Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Percentage Accurate: 99.8% → 99.8%

Time: 1.5s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \left(y + 0.5\right) \cdot \log y\\ t_1 := t\_0 - z\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+64}:\\ \;\;\;\;t\_0 + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- x (* (+ y 0.5) (log y)))) (t_1 (- t_0 z)))
   (if (<= z -9.5e+88) t_1 (if (<= z 6e+64) (+ t_0 y) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = x - ((y + 0.5) * log(y));
	double t_1 = t_0 - z;
	double tmp;
	if (z <= -9.5e+88) {
		tmp = t_1;
	} else if (z <= 6e+64) {
		tmp = t_0 + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x - ((y + 0.5d0) * log(y))
    t_1 = t_0 - z
    if (z <= (-9.5d+88)) then
        tmp = t_1
    else if (z <= 6d+64) then
        tmp = t_0 + y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x - ((y + 0.5) * Math.log(y));
	double t_1 = t_0 - z;
	double tmp;
	if (z <= -9.5e+88) {
		tmp = t_1;
	} else if (z <= 6e+64) {
		tmp = t_0 + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x - ((y + 0.5) * math.log(y))
	t_1 = t_0 - z
	tmp = 0
	if z <= -9.5e+88:
		tmp = t_1
	elif z <= 6e+64:
		tmp = t_0 + y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x - Float64(Float64(y + 0.5) * log(y)))
	t_1 = Float64(t_0 - z)
	tmp = 0.0
	if (z <= -9.5e+88)
		tmp = t_1;
	elseif (z <= 6e+64)
		tmp = Float64(t_0 + y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x - ((y + 0.5) * log(y));
	t_1 = t_0 - z;
	tmp = 0.0;
	if (z <= -9.5e+88)
		tmp = t_1;
	elseif (z <= 6e+64)
		tmp = t_0 + y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - z), $MachinePrecision]}, If[LessEqual[z, -9.5e+88], t$95$1, If[LessEqual[z, 6e+64], N[(t$95$0 + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.50000000000000059e88 or 6.0000000000000004e64 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 90.4%

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

   if -9.50000000000000059e88 < z < 6.0000000000000004e64

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 97.6%

      \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y - z\\ \mathbf{if}\;z \leq -1.22 \cdot 10^{+118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+125}:\\ \;\;\;\;\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log y) z)))
   (if (<= z -1.22e+118)
     t_0
     (if (<= z 1.65e+125) (+ (- x (* (+ y 0.5) (log y))) y) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) - z;
	double tmp;
	if (z <= -1.22e+118) {
		tmp = t_0;
	} else if (z <= 1.65e+125) {
		tmp = (x - ((y + 0.5) * log(y))) + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(y) - z
    if (z <= (-1.22d+118)) then
        tmp = t_0
    else if (z <= 1.65d+125) then
        tmp = (x - ((y + 0.5d0) * log(y))) + y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(y) - z;
	double tmp;
	if (z <= -1.22e+118) {
		tmp = t_0;
	} else if (z <= 1.65e+125) {
		tmp = (x - ((y + 0.5) * Math.log(y))) + y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) - z
	tmp = 0
	if z <= -1.22e+118:
		tmp = t_0
	elif z <= 1.65e+125:
		tmp = (x - ((y + 0.5) * math.log(y))) + y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) - z)
	tmp = 0.0
	if (z <= -1.22e+118)
		tmp = t_0;
	elseif (z <= 1.65e+125)
		tmp = Float64(Float64(x - Float64(Float64(y + 0.5) * log(y))) + y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) - z;
	tmp = 0.0;
	if (z <= -1.22e+118)
		tmp = t_0;
	elseif (z <= 1.65e+125)
		tmp = (x - ((y + 0.5) * log(y))) + y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[z, -1.22e+118], t$95$0, If[LessEqual[z, 1.65e+125], N[(N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2200000000000001e118 or 1.65000000000000003e125 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 92.5%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 77.6%

      \[\leadsto \log y - z \]

   if -1.2200000000000001e118 < z < 1.65000000000000003e125

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 94.5%

      \[\leadsto \color{blue}{\left(x - \left(y + 0.5\right) \cdot \log y\right) + y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 66.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log y - z\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{+109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+125}:\\ \;\;\;\;x - \left(y + 0.5\right) \cdot \log y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (log y) z)))
   (if (<= z -5.2e+109)
     t_0
     (if (<= z 1.1e+125) (- x (* (+ y 0.5) (log y))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = log(y) - z;
	double tmp;
	if (z <= -5.2e+109) {
		tmp = t_0;
	} else if (z <= 1.1e+125) {
		tmp = x - ((y + 0.5) * log(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = log(y) - z
    if (z <= (-5.2d+109)) then
        tmp = t_0
    else if (z <= 1.1d+125) then
        tmp = x - ((y + 0.5d0) * log(y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = Math.log(y) - z;
	double tmp;
	if (z <= -5.2e+109) {
		tmp = t_0;
	} else if (z <= 1.1e+125) {
		tmp = x - ((y + 0.5) * Math.log(y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = math.log(y) - z
	tmp = 0
	if z <= -5.2e+109:
		tmp = t_0
	elif z <= 1.1e+125:
		tmp = x - ((y + 0.5) * math.log(y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(log(y) - z)
	tmp = 0.0
	if (z <= -5.2e+109)
		tmp = t_0;
	elseif (z <= 1.1e+125)
		tmp = Float64(x - Float64(Float64(y + 0.5) * log(y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = log(y) - z;
	tmp = 0.0;
	if (z <= -5.2e+109)
		tmp = t_0;
	elseif (z <= 1.1e+125)
		tmp = x - ((y + 0.5) * log(y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[Log[y], $MachinePrecision] - z), $MachinePrecision]}, If[LessEqual[z, -5.2e+109], t$95$0, If[LessEqual[z, 1.1e+125], N[(x - N[(N[(y + 0.5), $MachinePrecision] * N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999997e109 or 1.09999999999999995e125 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 76.9%

      \[\leadsto \log y - z \]

   if -5.1999999999999997e109 < z < 1.09999999999999995e125

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 95.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 70.9%

      \[\leadsto x - \color{blue}{\left(y + 0.5\right) \cdot \log y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 28.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \log y - z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- (log y) z))

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = log(y) - z
end function

Java

public static double code(double x, double y, double z) {
	return Math.log(y) - z;
}

Python

def code(x, y, z):
	return math.log(y) - z

Julia

function code(x, y, z)
	return Float64(log(y) - z)
end

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[Log[y], $MachinePrecision] - z), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 80.9%

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

6. Taylor expanded in x around -inf

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

8. Applied rewrites 29.4%

   \[\leadsto \log y - z \]
9. Add Preprocessing

Alternative 6: 3.8% accurate, 29.5× speedup

Math

\[\begin{array}{l} \\ y + 0.5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ y 0.5))

C

double code(double x, double y, double z) {
	return y + 0.5;
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return y + 0.5;
}

Python

def code(x, y, z):
	return y + 0.5

Julia

function code(x, y, z)
	return Float64(y + 0.5)
end

MATLAB

function tmp = code(x, y, z)
	tmp = y + 0.5;
end

Wolfram

code[x_, y_, z_] := N[(y + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 71.4%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 3.8%

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

Developer Target 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

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

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

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