Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 17.2s

Alternatives: 12

Speedup: 1.0×

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

\[\begin{array}{l} \\ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log \left(\frac{1}{t}\right) \cdot \left(0.5 - a\right) \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (log (/ 1.0 t)) (- 0.5 a))))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + (log((1.0 / t)) * (0.5 - a));
}

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) + (log((1.0d0 / t)) * (0.5d0 - a))
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) + (Math.log((1.0 / t)) * (0.5 - a));
}

Python

def code(x, y, z, t, a):
	return ((math.log((x + y)) + math.log(z)) - t) + (math.log((1.0 / t)) * (0.5 - a))

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(log(Float64(1.0 / t)) * Float64(0.5 - a)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = ((log((x + y)) + log(z)) - t) + (log((1.0 / t)) * (0.5 - a));
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[Log[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] * N[(0.5 - a), $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. Taylor expanded in t around inf 99.6%

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

3. Final simplification 99.6%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 22.5:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 22.5)
   (+ (log z) (+ (log (+ x y)) (* (- a 0.5) (log t))))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 22.5) {
		tmp = log(z) + (log((x + y)) + ((a - 0.5) * log(t)));
	} else {
		tmp = (log(z) - t) + (a * log(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 <= 22.5d0) then
        tmp = log(z) + (log((x + y)) + ((a - 0.5d0) * log(t)))
    else
        tmp = (log(z) - t) + (a * log(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 <= 22.5) {
		tmp = Math.log(z) + (Math.log((x + y)) + ((a - 0.5) * Math.log(t)));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 22.5:
		tmp = math.log(z) + (math.log((x + y)) + ((a - 0.5) * math.log(t)))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 22.5)
		tmp = Float64(log(z) + Float64(log(Float64(x + y)) + Float64(Float64(a - 0.5) * log(t))));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 22.5)
		tmp = log(z) + (log((x + y)) + ((a - 0.5) * log(t)));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 22.5], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 22.5

   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. +-commutative 99.3%

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

      3. associate-+l+ 99.3%

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

      4. +-commutative 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in t around 0 98.1%

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

   if 22.5 < 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. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.0%

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

      1. *-commutative 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 22.5:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 80.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 22.5:\\ \;\;\;\;\log y + \left(\log z + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 22.5)
   (+ (log y) (+ (log z) (* (- a 0.5) (log t))))
   (+ (- (log z) t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 22.5) {
		tmp = log(y) + (log(z) + ((a - 0.5) * log(t)));
	} else {
		tmp = (log(z) - t) + (a * log(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 <= 22.5d0) then
        tmp = log(y) + (log(z) + ((a - 0.5d0) * log(t)))
    else
        tmp = (log(z) - t) + (a * log(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 <= 22.5) {
		tmp = Math.log(y) + (Math.log(z) + ((a - 0.5) * Math.log(t)));
	} else {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 22.5:
		tmp = math.log(y) + (math.log(z) + ((a - 0.5) * math.log(t)))
	else:
		tmp = (math.log(z) - t) + (a * math.log(t))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 22.5)
		tmp = Float64(log(y) + Float64(log(z) + Float64(Float64(a - 0.5) * log(t))));
	else
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 22.5)
		tmp = log(y) + (log(z) + ((a - 0.5) * log(t)));
	else
		tmp = (log(z) - t) + (a * log(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 22.5], N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 22.5

   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. Taylor expanded in x around 0 64.3%

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

   3. Taylor expanded in t around 0 63.9%

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

   if 22.5 < 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. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in a around inf 99.0%

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

      1. *-commutative 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 22.5:\\ \;\;\;\;\log y + \left(\log z + \left(a - 0.5\right) \cdot \log t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \end{array} \]

Alternative 5: 69.0% accurate, 1.0× speedup

Math

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

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(Float64(log(z) + 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[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $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. Taylor expanded in x around 0 67.2%

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

3. Final simplification 67.2%

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

Alternative 6: 72.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-15} \lor \neg \left(a \leq -1.65 \cdot 10^{-128} \lor \neg \left(a \leq 2.35 \cdot 10^{-112}\right) \land a \leq 1.8 \cdot 10^{-78}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.1e-15)
         (not
          (or (<= a -1.65e-128) (and (not (<= a 2.35e-112)) (<= a 1.8e-78)))))
   (+ (- (log z) t) (* a (log 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 <= -1.1e-15) || !((a <= -1.65e-128) || (!(a <= 2.35e-112) && (a <= 1.8e-78)))) {
		tmp = (log(z) - t) + (a * log(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 <= (-1.1d-15)) .or. (.not. (a <= (-1.65d-128)) .or. (.not. (a <= 2.35d-112)) .and. (a <= 1.8d-78))) then
        tmp = (log(z) - t) + (a * log(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 <= -1.1e-15) || !((a <= -1.65e-128) || (!(a <= 2.35e-112) && (a <= 1.8e-78)))) {
		tmp = (Math.log(z) - t) + (a * Math.log(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 <= -1.1e-15) or not ((a <= -1.65e-128) or (not (a <= 2.35e-112) and (a <= 1.8e-78))):
		tmp = (math.log(z) - t) + (a * math.log(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 <= -1.1e-15) || !((a <= -1.65e-128) || (!(a <= 2.35e-112) && (a <= 1.8e-78))))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(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 <= -1.1e-15) || ~(((a <= -1.65e-128) || (~((a <= 2.35e-112)) && (a <= 1.8e-78)))))
		tmp = (log(z) - t) + (a * log(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, -1.1e-15], N[Not[Or[LessEqual[a, -1.65e-128], And[N[Not[LessEqual[a, 2.35e-112]], $MachinePrecision], LessEqual[a, 1.8e-78]]]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $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 < -1.09999999999999993e-15 or -1.65e-128 < a < 2.3500000000000002e-112 or 1.8000000000000001e-78 < 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. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 81.4%

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

      1. *-commutative 81.4%

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

   6. Simplified 81.4%

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

   if -1.09999999999999993e-15 < a < -1.65e-128 or 2.3500000000000002e-112 < a < 1.8000000000000001e-78

   1. Initial program 99.1%

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

   2. Taylor expanded in x around 0 62.4%

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

   3. Taylor expanded in a around 0 62.4%

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

   4. Taylor expanded in t around 0 46.9%

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

      1. log-pow 46.9%

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

      2. associate-+r+ 46.9%

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

      3. log-prod 42.2%

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

      4. log-pow 42.2%

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

      5. *-commutative 42.2%

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

   6. Simplified 42.2%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.1 \cdot 10^{-15} \lor \neg \left(a \leq -1.65 \cdot 10^{-128} \lor \neg \left(a \leq 2.35 \cdot 10^{-112}\right) \land a \leq 1.8 \cdot 10^{-78}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \end{array} \]

Alternative 7: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-11} \lor \neg \left(a \leq 1.7 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -9.5e-11) (not (<= a 1.7e-11)))
   (+ (- (log z) t) (* a (log t)))
   (- (log (* (+ x y) (* z (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -9.5e-11) || !(a <= 1.7e-11)) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = log(((x + y) * (z * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-9.5d-11)) .or. (.not. (a <= 1.7d-11))) then
        tmp = (log(z) - t) + (a * log(t))
    else
        tmp = log(((x + y) * (z * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -9.5e-11) || !(a <= 1.7e-11)) {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	} else {
		tmp = Math.log(((x + y) * (z * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -9.5e-11) or not (a <= 1.7e-11):
		tmp = (math.log(z) - t) + (a * math.log(t))
	else:
		tmp = math.log(((x + y) * (z * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -9.5e-11) || !(a <= 1.7e-11))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(log(Float64(Float64(x + y) * Float64(z * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -9.5e-11) || ~((a <= 1.7e-11)))
		tmp = (log(z) - t) + (a * log(t));
	else
		tmp = log(((x + y) * (z * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -9.5e-11], N[Not[LessEqual[a, 1.7e-11]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(N[(x + y), $MachinePrecision] * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -9.49999999999999951e-11 or 1.6999999999999999e-11 < 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. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 97.1%

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

      1. *-commutative 97.1%

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

   6. Simplified 97.1%

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

   if -9.49999999999999951e-11 < a < 1.6999999999999999e-11

   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. Taylor expanded in t around inf 99.4%

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

      1. add-exp-log 59.1%

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

      2. log-rec 59.1%

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

   4. Applied egg-rr 59.1%

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

   5. Taylor expanded in a around 0 99.4%

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

      1. log-pow 99.4%

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

      2. log-prod 87.8%

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

      3. log-prod 76.5%

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

      4. *-commutative 76.5%

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

      5. associate-*l* 79.6%

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

      6. +-commutative 79.6%

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

   7. Simplified 79.6%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-11} \lor \neg \left(a \leq 1.7 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]

Alternative 8: 58.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{if}\;t \leq 6.8 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-57}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+51}:\\ \;\;\;\;\log \left(\frac{1}{t}\right) \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (* y z)) (* (log t) -0.5))))
   (if (<= t 6.8e-280)
     t_1
     (if (<= t 1.5e-57)
       (* a (log t))
       (if (<= t 1.3e-15)
         t_1
         (if (<= t 1.6e+51) (* (log (/ 1.0 t)) (- a)) (- t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((y * z)) + (log(t) * -0.5);
	double tmp;
	if (t <= 6.8e-280) {
		tmp = t_1;
	} else if (t <= 1.5e-57) {
		tmp = a * log(t);
	} else if (t <= 1.3e-15) {
		tmp = t_1;
	} else if (t <= 1.6e+51) {
		tmp = log((1.0 / t)) * -a;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = log((y * z)) + (log(t) * (-0.5d0))
    if (t <= 6.8d-280) then
        tmp = t_1
    else if (t <= 1.5d-57) then
        tmp = a * log(t)
    else if (t <= 1.3d-15) then
        tmp = t_1
    else if (t <= 1.6d+51) then
        tmp = log((1.0d0 / t)) * -a
    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 t_1 = Math.log((y * z)) + (Math.log(t) * -0.5);
	double tmp;
	if (t <= 6.8e-280) {
		tmp = t_1;
	} else if (t <= 1.5e-57) {
		tmp = a * Math.log(t);
	} else if (t <= 1.3e-15) {
		tmp = t_1;
	} else if (t <= 1.6e+51) {
		tmp = Math.log((1.0 / t)) * -a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((y * z)) + (math.log(t) * -0.5)
	tmp = 0
	if t <= 6.8e-280:
		tmp = t_1
	elif t <= 1.5e-57:
		tmp = a * math.log(t)
	elif t <= 1.3e-15:
		tmp = t_1
	elif t <= 1.6e+51:
		tmp = math.log((1.0 / t)) * -a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(y * z)) + Float64(log(t) * -0.5))
	tmp = 0.0
	if (t <= 6.8e-280)
		tmp = t_1;
	elseif (t <= 1.5e-57)
		tmp = Float64(a * log(t));
	elseif (t <= 1.3e-15)
		tmp = t_1;
	elseif (t <= 1.6e+51)
		tmp = Float64(log(Float64(1.0 / t)) * Float64(-a));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((y * z)) + (log(t) * -0.5);
	tmp = 0.0;
	if (t <= 6.8e-280)
		tmp = t_1;
	elseif (t <= 1.5e-57)
		tmp = a * log(t);
	elseif (t <= 1.3e-15)
		tmp = t_1;
	elseif (t <= 1.6e+51)
		tmp = log((1.0 / t)) * -a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 6.8e-280], t$95$1, If[LessEqual[t, 1.5e-57], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e-15], t$95$1, If[LessEqual[t, 1.6e+51], N[(N[Log[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] * (-a)), $MachinePrecision], (-t)]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 6.7999999999999995e-280 or 1.5e-57 < t < 1.30000000000000002e-15

   1. Initial program 98.9%

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

   2. Taylor expanded in x around 0 63.5%

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

   3. Taylor expanded in a around 0 41.7%

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

   4. Taylor expanded in t around 0 41.8%

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

      1. log-pow 41.8%

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

      2. associate-+r+ 41.7%

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

      3. log-prod 38.4%

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

      4. log-pow 38.4%

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

      5. *-commutative 38.4%

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

   6. Simplified 38.4%

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

   if 6.7999999999999995e-280 < t < 1.5e-57

   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. Taylor expanded in t around inf 99.4%

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

   3. Taylor expanded in a around inf 53.6%

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

      1. mul-1-neg 53.6%

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

      2. log-rec 53.6%

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

      3. distribute-rgt-neg-in 53.6%

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

      4. remove-double-neg 53.6%

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

   5. Simplified 53.6%

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

   if 1.30000000000000002e-15 < t < 1.6000000000000001e51

   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. Taylor expanded in t around inf 99.7%

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

   3. Taylor expanded in a around inf 64.2%

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

   if 1.6000000000000001e51 < 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. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 83.5%

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

      1. neg-mul-1 83.5%

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

   6. Simplified 83.5%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.8 \cdot 10^{-280}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-57}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-15}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot -0.5\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+51}:\\ \;\;\;\;\log \left(\frac{1}{t}\right) \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 9: 72.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-9} \lor \neg \left(a \leq 1.7 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.5e-9) (not (<= a 1.7e-11)))
   (+ (- (log z) t) (* a (log t)))
   (- (log (* y (* z (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.5e-9) || !(a <= 1.7e-11)) {
		tmp = (log(z) - t) + (a * log(t));
	} else {
		tmp = log((y * (z * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.5d-9)) .or. (.not. (a <= 1.7d-11))) then
        tmp = (log(z) - t) + (a * log(t))
    else
        tmp = log((y * (z * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.5e-9) || !(a <= 1.7e-11)) {
		tmp = (Math.log(z) - t) + (a * Math.log(t));
	} else {
		tmp = Math.log((y * (z * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.5e-9) or not (a <= 1.7e-11):
		tmp = (math.log(z) - t) + (a * math.log(t))
	else:
		tmp = math.log((y * (z * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.5e-9) || !(a <= 1.7e-11))
		tmp = Float64(Float64(log(z) - t) + Float64(a * log(t)));
	else
		tmp = Float64(log(Float64(y * Float64(z * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.5e-9) || ~((a <= 1.7e-11)))
		tmp = (log(z) - t) + (a * log(t));
	else
		tmp = log((y * (z * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.5e-9], N[Not[LessEqual[a, 1.7e-11]], $MachinePrecision]], N[(N[(N[Log[z], $MachinePrecision] - t), $MachinePrecision] + N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[N[(y * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.49999999999999999e-9 or 1.6999999999999999e-11 < 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. +-commutative 99.7%

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

      3. associate-+l+ 99.7%

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

      4. +-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. sub-neg 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a around inf 97.1%

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

      1. *-commutative 97.1%

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

   6. Simplified 97.1%

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

   if -1.49999999999999999e-9 < a < 1.6999999999999999e-11

   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. Taylor expanded in x around 0 62.8%

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

   3. Taylor expanded in a around 0 62.8%

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

   4. Taylor expanded in y around 0 62.8%

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

      1. log-pow 62.8%

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

      2. associate-+r+ 62.8%

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

      3. log-prod 48.4%

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

      4. associate--l+ 48.4%

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

      5. unsub-neg 48.4%

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

      6. +-commutative 48.4%

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

      7. neg-sub0 48.4%

         \[\leadsto \log \left(y \cdot z\right) + \left(\color{blue}{\left(0 - t\right)} + \log \left({t}^{-0.5}\right)\right) \]

      8. associate-+l- 48.4%

         \[\leadsto \log \left(y \cdot z\right) + \color{blue}{\left(0 - \left(t - \log \left({t}^{-0.5}\right)\right)\right)} \]

      9. neg-sub0 48.4%

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

      10. sub-neg 48.4%

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

      11. associate--r- 48.4%

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

      12. rem-log-exp 28.4%

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

      13. log-div 29.0%

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

      14. log-prod 26.6%

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

      15. *-commutative 26.6%

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

      16. associate-*r/ 26.6%

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

      17. log-div 26.2%

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

   6. Simplified 49.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{-9} \lor \neg \left(a \leq 1.7 \cdot 10^{-11}\right):\\ \;\;\;\;\left(\log z - t\right) + a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]

Alternative 10: 61.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;\log \left(\frac{1}{t}\right) \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.35e+51) (* (log (/ 1.0 t)) (- a)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.35e+51) {
		tmp = log((1.0 / t)) * -a;
	} 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 <= 2.35d+51) then
        tmp = log((1.0d0 / t)) * -a
    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 <= 2.35e+51) {
		tmp = Math.log((1.0 / t)) * -a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.35e+51:
		tmp = math.log((1.0 / t)) * -a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.35e+51)
		tmp = Float64(log(Float64(1.0 / t)) * Float64(-a));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.35e+51)
		tmp = log((1.0 / t)) * -a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.35e+51], N[(N[Log[N[(1.0 / t), $MachinePrecision]], $MachinePrecision] * (-a)), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.3500000000000001e51

   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. Taylor expanded in t around inf 99.4%

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

   3. Taylor expanded in a around inf 49.6%

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

   if 2.3500000000000001e51 < 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. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 83.5%

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

      1. neg-mul-1 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.35 \cdot 10^{+51}:\\ \;\;\;\;\log \left(\frac{1}{t}\right) \cdot \left(-a\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 61.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.7e+51) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.7e+51) {
		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 <= 2.7d+51) 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 <= 2.7e+51) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.7e+51:
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.7e+51)
		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 <= 2.7e+51)
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.7e+51], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.69999999999999992e51

   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. Taylor expanded in t around inf 99.4%

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

   3. Taylor expanded in a around inf 49.6%

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

      1. mul-1-neg 49.6%

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

      2. log-rec 49.6%

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

      3. distribute-rgt-neg-in 49.6%

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

      4. remove-double-neg 49.6%

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

   5. Simplified 49.6%

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

   if 2.69999999999999992e51 < 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. +-commutative 99.9%

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

      3. associate-+l+ 99.9%

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

      4. +-commutative 99.9%

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

      5. fma-def 99.9%

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

      6. sub-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in t around inf 83.5%

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

      1. neg-mul-1 83.5%

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

   6. Simplified 83.5%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.7 \cdot 10^{+51}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 12: 37.3% 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. +-commutative 99.6%

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

   3. associate-+l+ 99.6%

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

   4. +-commutative 99.6%

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

   5. fma-def 99.6%

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

   6. sub-neg 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 33.5%

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

   1. neg-mul-1 33.5%

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

6. Simplified 33.5%

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

7. Final simplification 33.5%

   \[\leadsto -t \]

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023320 
(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))))