Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.3%

Time: 20.5s

Alternatives: 12

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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (- (+ (log y) (+ (log z) (* (log t) (- a 0.5)))) t))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

assert x < y;
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

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

Julia

x, y = sort([x, y])
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

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z, t, a)
	tmp = (log(y) + (log(z) + (log(t) * (a - 0.5)))) - t;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
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 99.5%

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

   1. associate--l+ 99.5%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 69.3%

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

5. Final simplification 69.3%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+18} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log z\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -1e+18) (not (<= (- a 0.5) -0.4)))
   (- (* (log t) a) t)
   (- (+ (log y) (log z)) (+ t (* (log t) 0.5)))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -1e+18) || !((a - 0.5) <= -0.4)) {
		tmp = (log(t) * a) - t;
	} else {
		tmp = (log(y) + log(z)) - (t + (log(t) * 0.5));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-1d+18)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (log(t) * a) - t
    else
        tmp = (log(y) + log(z)) - (t + (log(t) * 0.5d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -1e+18) || !((a - 0.5) <= -0.4)) {
		tmp = (Math.log(t) * a) - t;
	} else {
		tmp = (Math.log(y) + Math.log(z)) - (t + (Math.log(t) * 0.5));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -1e+18) or not ((a - 0.5) <= -0.4):
		tmp = (math.log(t) * a) - t
	else:
		tmp = (math.log(y) + math.log(z)) - (t + (math.log(t) * 0.5))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -1e+18) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(log(t) * a) - t);
	else
		tmp = Float64(Float64(log(y) + log(z)) - Float64(t + Float64(log(t) * 0.5)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -1e+18) || ~(((a - 0.5) <= -0.4)))
		tmp = (log(t) * a) - t;
	else
		tmp = (log(y) + log(z)) - (t + (log(t) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -1e+18], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - N[(t + N[(N[Log[t], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -1e18 or -0.40000000000000002 < (-.f64 a 1/2)

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 74.7%

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

   5. Taylor expanded in a around inf 98.8%

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

      1. *-commutative 98.8%

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

   7. Simplified 98.8%

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

   if -1e18 < (-.f64 a 1/2) < -0.40000000000000002

   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%

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

      2. +-commutative 99.4%

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

      3. associate--l+ 99.4%

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

      4. sub-neg 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. distribute-rgt-neg-in 99.4%

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

      8. fma-def 99.4%

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

      9. sub-neg 99.4%

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

      10. +-commutative 99.4%

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

      11. distribute-neg-in 99.4%

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

      12. metadata-eval 99.4%

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

      13. metadata-eval 99.4%

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

      14. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 64.2%

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

   5. Taylor expanded in a around 0 64.2%

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

      1. *-commutative 47.5%

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

   7. Simplified 64.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+18} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log z\right) - \left(t + \log t \cdot 0.5\right)\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+18} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -1e+18) (not (<= (- a 0.5) -0.4)))
   (- (* (log t) a) t)
   (- (+ (log y) (+ (log z) (* (log t) -0.5))) t)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -1e+18) || !((a - 0.5) <= -0.4)) {
		tmp = (log(t) * a) - t;
	} else {
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((a - 0.5d0) <= (-1d+18)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (log(t) * a) - t
    else
        tmp = (log(y) + (log(z) + (log(t) * (-0.5d0)))) - t
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -1e+18) || !((a - 0.5) <= -0.4)) {
		tmp = (Math.log(t) * a) - t;
	} else {
		tmp = (Math.log(y) + (Math.log(z) + (Math.log(t) * -0.5))) - t;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -1e+18) or not ((a - 0.5) <= -0.4):
		tmp = (math.log(t) * a) - t
	else:
		tmp = (math.log(y) + (math.log(z) + (math.log(t) * -0.5))) - t
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -1e+18) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(log(t) * a) - t);
	else
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(log(t) * -0.5))) - t);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -1e+18) || ~(((a - 0.5) <= -0.4)))
		tmp = (log(t) * a) - t;
	else
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -1e+18], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -1e18 or -0.40000000000000002 < (-.f64 a 1/2)

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 74.7%

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

   5. Taylor expanded in a around inf 98.8%

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

      1. *-commutative 98.8%

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

   7. Simplified 98.8%

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

   if -1e18 < (-.f64 a 1/2) < -0.40000000000000002

   1. Initial program 99.4%

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

      1. associate--l+ 99.4%

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

      2. sub-neg 99.4%

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

      3. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 64.3%

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

   5. Taylor expanded in a around 0 64.3%

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

      1. *-commutative 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -1 \cdot 10^{+18} \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \end{array} \]

Alternative 4: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 490:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 490.0)
   (+ (log y) (+ (log z) (* (log t) (- a 0.5))))
   (- (* (log t) a) t)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 490.0) {
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	} else {
		tmp = (log(t) * a) - t;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 490.0d0) then
        tmp = log(y) + (log(z) + (log(t) * (a - 0.5d0)))
    else
        tmp = (log(t) * a) - t
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 490.0) {
		tmp = Math.log(y) + (Math.log(z) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (Math.log(t) * a) - t;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if t <= 490.0:
		tmp = math.log(y) + (math.log(z) + (math.log(t) * (a - 0.5)))
	else:
		tmp = (math.log(t) * a) - t
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 490.0)
		tmp = Float64(log(y) + Float64(log(z) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 490.0)
		tmp = log(y) + (log(z) + (log(t) * (a - 0.5)));
	else
		tmp = (log(t) * a) - t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 490.0], 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[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 490

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-def 99.2%

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

      3. sub-neg 99.2%

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

      4. metadata-eval 99.2%

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

      5. associate--l+ 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 65.2%

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

   5. Taylor expanded in t around 0 63.7%

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

   if 490 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 73.2%

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

   5. Taylor expanded in a around inf 97.1%

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

      1. *-commutative 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 490:\\ \;\;\;\;\log y + \left(\log z + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]

Alternative 5: 99.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (+ (+ (log y) (log z)) (- (* (log t) (- a 0.5)) t)))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

assert x < y;
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

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z, t, a)
	tmp = (log(y) + log(z)) + ((log(t) * (a - 0.5)) - t);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(N[Log[y], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Derivation

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. +-commutative 99.5%

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

   3. associate--l+ 99.5%

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

   4. sub-neg 99.5%

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

   5. +-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. fma-def 99.5%

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

   9. sub-neg 99.5%

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

   10. +-commutative 99.5%

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

   11. distribute-neg-in 99.5%

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

   12. metadata-eval 99.5%

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

   13. metadata-eval 99.5%

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

   14. unsub-neg 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 69.3%

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

5. Final simplification 69.3%

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

Alternative 6: 87.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+14}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.15e+14)
   (- (+ (log (* y z)) (* (log t) (- a 0.5))) t)
   (- (* (log t) a) t)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.15e+14) {
		tmp = (log((y * z)) + (log(t) * (a - 0.5))) - t;
	} else {
		tmp = (log(t) * a) - t;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 2.15d+14) then
        tmp = (log((y * z)) + (log(t) * (a - 0.5d0))) - t
    else
        tmp = (log(t) * a) - t
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.15e+14) {
		tmp = (Math.log((y * z)) + (Math.log(t) * (a - 0.5))) - t;
	} else {
		tmp = (Math.log(t) * a) - t;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.15e+14:
		tmp = (math.log((y * z)) + (math.log(t) * (a - 0.5))) - t
	else:
		tmp = (math.log(t) * a) - t
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.15e+14)
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5))) - t);
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.15e+14)
		tmp = (log((y * z)) + (log(t) * (a - 0.5))) - t;
	else
		tmp = (log(t) * a) - t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.15e+14], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.15e14

   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. +-commutative 99.3%

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

      3. associate--l+ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. fma-def 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. metadata-eval 99.3%

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

      14. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

      1. associate-+r- 99.3%

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

      2. +-commutative 99.3%

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

      3. fma-udef 99.3%

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

      4. associate--r+ 99.3%

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

      5. +-commutative 99.3%

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

      6. sum-log 77.6%

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

   5. Applied egg-rr 77.6%

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

   6. Taylor expanded in x around 0 51.4%

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

      1. *-commutative 51.4%

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

   8. Simplified 51.4%

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

   if 2.15e14 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 72.2%

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

   5. Taylor expanded in a around inf 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.15 \cdot 10^{+14}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]

Alternative 7: 79.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -0.0076 \lor \neg \left(a \leq 4.2\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\log z + \left(\log \left(y + x\right) - t\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.0076) (not (<= a 4.2)))
   (- (* (log t) a) t)
   (+ (log z) (- (log (+ y x)) t))))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0076) || !(a <= 4.2)) {
		tmp = (log(t) * a) - t;
	} else {
		tmp = log(z) + (log((y + x)) - t);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.0076d0)) .or. (.not. (a <= 4.2d0))) then
        tmp = (log(t) * a) - t
    else
        tmp = log(z) + (log((y + x)) - t)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.0076) || !(a <= 4.2)) {
		tmp = (Math.log(t) * a) - t;
	} else {
		tmp = Math.log(z) + (Math.log((y + x)) - t);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.0076) or not (a <= 4.2):
		tmp = (math.log(t) * a) - t
	else:
		tmp = math.log(z) + (math.log((y + x)) - t)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.0076) || !(a <= 4.2))
		tmp = Float64(Float64(log(t) * a) - t);
	else
		tmp = Float64(log(z) + Float64(log(Float64(y + x)) - t));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.0076) || ~((a <= 4.2)))
		tmp = (log(t) * a) - t;
	else
		tmp = log(z) + (log((y + x)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.0076], N[Not[LessEqual[a, 4.2]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(y + x), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.00759999999999999998 or 4.20000000000000018 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. sub-neg 99.6%

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

      3. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 74.0%

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

   5. Taylor expanded in a around inf 98.2%

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

      1. *-commutative 98.2%

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

   7. Simplified 98.2%

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

   if -0.00759999999999999998 < a < 4.20000000000000018

   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%

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

      2. +-commutative 99.4%

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

      3. associate--l+ 99.4%

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

      4. sub-neg 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. distribute-rgt-neg-in 99.4%

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

      8. fma-def 99.4%

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

      9. sub-neg 99.4%

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

      10. +-commutative 99.4%

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

      11. distribute-neg-in 99.4%

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

      12. metadata-eval 99.4%

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

      13. metadata-eval 99.4%

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

      14. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t around inf 60.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.0076 \lor \neg \left(a \leq 4.2\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\log z + \left(\log \left(y + x\right) - t\right)\\ \end{array} \]

Alternative 8: 83.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -0.098 \lor \neg \left(a \leq 8.5 \cdot 10^{-40}\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right) - t\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.098) (not (<= a 8.5e-40)))
   (- (* (log t) a) t)
   (- (log (* y (/ z (sqrt t)))) t)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.098) || !(a <= 8.5e-40)) {
		tmp = (log(t) * a) - t;
	} else {
		tmp = log((y * (z / sqrt(t)))) - t;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.098d0)) .or. (.not. (a <= 8.5d-40))) then
        tmp = (log(t) * a) - t
    else
        tmp = log((y * (z / sqrt(t)))) - t
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.098) || !(a <= 8.5e-40)) {
		tmp = (Math.log(t) * a) - t;
	} else {
		tmp = Math.log((y * (z / Math.sqrt(t)))) - t;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.098) or not (a <= 8.5e-40):
		tmp = (math.log(t) * a) - t
	else:
		tmp = math.log((y * (z / math.sqrt(t)))) - t
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.098) || !(a <= 8.5e-40))
		tmp = Float64(Float64(log(t) * a) - t);
	else
		tmp = Float64(log(Float64(y * Float64(z / sqrt(t)))) - t);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.098) || ~((a <= 8.5e-40)))
		tmp = (log(t) * a) - t;
	else
		tmp = log((y * (z / sqrt(t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.098], N[Not[LessEqual[a, 8.5e-40]], $MachinePrecision]], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * N[(z / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.098000000000000004 or 8.4999999999999998e-40 < 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. Taylor expanded in x around 0 74.2%

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

   5. Taylor expanded in a around inf 98.1%

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

      1. *-commutative 98.1%

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

   7. Simplified 98.1%

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

   if -0.098000000000000004 < a < 8.4999999999999998e-40

   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%

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

      2. +-commutative 99.4%

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

      3. associate--l+ 99.4%

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

      4. sub-neg 99.4%

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

      5. +-commutative 99.4%

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

      6. *-commutative 99.4%

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

      7. distribute-rgt-neg-in 99.4%

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

      8. fma-def 99.4%

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

      9. sub-neg 99.4%

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

      10. +-commutative 99.4%

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

      11. distribute-neg-in 99.4%

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

      12. metadata-eval 99.4%

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

      13. metadata-eval 99.4%

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

      14. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

      1. associate-+r- 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-udef 99.4%

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

      4. associate--r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. sum-log 72.5%

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

   5. Applied egg-rr 72.5%

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

   6. Taylor expanded in x around 0 47.7%

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

      1. *-commutative 47.7%

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

   8. Simplified 47.7%

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

   9. Taylor expanded in a around 0 47.7%

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

       1. *-commutative 47.7%

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

   11. Simplified 47.7%

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

       1. log-prod 64.1%

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

       2. +-commutative 64.1%

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

       3. associate--l+ 64.1%

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

       4. add-log-exp 64.1%

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

       5. exp-to-pow 64.1%

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

       6. pow1/2 64.1%

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

   13. Applied egg-rr 64.1%

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

       1. associate-+r- 64.1%

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

       2. +-commutative 64.1%

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

       3. log-prod 47.7%

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

       4. log-div 44.0%

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

       5. associate-/l* 46.7%

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

       6. associate-/r/ 47.2%

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

   15. Simplified 47.2%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.098 \lor \neg \left(a \leq 8.5 \cdot 10^{-40}\right):\\ \;\;\;\;\log t \cdot a - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \frac{z}{\sqrt{t}}\right) - t\\ \end{array} \]

Alternative 9: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 330:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (<= t 330.0) (+ (log (* y z)) (* (log t) (- a 0.5))) (- (* (log t) a) t)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 330.0) {
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	} else {
		tmp = (log(t) * a) - t;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 330.0d0) then
        tmp = log((y * z)) + (log(t) * (a - 0.5d0))
    else
        tmp = (log(t) * a) - t
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 330.0) {
		tmp = Math.log((y * z)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(t) * a) - t;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if t <= 330.0:
		tmp = math.log((y * z)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(t) * a) - t
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 330.0)
		tmp = Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(t) * a) - t);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 330.0)
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	else
		tmp = (log(t) * a) - t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[LessEqual[t, 330.0], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 330

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. +-commutative 99.2%

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

      3. associate--l+ 99.2%

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

      4. sub-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. *-commutative 99.2%

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

      7. distribute-rgt-neg-in 99.2%

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

      8. fma-def 99.2%

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

      9. sub-neg 99.2%

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

      10. +-commutative 99.2%

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

      11. distribute-neg-in 99.2%

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

      12. metadata-eval 99.2%

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

      13. metadata-eval 99.2%

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

      14. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 65.2%

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

   5. Taylor expanded in t around 0 63.6%

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

      1. log-prod 49.6%

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

      2. *-commutative 49.6%

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

   7. Simplified 49.6%

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

   if 330 < t

   1. Initial program 99.8%

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

      1. associate--l+ 99.8%

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

      2. sub-neg 99.8%

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

      3. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 73.2%

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

   5. Taylor expanded in a around inf 97.1%

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

      1. *-commutative 97.1%

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

   7. Simplified 97.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 330:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\log t \cdot a - t\\ \end{array} \]

Alternative 10: 62.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;a \leq -38000000000000 \lor \neg \left(a \leq 6.3 \cdot 10^{+16}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -38000000000000.0) (not (<= a 6.3e+16))) (* (log t) a) (- t)))

C

assert(x < y);
double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -38000000000000.0) || !(a <= 6.3e+16)) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= (-38000000000000.0d0)) .or. (.not. (a <= 6.3d+16))) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -38000000000000.0) || !(a <= 6.3e+16)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	tmp = 0
	if (a <= -38000000000000.0) or not (a <= 6.3e+16):
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -38000000000000.0) || !(a <= 6.3e+16))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -38000000000000.0) || ~((a <= 6.3e+16)))
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -38000000000000.0], N[Not[LessEqual[a, 6.3e+16]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -3.8e13 or 6.3e16 < 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%

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

      2. +-commutative 99.6%

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

      3. associate--l+ 99.6%

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

      4. sub-neg 99.6%

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

      5. +-commutative 99.6%

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

      6. *-commutative 99.6%

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

      7. distribute-rgt-neg-in 99.6%

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

      8. fma-def 99.6%

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

      9. sub-neg 99.6%

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

      10. +-commutative 99.6%

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

      11. distribute-neg-in 99.6%

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

      12. metadata-eval 99.6%

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

      13. metadata-eval 99.6%

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

      14. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a around inf 79.2%

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

      1. *-commutative 79.2%

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

   6. Simplified 79.2%

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

   if -3.8e13 < a < 6.3e16

   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. +-commutative 99.5%

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

      3. associate--l+ 99.5%

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

      4. sub-neg 99.5%

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

      5. +-commutative 99.5%

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

      6. *-commutative 99.5%

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

      7. distribute-rgt-neg-in 99.5%

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

      8. fma-def 99.5%

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

      9. sub-neg 99.5%

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

      10. +-commutative 99.5%

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

      11. distribute-neg-in 99.5%

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

      12. metadata-eval 99.5%

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

      13. metadata-eval 99.5%

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

      14. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in t around inf 52.1%

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

      1. neg-mul-1 52.1%

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

   6. Simplified 52.1%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -38000000000000 \lor \neg \left(a \leq 6.3 \cdot 10^{+16}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 11: 75.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \log t \cdot a - t \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (- (* (log t) a) t))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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(t) * a) - t
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	return (math.log(t) * a) - t

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	return Float64(Float64(log(t) * a) - t)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z, t, a)
	tmp = (log(t) * a) - t;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := N[(N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision] - t), $MachinePrecision]

Derivation

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%

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

   2. sub-neg 99.5%

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

   3. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 69.3%

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

5. Taylor expanded in a around inf 75.1%

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

   1. *-commutative 75.1%

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

7. Simplified 75.1%

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

8. Final simplification 75.1%

   \[\leadsto \log t \cdot a - t \]

Alternative 12: 38.2% accurate, 156.5× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ -t \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y z t a) :precision binary64 (- t))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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

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

Python

[x, y] = sort([x, y])
def code(x, y, z, t, a):
	return -t

Julia

x, y = sort([x, y])
function code(x, y, z, t, a)
	return Float64(-t)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y, z, t, a)
	tmp = -t;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_, a_] := (-t)

Derivation

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. +-commutative 99.5%

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

   3. associate--l+ 99.5%

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

   4. sub-neg 99.5%

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

   5. +-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. fma-def 99.5%

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

   9. sub-neg 99.5%

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

   10. +-commutative 99.5%

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

   11. distribute-neg-in 99.5%

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

   12. metadata-eval 99.5%

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

   13. metadata-eval 99.5%

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

   14. unsub-neg 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in t around inf 36.7%

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

   1. neg-mul-1 36.7%

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

6. Simplified 36.7%

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

7. Final simplification 36.7%

   \[\leadsto -t \]

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 2023308 
(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))))