \left(\left(a - 0.5\right) \cdot \log t\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \log y + \color{blue}{\frac{x}{y}} \]

   11. Applied rewrites 56.2%

       \[\leadsto \left(a - 0.5\right) \cdot \color{blue}{\log t} \]

   if 1.05e17 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\log y + \log z\right) - t \]

   8. Applied rewrites 79.4%

      \[\leadsto \left(a - 0.5\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 41.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.0) (log (+ x y)) (- (- a 0.5) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.0) {
		tmp = log((x + y));
	} else {
		tmp = (a - 0.5) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.0d0) then
        tmp = log((x + y))
    else
        tmp = (a - 0.5d0) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = (a - 0.5) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.0:
		tmp = math.log((x + y))
	else:
		tmp = (a - 0.5) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.0)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(Float64(a - 0.5) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.0)
		tmp = log((x + y));
	else
		tmp = (a - 0.5) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + \left(x \cdot \left(x \cdot \left(\frac{1}{3} \cdot \frac{x}{{y}^{3}} - \frac{1}{2} \cdot \frac{1}{{y}^{2}}\right) + \frac{1}{y}\right) + \log t \cdot \left(a - \frac{1}{2}\right)\right)\right)\right) - t} \]

   5. Applied rewrites 10.2%

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

   if 1 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\log y + \log z\right) - t \]

   8. Applied rewrites 78.4%

      \[\leadsto \left(a - 0.5\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 41.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1:\\ \;\;\;\;\log z\\ \mathbf{else}:\\ \;\;\;\;\left(a - 0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.0) (log z) (- (- a 0.5) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.0) {
		tmp = log(z);
	} else {
		tmp = (a - 0.5) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.0d0) then
        tmp = log(z)
    else
        tmp = (a - 0.5d0) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.0) {
		tmp = Math.log(z);
	} else {
		tmp = (a - 0.5) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.0:
		tmp = math.log(z)
	else:
		tmp = (a - 0.5) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.0)
		tmp = log(z);
	else
		tmp = Float64(Float64(a - 0.5) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.0)
		tmp = log(z);
	else
		tmp = (a - 0.5) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.0], N[Log[z], $MachinePrecision], N[(N[(a - 0.5), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1

   1. Initial program 99.3%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 10.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 6.4%

      \[\leadsto \log z \]

   if 1 < t

   1. Initial program 99.9%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(\log y + \log z\right) - t \]

   8. Applied rewrites 78.4%

      \[\leadsto \left(a - 0.5\right) - t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 38.3% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \left(a - 0.5\right) - t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- (- a 0.5) t))

C

double code(double x, double y, double z, double t, double a) {
	return (a - 0.5) - t;
}

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return (a - 0.5) - t;
}

Python

def code(x, y, z, t, a):
	return (a - 0.5) - t

Julia

function code(x, y, z, t, a)
	return Float64(Float64(a - 0.5) - t)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (a - 0.5) - t;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(a - 0.5), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 41.2%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\log y + \log z\right) - t \]

8. Applied rewrites 38.2%

   \[\leadsto \left(a - 0.5\right) - t \]
9. Add Preprocessing

Alternative 6: 4.9% accurate, 80.3× speedup

Math

\[\begin{array}{l} \\ a - 0.5 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (- a 0.5))

C

double code(double x, double y, double z, double t, double a) {
	return a - 0.5;
}

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return a - 0.5;
}

Python

def code(x, y, z, t, a):
	return a - 0.5

Julia

function code(x, y, z, t, a)
	return Float64(a - 0.5)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = a - 0.5;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(a - 0.5), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 6.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \log y + \color{blue}{\log z} \]

8. Applied rewrites 4.4%

   \[\leadsto a - \color{blue}{0.5} \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	return log((x + y)) + ((log(z) - t) + ((a - 0.5) * log(t)));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = log((x + y)) + ((log(z) - t) + ((a - 0.5d0) * log(t)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return Math.log((x + y)) + ((Math.log(z) - t) + ((a - 0.5) * Math.log(t)));
}

Python

def code(x, y, z, t, a):
	return math.log((x + y)) + ((math.log(z) - t) + ((a - 0.5) * math.log(t)))

Julia

function code(x, y, z, t, a)
	return Float64(log(Float64(x + y)) + Float64(Float64(log(z) - t) + Float64(Float64(a - 0.5) * log(t))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = log((x + y)) + ((log(z) - t) + ((a - 0.5) * log(t)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024321 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64
  :pre (TRUE)

  :alt
  (! :herbie-platform default (+ (log (+ x y)) (+ (- (log z) t) (* (- a 1/2) (log t)))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))