Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 22.7s

Alternatives: 16

Speedup: N/A×

Specification

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

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

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(log(Float64(x + y)) + 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[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision]), $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. Step-by-step derivation

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

Alternative 2: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + -0.5\right) \cdot \log t - t\\ \mathbf{if}\;\log z \leq 310:\\ \;\;\;\;t_1 + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (+ a -0.5) (log t)) t)))
   (if (<= (log z) 310.0) (+ t_1 (log (* (+ x y) z))) t_1)))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((a + -0.5) * log(t)) - t;
	double tmp;
	if (log(z) <= 310.0) {
		tmp = t_1 + log(((x + y) * z));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = ((a + (-0.5d0)) * log(t)) - t
    if (log(z) <= 310.0d0) then
        tmp = t_1 + log(((x + y) * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((a + -0.5) * Math.log(t)) - t;
	double tmp;
	if (Math.log(z) <= 310.0) {
		tmp = t_1 + Math.log(((x + y) * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((a + -0.5) * math.log(t)) - t
	tmp = 0
	if math.log(z) <= 310.0:
		tmp = t_1 + math.log(((x + y) * z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(a + -0.5) * log(t)) - t)
	tmp = 0.0
	if (log(z) <= 310.0)
		tmp = Float64(t_1 + log(Float64(Float64(x + y) * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((a + -0.5) * log(t)) - t;
	tmp = 0.0;
	if (log(z) <= 310.0)
		tmp = t_1 + log(((x + y) * z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[N[Log[z], $MachinePrecision], 310.0], N[(t$95$1 + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 z) < 310

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log t} \]
   4. Step-by-step derivation

      1. associate-+r- 99.6%

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

      2. associate-+l- 99.6%

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

      3. sum-log 94.1%

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

   5. Applied egg-rr 94.1%

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

   if 310 < (log.f64 z)

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 75.7%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 75.7%

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

   6. Simplified 75.7%

      \[\leadsto \color{blue}{\left(-t\right)} + \left(a + -0.5\right) \cdot \log t \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\log z \leq 310:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t - t\right) + \log \left(\left(x + y\right) \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]

Alternative 3: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 255:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 255.0)
   (+ (log z) (+ (log (+ x y)) (* (log t) (- a 0.5))))
   (- (* (+ a -0.5) (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 255.0) {
		tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5)));
	} else {
		tmp = ((a + -0.5) * log(t)) - 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 <= 255.0d0) then
        tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5d0)))
    else
        tmp = ((a + (-0.5d0)) * log(t)) - 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 <= 255.0) {
		tmp = Math.log(z) + (Math.log((x + y)) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 255.0:
		tmp = math.log(z) + (math.log((x + y)) + (math.log(t) * (a - 0.5)))
	else:
		tmp = ((a + -0.5) * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 255.0)
		tmp = Float64(log(z) + Float64(log(Float64(x + y)) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 255.0)
		tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5)));
	else
		tmp = ((a + -0.5) * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 255.0], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 255

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t around 0 98.0%

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

   if 255 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 98.7%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 98.7%

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

   6. Simplified 98.7%

      \[\leadsto \color{blue}{\left(-t\right)} + \left(a + -0.5\right) \cdot \log t \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 255:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]

Alternative 4: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 290:\\ \;\;\;\;\log y + \left(\log z - \log t \cdot \left(0.5 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 290.0)
   (+ (log y) (- (log z) (* (log t) (- 0.5 a))))
   (- (* (+ a -0.5) (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 290.0) {
		tmp = log(y) + (log(z) - (log(t) * (0.5 - a)));
	} else {
		tmp = ((a + -0.5) * log(t)) - 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 <= 290.0d0) then
        tmp = log(y) + (log(z) - (log(t) * (0.5d0 - a)))
    else
        tmp = ((a + (-0.5d0)) * log(t)) - 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 <= 290.0) {
		tmp = Math.log(y) + (Math.log(z) - (Math.log(t) * (0.5 - a)));
	} else {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 290.0:
		tmp = math.log(y) + (math.log(z) - (math.log(t) * (0.5 - a)))
	else:
		tmp = ((a + -0.5) * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 290.0)
		tmp = Float64(log(y) + Float64(log(z) - Float64(log(t) * Float64(0.5 - a))));
	else
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 290.0)
		tmp = log(y) + (log(z) - (log(t) * (0.5 - a)));
	else
		tmp = ((a + -0.5) * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 290.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * N[(0.5 - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 290

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t around 0 98.0%

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

   5. Taylor expanded in x around 0 62.3%

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

   if 290 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 98.7%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 98.7%

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

   6. Simplified 98.7%

      \[\leadsto \color{blue}{\left(-t\right)} + \left(a + -0.5\right) \cdot \log t \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 290:\\ \;\;\;\;\log y + \left(\log z - \log t \cdot \left(0.5 - a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]

Alternative 5: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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)) * log(t)) + ((log(z) - t) + log(y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 68.1%

   \[\leadsto \color{blue}{\left(\left(\log y + \log z\right) - t\right)} + \left(a + -0.5\right) \cdot \log t \]
5. Step-by-step derivation

   1. associate--l+ 68.1%

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

6. Simplified 68.1%

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

7. Final simplification 68.1%

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

Alternative 6: 69.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (log(y) + (log(z) - (log(t) * (0.5 - a)))) - 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(y) + (log(z) - (log(t) * (0.5d0 - a)))) - t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 68.0%

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

5. Final simplification 68.0%

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

Alternative 7: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.499999999\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -2e+14) (not (<= (- a 0.5) -0.499999999)))
   (- (* (+ a -0.5) (log t)) t)
   (- (+ (log (* y z)) (* -0.5 (log t))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -2e+14) || !((a - 0.5) <= -0.499999999)) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else {
		tmp = (log((y * z)) + (-0.5 * log(t))) - 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 (((a - 0.5d0) <= (-2d+14)) .or. (.not. ((a - 0.5d0) <= (-0.499999999d0)))) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else
        tmp = (log((y * z)) + ((-0.5d0) * log(t))) - 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 (((a - 0.5) <= -2e+14) || !((a - 0.5) <= -0.499999999)) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else {
		tmp = (Math.log((y * z)) + (-0.5 * Math.log(t))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -2e+14) or not ((a - 0.5) <= -0.499999999):
		tmp = ((a + -0.5) * math.log(t)) - t
	else:
		tmp = (math.log((y * z)) + (-0.5 * math.log(t))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -2e+14) || !(Float64(a - 0.5) <= -0.499999999))
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	else
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(-0.5 * log(t))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -2e+14) || ~(((a - 0.5) <= -0.499999999)))
		tmp = ((a + -0.5) * log(t)) - t;
	else
		tmp = (log((y * z)) + (-0.5 * log(t))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -2e+14], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.499999999]], $MachinePrecision]], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -2e14 or -0.499999998999999973 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 97.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 97.4%

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

   6. Simplified 97.4%

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

   if -2e14 < (-.f64 a 1/2) < -0.499999998999999973

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log t} \]
   4. Step-by-step derivation

      1. add-exp-log 35.1%

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

      2. associate-+r- 35.1%

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

      3. sum-log 21.2%

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

   5. Applied egg-rr 21.2%

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

   6. Taylor expanded in a around 0 75.1%

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

   7. Taylor expanded in x around 0 48.4%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} + \left(-0.5 \cdot \log t + a \cdot \log t\right)\right) - t \]
   8. Step-by-step derivation

      1. *-commutative 48.4%

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

   9. Simplified 48.4%

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

   10. Taylor expanded in a around 0 47.6%

       \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 47.6%

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

       2. +-commutative 47.6%

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

       3. log-prod 61.3%

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

       4. +-commutative 61.3%

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

       5. log-prod 47.6%

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

   12. Simplified 47.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.499999999\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \end{array} \]

Alternative 8: 72.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.5 \cdot \log t\\ \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.499999999\right):\\ \;\;\;\;\left(t_1 + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + t_1\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* -0.5 (log t))))
   (if (or (<= (- a 0.5) -2e+14) (not (<= (- a 0.5) -0.499999999)))
     (- (+ t_1 (* a (log t))) t)
     (- (+ (log (* y z)) t_1) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -0.5 * log(t);
	double tmp;
	if (((a - 0.5) <= -2e+14) || !((a - 0.5) <= -0.499999999)) {
		tmp = (t_1 + (a * log(t))) - t;
	} else {
		tmp = (log((y * z)) + t_1) - 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) :: t_1
    real(8) :: tmp
    t_1 = (-0.5d0) * log(t)
    if (((a - 0.5d0) <= (-2d+14)) .or. (.not. ((a - 0.5d0) <= (-0.499999999d0)))) then
        tmp = (t_1 + (a * log(t))) - t
    else
        tmp = (log((y * z)) + t_1) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = -0.5 * Math.log(t);
	double tmp;
	if (((a - 0.5) <= -2e+14) || !((a - 0.5) <= -0.499999999)) {
		tmp = (t_1 + (a * Math.log(t))) - t;
	} else {
		tmp = (Math.log((y * z)) + t_1) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -0.5 * math.log(t)
	tmp = 0
	if ((a - 0.5) <= -2e+14) or not ((a - 0.5) <= -0.499999999):
		tmp = (t_1 + (a * math.log(t))) - t
	else:
		tmp = (math.log((y * z)) + t_1) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(-0.5 * log(t))
	tmp = 0.0
	if ((Float64(a - 0.5) <= -2e+14) || !(Float64(a - 0.5) <= -0.499999999))
		tmp = Float64(Float64(t_1 + Float64(a * log(t))) - t);
	else
		tmp = Float64(Float64(log(Float64(y * z)) + t_1) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = -0.5 * log(t);
	tmp = 0.0;
	if (((a - 0.5) <= -2e+14) || ~(((a - 0.5) <= -0.499999999)))
		tmp = (t_1 + (a * log(t))) - t;
	else
		tmp = (log((y * z)) + t_1) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -2e+14], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.499999999]], $MachinePrecision]], N[(N[(t$95$1 + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -2e14 or -0.499999998999999973 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 97.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in a around 0 97.4%

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

   if -2e14 < (-.f64 a 1/2) < -0.499999998999999973

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log t} \]
   4. Step-by-step derivation

      1. add-exp-log 35.1%

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

      2. associate-+r- 35.1%

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

      3. sum-log 21.2%

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

   5. Applied egg-rr 21.2%

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

   6. Taylor expanded in a around 0 75.1%

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

   7. Taylor expanded in x around 0 48.4%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} + \left(-0.5 \cdot \log t + a \cdot \log t\right)\right) - t \]
   8. Step-by-step derivation

      1. *-commutative 48.4%

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

   9. Simplified 48.4%

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

   10. Taylor expanded in a around 0 47.6%

       \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 47.6%

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

       2. +-commutative 47.6%

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

       3. log-prod 61.3%

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

       4. +-commutative 61.3%

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

       5. log-prod 47.6%

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

   12. Simplified 47.6%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -2 \cdot 10^{+14} \lor \neg \left(a - 0.5 \leq -0.499999999\right):\\ \;\;\;\;\left(-0.5 \cdot \log t + a \cdot \log t\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right) - t\\ \end{array} \]

Alternative 9: 64.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(a + -0.5\right) \cdot \log t - t\\ \mathbf{if}\;z \leq 3.25 \cdot 10^{+134}:\\ \;\;\;\;t_1 + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (+ a -0.5) (log t)) t)))
   (if (<= z 3.25e+134) (+ t_1 (log (* y z))) t_1)))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((a + -0.5) * log(t)) - t;
	double tmp;
	if (z <= 3.25e+134) {
		tmp = t_1 + log((y * z));
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = ((a + (-0.5d0)) * log(t)) - t
    if (z <= 3.25d+134) then
        tmp = t_1 + log((y * z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((a + -0.5) * Math.log(t)) - t;
	double tmp;
	if (z <= 3.25e+134) {
		tmp = t_1 + Math.log((y * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((a + -0.5) * math.log(t)) - t
	tmp = 0
	if z <= 3.25e+134:
		tmp = t_1 + math.log((y * z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(a + -0.5) * log(t)) - t)
	tmp = 0.0
	if (z <= 3.25e+134)
		tmp = Float64(t_1 + log(Float64(y * z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((a + -0.5) * log(t)) - t;
	tmp = 0.0;
	if (z <= 3.25e+134)
		tmp = t_1 + log((y * z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[z, 3.25e+134], N[(t$95$1 + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]

Derivation

1. Split input into 2 regimes

2. if z < 3.25e134

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log t} \]
   4. Step-by-step derivation

      1. add-exp-log 26.0%

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

      2. associate-+r- 26.0%

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

      3. sum-log 23.8%

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

   5. Applied egg-rr 23.8%

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

   6. Taylor expanded in x around 0 65.5%

      \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t} \]
   7. Step-by-step derivation

      1. associate--l+ 65.5%

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

      2. sub-neg 65.5%

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

      3. metadata-eval 65.5%

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

   8. Simplified 65.5%

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

   if 3.25e134 < z

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 75.7%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 75.7%

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

   6. Simplified 75.7%

      \[\leadsto \color{blue}{\left(-t\right)} + \left(a + -0.5\right) \cdot \log t \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.25 \cdot 10^{+134}:\\ \;\;\;\;\left(\left(a + -0.5\right) \cdot \log t - t\right) + \log \left(y \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]

Alternative 10: 72.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-162} \lor \neg \left(a \leq 1.25 \cdot 10^{-9}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6e-162) (not (<= a 1.25e-9)))
   (- (* (+ a -0.5) (log t)) t)
   (- (log (* (* y z) (pow t -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6e-162) || !(a <= 1.25e-9)) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else {
		tmp = log(((y * z) * pow(t, -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 ((a <= (-6d-162)) .or. (.not. (a <= 1.25d-9))) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else
        tmp = log(((y * z) * (t ** (-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 ((a <= -6e-162) || !(a <= 1.25e-9)) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else {
		tmp = Math.log(((y * z) * Math.pow(t, -0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6e-162) or not (a <= 1.25e-9):
		tmp = ((a + -0.5) * math.log(t)) - t
	else:
		tmp = math.log(((y * z) * math.pow(t, -0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6e-162) || !(a <= 1.25e-9))
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	else
		tmp = Float64(log(Float64(Float64(y * z) * (t ^ -0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -6e-162) || ~((a <= 1.25e-9)))
		tmp = ((a + -0.5) * log(t)) - t;
	else
		tmp = log(((y * z) * (t ^ -0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6e-162], N[Not[LessEqual[a, 1.25e-9]], $MachinePrecision]], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(N[(y * z), $MachinePrecision] * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.99999999999999997e-162 or 1.25e-9 < a

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 88.4%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 88.4%

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

   6. Simplified 88.4%

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

   if -5.99999999999999997e-162 < a < 1.25e-9

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(\log \left(x + y\right) + \left(\log z - t\right)\right) + \left(a + -0.5\right) \cdot \log t} \]
   4. Step-by-step derivation

      1. add-exp-log 40.8%

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

      2. associate-+r- 40.8%

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

      3. sum-log 24.2%

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

   5. Applied egg-rr 24.2%

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

   6. Taylor expanded in a around 0 74.4%

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

   7. Taylor expanded in x around 0 47.8%

      \[\leadsto \left(\color{blue}{\log \left(y \cdot z\right)} + \left(-0.5 \cdot \log t + a \cdot \log t\right)\right) - t \]
   8. Step-by-step derivation

      1. *-commutative 47.8%

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

   9. Simplified 47.8%

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

   10. Taylor expanded in a around 0 47.8%

       \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) + -0.5 \cdot \log t\right)} - t \]
   11. Step-by-step derivation

       1. *-commutative 47.8%

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

       2. log-pow 47.8%

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

       3. log-prod 40.4%

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

       4. *-commutative 40.4%

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

   12. Simplified 40.4%

       \[\leadsto \color{blue}{\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6 \cdot 10^{-162} \lor \neg \left(a \leq 1.25 \cdot 10^{-9}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(y \cdot z\right) \cdot {t}^{-0.5}\right) - t\\ \end{array} \]

Alternative 11: 64.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -7.5 \cdot 10^{+82} \lor \neg \left(a - 0.5 \leq 2.4\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -7.5e+82) (not (<= (- a 0.5) 2.4)))
   (* (log t) (- a 0.5))
   (- (* -0.5 (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -7.5e+82) || !((a - 0.5) <= 2.4)) {
		tmp = log(t) * (a - 0.5);
	} else {
		tmp = (-0.5 * log(t)) - 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 (((a - 0.5d0) <= (-7.5d+82)) .or. (.not. ((a - 0.5d0) <= 2.4d0))) then
        tmp = log(t) * (a - 0.5d0)
    else
        tmp = ((-0.5d0) * log(t)) - 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 (((a - 0.5) <= -7.5e+82) || !((a - 0.5) <= 2.4)) {
		tmp = Math.log(t) * (a - 0.5);
	} else {
		tmp = (-0.5 * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -7.5e+82) or not ((a - 0.5) <= 2.4):
		tmp = math.log(t) * (a - 0.5)
	else:
		tmp = (-0.5 * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -7.5e+82) || !(Float64(a - 0.5) <= 2.4))
		tmp = Float64(log(t) * Float64(a - 0.5));
	else
		tmp = Float64(Float64(-0.5 * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -7.5e+82) || ~(((a - 0.5) <= 2.4)))
		tmp = log(t) * (a - 0.5);
	else
		tmp = (-0.5 * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -7.5e+82], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], 2.4]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -7.4999999999999999e82 or 2.39999999999999991 < (-.f64 a 1/2)

   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. Step-by-step derivation

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in t around inf 97.9%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 97.9%

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

   6. Simplified 97.9%

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

   7. Taylor expanded in t around 0 81.1%

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

   if -7.4999999999999999e82 < (-.f64 a 1/2) < 2.39999999999999991

   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. Step-by-step derivation

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around inf 57.5%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 57.5%

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

   6. Simplified 57.5%

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

   7. Taylor expanded in a around 0 54.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -7.5 \cdot 10^{+82} \lor \neg \left(a - 0.5 \leq 2.4\right):\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \log t - t\\ \end{array} \]

Alternative 12: 77.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 430:\\ \;\;\;\;t + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 430.0) (+ t (* (+ a -0.5) (log t))) (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 430.0) {
		tmp = t + ((a + -0.5) * log(t));
	} else {
		tmp = (a * log(t)) - 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 <= 430.0d0) then
        tmp = t + ((a + (-0.5d0)) * log(t))
    else
        tmp = (a * log(t)) - 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 <= 430.0) {
		tmp = t + ((a + -0.5) * Math.log(t));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 430.0:
		tmp = t + ((a + -0.5) * math.log(t))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 430.0)
		tmp = Float64(t + Float64(Float64(a + -0.5) * log(t)));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 430.0)
		tmp = t + ((a + -0.5) * log(t));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 430.0], N[(t + N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 430

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t around inf 49.8%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in a around 0 49.8%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \log t + a \cdot \log t\right) - t} \]
   8. Step-by-step derivation

      1. distribute-rgt-out 49.8%

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

      2. +-commutative 49.8%

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

      3. metadata-eval 49.8%

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

      4. sub-neg 49.8%

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

      5. add-sqr-sqrt 27.9%

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

      6. sqrt-unprod 17.4%

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

      7. pow2 17.4%

         \[\leadsto \sqrt{\color{blue}{{\left(\log t \cdot \left(a - 0.5\right)\right)}^{2}}} - t \]

      8. sub-neg 17.4%

         \[\leadsto \sqrt{{\left(\log t \cdot \color{blue}{\left(a + \left(-0.5\right)\right)}\right)}^{2}} - t \]

      9. metadata-eval 17.4%

         \[\leadsto \sqrt{{\left(\log t \cdot \left(a + \color{blue}{-0.5}\right)\right)}^{2}} - t \]

   9. Applied egg-rr 17.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\log t \cdot \left(a + -0.5\right)\right)}^{2}}} - t \]
   10. Step-by-step derivation

       1. unpow2 17.4%

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

       2. rem-sqrt-square 28.2%

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

   11. Simplified 28.2%

       \[\leadsto \color{blue}{\left|\log t \cdot \left(a + -0.5\right)\right|} - t \]
   12. Step-by-step derivation

       1. sub-neg 28.2%

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

       2. distribute-rgt-in 28.2%

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

       3. +-commutative 28.2%

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

       4. add-sqr-sqrt 27.9%

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

       5. fabs-sqr 27.9%

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

       6. add-sqr-sqrt 49.8%

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

       7. distribute-rgt-out 49.8%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 50.0%

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

       10. sqr-neg 50.0%

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

       11. sqrt-unprod 50.0%

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

       12. add-sqr-sqrt 50.0%

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

   13. Applied egg-rr 50.0%

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

   if 430 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 99.9%

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

   5. Taylor expanded in a around inf 98.7%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 430:\\ \;\;\;\;t + \left(a + -0.5\right) \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 13: 64.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{+78}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.75e+78) (* (log t) (- a 0.5)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.75e+78) {
		tmp = log(t) * (a - 0.5);
	} else {
		tmp = -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 <= 1.75d+78) then
        tmp = log(t) * (a - 0.5d0)
    else
        tmp = -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 <= 1.75e+78) {
		tmp = Math.log(t) * (a - 0.5);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.75e+78:
		tmp = math.log(t) * (a - 0.5)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.75e+78)
		tmp = Float64(log(t) * Float64(a - 0.5));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.75e+78)
		tmp = log(t) * (a - 0.5);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.75e+78], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.7500000000000001e78

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.5%

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

      2. sub-neg 99.5%

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

      3. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t around inf 57.7%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 57.7%

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

   6. Simplified 57.7%

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

   7. Taylor expanded in t around 0 51.5%

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

   if 1.7500000000000001e78 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 99.9%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in t around inf 79.7%

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

      1. mul-1-neg 79.7%

         \[\leadsto \color{blue}{-t} \]

   9. Simplified 79.7%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.75 \cdot 10^{+78}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14: 77.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 82:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 82.0) (* (log t) (- a 0.5)) (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 82.0) {
		tmp = log(t) * (a - 0.5);
	} else {
		tmp = (a * log(t)) - 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 <= 82.0d0) then
        tmp = log(t) * (a - 0.5d0)
    else
        tmp = (a * log(t)) - 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 <= 82.0) {
		tmp = Math.log(t) * (a - 0.5);
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 82.0:
		tmp = math.log(t) * (a - 0.5)
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 82.0)
		tmp = Float64(log(t) * Float64(a - 0.5));
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 82.0)
		tmp = log(t) * (a - 0.5);
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 82.0], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 82

   1. Initial program 99.4%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. Step-by-step derivation

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t around inf 49.8%

      \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
   5. Step-by-step derivation

      1. neg-mul-1 49.8%

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

   6. Simplified 49.8%

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

   7. Taylor expanded in t around 0 49.9%

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

   if 82 < 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. Step-by-step derivation

      1. associate--l+ 99.9%

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

      2. sub-neg 99.9%

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

      3. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around 0 99.9%

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

   5. Taylor expanded in a around inf 98.7%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 82:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 15: 77.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $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. Step-by-step derivation

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 74.0%

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
5. Step-by-step derivation

   1. neg-mul-1 74.0%

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

6. Simplified 74.0%

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

7. Final simplification 74.0%

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

Alternative 16: 38.2% accurate, 156.5× speedup

Math

\[\begin{array}{l} \\ -t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return -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 = -t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := (-t)

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. Step-by-step derivation

   1. associate--l+ 99.6%

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

   2. sub-neg 99.6%

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

   3. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 74.0%

   \[\leadsto \color{blue}{-1 \cdot t} + \left(a + -0.5\right) \cdot \log t \]
5. Step-by-step derivation

   1. neg-mul-1 74.0%

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

6. Simplified 74.0%

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

7. Taylor expanded in t around inf 36.6%

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

   1. mul-1-neg 36.6%

      \[\leadsto \color{blue}{-t} \]

9. Simplified 36.6%

   \[\leadsto \color{blue}{-t} \]

10. Final simplification 36.6%

    \[\leadsto -t \]

Developer target: 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 2023311 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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