Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 17.1s

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. Add Preprocessing

5. Final simplification 99.6%

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

Alternative 2: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 200:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 200.0)
   (+ (log y) (+ (log z) (* (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 <= 200.0) {
		tmp = log(y) + (log(z) + (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 <= 200.0d0) then
        tmp = log(y) + (log(z) + (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 <= 200.0) {
		tmp = Math.log(y) + (Math.log(z) + (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 <= 200.0:
		tmp = math.log(y) + (math.log(z) + (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 <= 200.0)
		tmp = Float64(log(y) + Float64(log(z) + 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 <= 200.0)
		tmp = log(y) + (log(z) + (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, 200.0], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 200

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 61.6%

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

   6. Taylor expanded in t around 0 60.9%

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

   if 200 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 75.5%

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

   6. Taylor expanded in a around inf 98.5%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   7. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   8. Simplified 98.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 200:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 210:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 210.0)
   (+ (* (+ a -0.5) (log t)) (+ (log z) (log y)))
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 210.0) {
		tmp = ((a + -0.5) * log(t)) + (log(z) + log(y));
	} 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 <= 210.0d0) then
        tmp = ((a + (-0.5d0)) * log(t)) + (log(z) + log(y))
    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 <= 210.0) {
		tmp = ((a + -0.5) * Math.log(t)) + (Math.log(z) + Math.log(y));
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 210.0:
		tmp = ((a + -0.5) * math.log(t)) + (math.log(z) + math.log(y))
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 210.0)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) + Float64(log(z) + log(y)));
	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 <= 210.0)
		tmp = ((a + -0.5) * log(t)) + (log(z) + log(y));
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 210.0], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 210

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 61.6%

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

      1. remove-double-neg 61.6%

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

      2. log-rec 61.6%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 61.6%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 61.6%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 61.6%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 61.6%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 61.6%

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

      8. remove-double-neg 61.6%

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

   7. Simplified 61.6%

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

   8. Taylor expanded in t around 0 60.9%

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

   if 210 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 75.5%

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

   6. Taylor expanded in a around inf 98.5%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   7. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   8. Simplified 98.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 210:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t + \left(\log z + \log y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

5. Taylor expanded in x around 0 68.5%

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

   1. remove-double-neg 68.5%

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

   2. log-rec 68.5%

      \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

   3. mul-1-neg 68.5%

      \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

   4. +-commutative 68.5%

      \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

   5. associate--l+ 68.5%

      \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

   6. mul-1-neg 68.5%

      \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

   7. log-rec 68.5%

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

   8. remove-double-neg 68.5%

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

7. Simplified 68.5%

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

8. Final simplification 68.5%

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

Alternative 5: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (log(y) + (log(z) + (log(t) * (a - 0.5)))) - 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[(a - 0.5), $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. Add Preprocessing

5. Taylor expanded in x around 0 68.5%

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

6. Final simplification 68.5%

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

Alternative 6: 87.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 2100000000:\\ \;\;\;\;\left(\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.4e-15)
   (- (* (+ a -0.5) (log t)) t)
   (if (<= a 2100000000.0)
     (- (+ (log (* z (+ x y))) (* (log t) (- a 0.5))) t)
     (- (* a (log t)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.4e-15) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else if (a <= 2100000000.0) {
		tmp = (log((z * (x + y))) + (log(t) * (a - 0.5))) - 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 (a <= (-1.4d-15)) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else if (a <= 2100000000.0d0) then
        tmp = (log((z * (x + y))) + (log(t) * (a - 0.5d0))) - 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 (a <= -1.4e-15) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else if (a <= 2100000000.0) {
		tmp = (Math.log((z * (x + y))) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.4e-15:
		tmp = ((a + -0.5) * math.log(t)) - t
	elif a <= 2100000000.0:
		tmp = (math.log((z * (x + y))) + (math.log(t) * (a - 0.5))) - t
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.4e-15)
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	elseif (a <= 2100000000.0)
		tmp = Float64(Float64(log(Float64(z * Float64(x + y))) + Float64(log(t) * Float64(a - 0.5))) - 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 (a <= -1.4e-15)
		tmp = ((a + -0.5) * log(t)) - t;
	elseif (a <= 2100000000.0)
		tmp = (log((z * (x + y))) + (log(t) * (a - 0.5))) - t;
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.4e-15], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, 2100000000.0], N[(N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.40000000000000007e-15

   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. Add Preprocessing

   5. Taylor expanded in x around 0 80.4%

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

      1. remove-double-neg 80.4%

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

      2. log-rec 80.4%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 80.4%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 80.4%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 80.4%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 80.4%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 80.4%

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

      8. remove-double-neg 80.4%

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

   7. Simplified 80.4%

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

   8. Taylor expanded in t around inf 98.0%

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

      1. neg-mul-1 98.0%

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

   10. Simplified 98.0%

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

   if -1.40000000000000007e-15 < a < 2.1e9

   1. Initial program 99.5%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-+r- 99.6%

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

      2. fma-udef 99.6%

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

      3. associate--r+ 99.5%

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

      4. sum-log 81.2%

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

   6. Applied egg-rr 81.2%

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

   if 2.1e9 < a

   1. Initial program 99.7%

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

   5. Taylor expanded in x around 0 72.2%

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

   6. Taylor expanded in a around inf 99.7%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   7. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   8. Simplified 99.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{-15}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{elif}\;a \leq 2100000000:\\ \;\;\;\;\left(\log \left(z \cdot \left(x + y\right)\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-15} \lor \neg \left(a \leq 1.85 \cdot 10^{-50}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.45e-15) (not (<= a 1.85e-50)))
   (- (* (+ a -0.5) (log t)) t)
   (- (log (* z (+ x y))) (+ t (* (log t) 0.5)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.45e-15) || !(a <= 1.85e-50)) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else {
		tmp = log((z * (x + y))) - (t + (log(t) * 0.5));
	}
	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 <= (-1.45d-15)) .or. (.not. (a <= 1.85d-50))) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else
        tmp = log((z * (x + y))) - (t + (log(t) * 0.5d0))
    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 <= -1.45e-15) || !(a <= 1.85e-50)) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else {
		tmp = Math.log((z * (x + y))) - (t + (Math.log(t) * 0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.45e-15) or not (a <= 1.85e-50):
		tmp = ((a + -0.5) * math.log(t)) - t
	else:
		tmp = math.log((z * (x + y))) - (t + (math.log(t) * 0.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.45e-15) || !(a <= 1.85e-50))
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	else
		tmp = Float64(log(Float64(z * Float64(x + y))) - Float64(t + Float64(log(t) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.45e-15) || ~((a <= 1.85e-50)))
		tmp = ((a + -0.5) * log(t)) - t;
	else
		tmp = log((z * (x + y))) - (t + (log(t) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.45e-15], N[Not[LessEqual[a, 1.85e-50]], $MachinePrecision]], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[(t + N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.45000000000000009e-15 or 1.85e-50 < a

   1. Initial program 99.7%

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

   5. Taylor expanded in x around 0 74.3%

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

      1. remove-double-neg 74.3%

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

      2. log-rec 74.3%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 74.3%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 74.3%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 74.3%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 74.3%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 74.3%

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

      8. remove-double-neg 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in t around inf 97.4%

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

      1. neg-mul-1 97.4%

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

   10. Simplified 97.4%

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

   if -1.45000000000000009e-15 < a < 1.85e-50

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 99.5%

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

      1. log-prod 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

   7. Simplified 82.2%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{-15} \lor \neg \left(a \leq 1.85 \cdot 10^{-50}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(x + y\right)\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{-15} \lor \neg \left(a \leq 1.65 \cdot 10^{-52}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.25e-15) (not (<= a 1.65e-52)))
   (- (* (+ a -0.5) (log t)) t)
   (- (log (* y z)) (+ t (* (log t) 0.5)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.25e-15) || !(a <= 1.65e-52)) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else {
		tmp = log((y * z)) - (t + (log(t) * 0.5));
	}
	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 <= (-1.25d-15)) .or. (.not. (a <= 1.65d-52))) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else
        tmp = log((y * z)) - (t + (log(t) * 0.5d0))
    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 <= -1.25e-15) || !(a <= 1.65e-52)) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else {
		tmp = Math.log((y * z)) - (t + (Math.log(t) * 0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.25e-15) or not (a <= 1.65e-52):
		tmp = ((a + -0.5) * math.log(t)) - t
	else:
		tmp = math.log((y * z)) - (t + (math.log(t) * 0.5))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.25e-15) || !(a <= 1.65e-52))
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	else
		tmp = Float64(log(Float64(y * z)) - Float64(t + Float64(log(t) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.25e-15) || ~((a <= 1.65e-52)))
		tmp = ((a + -0.5) * log(t)) - t;
	else
		tmp = log((y * z)) - (t + (log(t) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.25e-15], N[Not[LessEqual[a, 1.65e-52]], $MachinePrecision]], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - N[(t + N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.25e-15 or 1.64999999999999998e-52 < a

   1. Initial program 99.7%

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

   5. Taylor expanded in x around 0 74.3%

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

      1. remove-double-neg 74.3%

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

      2. log-rec 74.3%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 74.3%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 74.3%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 74.3%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 74.3%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 74.3%

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

      8. remove-double-neg 74.3%

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

   7. Simplified 74.3%

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

   8. Taylor expanded in t around inf 97.4%

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

      1. neg-mul-1 97.4%

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

   10. Simplified 97.4%

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

   if -1.25e-15 < a < 1.64999999999999998e-52

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 99.5%

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

      1. log-prod 82.2%

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

      2. +-commutative 82.2%

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

      3. *-commutative 82.2%

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

   7. Simplified 82.2%

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

   8. Taylor expanded in y around inf 60.4%

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

      1. mul-1-neg 60.4%

         \[\leadsto \left(\log z + \color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)}\right) - \left(t + \log t \cdot 0.5\right) \]

      2. log-rec 60.4%

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

      3. remove-double-neg 60.4%

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

      4. log-prod 51.4%

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

   10. Simplified 51.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{-15} \lor \neg \left(a \leq 1.65 \cdot 10^{-52}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-122} \lor \neg \left(a \leq -1.5 \cdot 10^{-207}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \log t \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.6e-122) (not (<= a -1.5e-207)))
   (- (* (+ a -0.5) (log t)) t)
   (- (log (* y z)) (* (log t) 0.5))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.6e-122) || !(a <= -1.5e-207)) {
		tmp = ((a + -0.5) * log(t)) - t;
	} else {
		tmp = log((y * z)) - (log(t) * 0.5);
	}
	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 <= (-6.6d-122)) .or. (.not. (a <= (-1.5d-207)))) then
        tmp = ((a + (-0.5d0)) * log(t)) - t
    else
        tmp = log((y * z)) - (log(t) * 0.5d0)
    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 <= -6.6e-122) || !(a <= -1.5e-207)) {
		tmp = ((a + -0.5) * Math.log(t)) - t;
	} else {
		tmp = Math.log((y * z)) - (Math.log(t) * 0.5);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6.6e-122) or not (a <= -1.5e-207):
		tmp = ((a + -0.5) * math.log(t)) - t
	else:
		tmp = math.log((y * z)) - (math.log(t) * 0.5)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.6e-122) || !(a <= -1.5e-207))
		tmp = Float64(Float64(Float64(a + -0.5) * log(t)) - t);
	else
		tmp = Float64(log(Float64(y * z)) - Float64(log(t) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -6.6e-122) || ~((a <= -1.5e-207)))
		tmp = ((a + -0.5) * log(t)) - t;
	else
		tmp = log((y * z)) - (log(t) * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.6e-122], N[Not[LessEqual[a, -1.5e-207]], $MachinePrecision]], N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] - N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.59999999999999999e-122 or -1.5e-207 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 70.3%

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

      1. remove-double-neg 70.3%

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

      2. log-rec 70.3%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 70.3%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 70.3%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 70.3%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 70.3%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 70.3%

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

      8. remove-double-neg 70.3%

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

   7. Simplified 70.3%

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

   8. Taylor expanded in t around inf 84.0%

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

      1. neg-mul-1 84.0%

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

   10. Simplified 84.0%

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

   if -6.59999999999999999e-122 < a < -1.5e-207

   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%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-udef 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in a around 0 99.5%

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

      1. log-prod 94.8%

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

      2. +-commutative 94.8%

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

      3. *-commutative 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in y around inf 48.0%

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

      1. mul-1-neg 48.0%

         \[\leadsto \left(\log z + \color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)}\right) - \left(t + \log t \cdot 0.5\right) \]

      2. log-rec 48.0%

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

      3. remove-double-neg 48.0%

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

      4. log-prod 42.6%

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

   10. Simplified 42.6%

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

   11. Taylor expanded in t around 0 29.2%

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

       1. *-commutative 29.2%

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

   13. Simplified 29.2%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-122} \lor \neg \left(a \leq -1.5 \cdot 10^{-207}\right):\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) - \log t \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.55e-17)
   (+ (log (* y z)) (* (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 <= 1.55e-17) {
		tmp = log((y * z)) + (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 <= 1.55d-17) then
        tmp = log((y * z)) + (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 <= 1.55e-17) {
		tmp = Math.log((y * z)) + (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 <= 1.55e-17:
		tmp = math.log((y * z)) + (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 <= 1.55e-17)
		tmp = Float64(log(Float64(y * z)) + 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 <= 1.55e-17)
		tmp = log((y * z)) + (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, 1.55e-17], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $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 < 1.5499999999999999e-17

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-udef 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 99.3%

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

      1. log-prod 78.5%

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

      2. +-commutative 78.5%

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

   7. Simplified 78.5%

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

   8. Taylor expanded in y around inf 61.3%

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

      1. mul-1-neg 23.2%

         \[\leadsto \left(\log z + \color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)}\right) - \left(t + \log t \cdot 0.5\right) \]

      2. log-rec 23.2%

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

      3. remove-double-neg 23.2%

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

      4. log-prod 20.7%

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

   10. Simplified 46.8%

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

   if 1.5499999999999999e-17 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 74.9%

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

      1. remove-double-neg 74.9%

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

      2. log-rec 74.9%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 74.9%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 74.9%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 74.9%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 74.9%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 74.9%

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

      8. remove-double-neg 74.9%

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

   7. Simplified 74.9%

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

   8. Taylor expanded in t around inf 96.5%

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

      1. neg-mul-1 96.5%

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

   10. Simplified 96.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a + -0.5\right) \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{+69}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 4.2e+69) (* (log t) (- a 0.5)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 4.2e+69) {
		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 <= 4.2d+69) 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 <= 4.2e+69) {
		tmp = Math.log(t) * (a - 0.5);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 4.2e+69:
		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 <= 4.2e+69)
		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 <= 4.2e+69)
		tmp = log(t) * (a - 0.5);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 4.2e+69], N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 4.2000000000000003e69

   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. Add Preprocessing

   5. Taylor expanded in x around 0 65.0%

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

      1. remove-double-neg 65.0%

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

      2. log-rec 65.0%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 65.0%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 65.0%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 65.0%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 65.0%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 65.0%

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

      8. remove-double-neg 65.0%

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

   7. Simplified 65.0%

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

   8. Taylor expanded in t around inf 66.7%

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

      1. neg-mul-1 66.7%

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

   10. Simplified 66.7%

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

   11. Taylor expanded in t around 0 60.9%

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

   if 4.2000000000000003e69 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 74.2%

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

   6. Taylor expanded in t around inf 82.4%

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

      1. neg-mul-1 82.4%

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

   8. Simplified 82.4%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.2 \cdot 10^{+69}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.000155:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.000155) (* (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 <= 0.000155) {
		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 <= 0.000155d0) 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 <= 0.000155) {
		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 <= 0.000155:
		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 <= 0.000155)
		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 <= 0.000155)
		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, 0.000155], 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 < 1.55e-4

   1. Initial program 99.3%

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

      1. associate--l+ 99.3%

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

      2. sub-neg 99.3%

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

      3. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 62.0%

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

      1. remove-double-neg 62.0%

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

      2. log-rec 62.0%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 62.0%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 62.0%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 62.0%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 62.0%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 62.0%

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

      8. remove-double-neg 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in t around inf 60.8%

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

      1. neg-mul-1 60.8%

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

   10. Simplified 60.8%

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

   11. Taylor expanded in t around 0 60.8%

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

   if 1.55e-4 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 74.7%

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

   6. Taylor expanded in a around inf 97.1%

      \[\leadsto \color{blue}{a \cdot \log t} - t \]
   7. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \color{blue}{\log t \cdot a} - t \]

   8. Simplified 97.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.000155:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 800000:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 800000.0) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 800000.0) {
		tmp = log((x + y));
	} 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 <= 800000.0d0) then
        tmp = log((x + y))
    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 <= 800000.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 800000.0:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 800000.0)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 800000.0)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 800000.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 8e5

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-udef 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\log z - \mathsf{fma}\left(\log t, 0.5 - a, t\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 8.3%

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

      1. neg-mul-1 8.3%

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

   7. Simplified 8.3%

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

   8. Taylor expanded in t around 0 8.4%

      \[\leadsto \color{blue}{\log \left(x + y\right)} \]
   9. Step-by-step derivation

      1. +-commutative 8.4%

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

   10. Simplified 8.4%

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

   if 8e5 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 75.7%

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

   6. Taylor expanded in t around inf 72.9%

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

      1. neg-mul-1 72.9%

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

   8. Simplified 72.9%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 800000:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{+69}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.4e+69) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.4e+69) {
		tmp = a * log(t);
	} 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 <= 2.4d+69) then
        tmp = a * log(t)
    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 <= 2.4e+69) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.4e+69:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.4e+69)
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.4e+69)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.4e+69], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.4000000000000002e69

   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. Add Preprocessing

   5. Taylor expanded in x around 0 65.0%

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

      1. remove-double-neg 65.0%

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

      2. log-rec 65.0%

         \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      3. mul-1-neg 65.0%

         \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

      4. +-commutative 65.0%

         \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

      5. associate--l+ 65.0%

         \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

      6. mul-1-neg 65.0%

         \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

      7. log-rec 65.0%

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

      8. remove-double-neg 65.0%

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

   7. Simplified 65.0%

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

   8. Taylor expanded in a around inf 56.9%

      \[\leadsto \color{blue}{a \cdot \log t} \]
   9. Step-by-step derivation

      1. *-commutative 56.9%

         \[\leadsto \color{blue}{\log t \cdot a} \]

   10. Simplified 56.9%

       \[\leadsto \color{blue}{\log t \cdot a} \]

   if 2.4000000000000002e69 < 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. Add Preprocessing

   5. Taylor expanded in x around 0 74.2%

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

   6. Taylor expanded in t around inf 82.4%

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

      1. neg-mul-1 82.4%

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

   8. Simplified 82.4%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.4 \cdot 10^{+69}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.3% 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. Add Preprocessing

5. Taylor expanded in x around 0 68.5%

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

   1. remove-double-neg 68.5%

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

   2. log-rec 68.5%

      \[\leadsto \left(\left(\left(-\color{blue}{\log \left(\frac{1}{y}\right)}\right) + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

   3. mul-1-neg 68.5%

      \[\leadsto \left(\left(\color{blue}{-1 \cdot \log \left(\frac{1}{y}\right)} + \log z\right) - t\right) + \left(a + -0.5\right) \cdot \log t \]

   4. +-commutative 68.5%

      \[\leadsto \left(\color{blue}{\left(\log z + -1 \cdot \log \left(\frac{1}{y}\right)\right)} - t\right) + \left(a + -0.5\right) \cdot \log t \]

   5. associate--l+ 68.5%

      \[\leadsto \color{blue}{\left(\log z + \left(-1 \cdot \log \left(\frac{1}{y}\right) - t\right)\right)} + \left(a + -0.5\right) \cdot \log t \]

   6. mul-1-neg 68.5%

      \[\leadsto \left(\log z + \left(\color{blue}{\left(-\log \left(\frac{1}{y}\right)\right)} - t\right)\right) + \left(a + -0.5\right) \cdot \log t \]

   7. log-rec 68.5%

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

   8. remove-double-neg 68.5%

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

7. Simplified 68.5%

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

8. Taylor expanded in t around inf 79.3%

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

   1. neg-mul-1 79.3%

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

10. Simplified 79.3%

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

11. Final simplification 79.3%

    \[\leadsto \left(a + -0.5\right) \cdot \log t - t \]
12. Add Preprocessing

Alternative 16: 37.7% 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. Add Preprocessing

5. Taylor expanded in x around 0 68.5%

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

6. Taylor expanded in t around inf 36.8%

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

   1. neg-mul-1 36.8%

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

8. Simplified 36.8%

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

9. Final simplification 36.8%

   \[\leadsto -t \]
10. Add Preprocessing

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 2024031 
(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))))