Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 17.5s

Alternatives: 18

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(x + y), $MachinePrecision]], $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%

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

Alternative 2: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log \left(x + y\right) + \log t \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -20000000.0) (not (<= (- a 0.5) -0.4)))
   (- (* a (log t)) t)
   (- (+ (log z) (+ (log (+ x y)) (* (log t) -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + (log((x + y)) + (log(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 - 0.5d0) <= (-20000000.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + (log((x + y)) + (log(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 - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + (Math.log((x + y)) + (Math.log(t) * -0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -20000000.0) or not ((a - 0.5) <= -0.4):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + (math.log((x + y)) + (math.log(t) * -0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -20000000.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + Float64(log(Float64(x + y)) + Float64(log(t) * -0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -20000000.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + (log((x + y)) + (log(t) * -0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -20000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $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) < -2e7 or -0.40000000000000002 < (-.f64 a 1/2)

   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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.5%

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

   5. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   if -2e7 < (-.f64 a 1/2) < -0.40000000000000002

   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+ 99.4%

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

      4. +-commutative 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around 0 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log \left(x + y\right) + \log t \cdot -0.5\right)\right) - t\\ \end{array} \]

Alternative 3: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(x + y\right) + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -20000000.0) (not (<= (- a 0.5) -0.4)))
   (- (* a (log t)) t)
   (- (+ (log (+ x y)) (+ (log z) (* (log t) -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log((x + y)) + (log(z) + (log(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 - 0.5d0) <= (-20000000.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log((x + y)) + (log(z) + (log(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 - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log((x + y)) + (Math.log(z) + (Math.log(t) * -0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -20000000.0) or not ((a - 0.5) <= -0.4):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log((x + y)) + (math.log(z) + (math.log(t) * -0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -20000000.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(Float64(x + y)) + Float64(log(z) + Float64(log(t) * -0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -20000000.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log((x + y)) + (log(z) + (log(t) * -0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -20000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(x + y), $MachinePrecision]], $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) < -2e7 or -0.40000000000000002 < (-.f64 a 1/2)

   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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.5%

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

   5. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   if -2e7 < (-.f64 a 1/2) < -0.40000000000000002

   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. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 98.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(x + y\right) + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \end{array} \]

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + \log t \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -20000000.0) (not (<= (- a 0.5) -0.4)))
   (- (* a (log t)) t)
   (- (+ (log z) (+ (log y) (* (log t) -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + (log(y) + (log(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 - 0.5d0) <= (-20000000.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + (log(y) + (log(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 - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + (Math.log(y) + (Math.log(t) * -0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -20000000.0) or not ((a - 0.5) <= -0.4):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + (math.log(y) + (math.log(t) * -0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -20000000.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + Float64(log(y) + Float64(log(t) * -0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -20000000.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + (log(y) + (log(t) * -0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -20000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $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) < -2e7 or -0.40000000000000002 < (-.f64 a 1/2)

   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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.5%

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

   5. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   if -2e7 < (-.f64 a 1/2) < -0.40000000000000002

   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. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 67.4%

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

   5. Taylor expanded in a around 0 66.6%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + \log t \cdot -0.5\right)\right) - t\\ \end{array} \]

Alternative 5: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log z + \log y\right) + \log t \cdot -0.5\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -20000000.0) (not (<= (- a 0.5) -0.4)))
   (- (* a (log t)) t)
   (- (+ (+ (log z) (log y)) (* (log t) -0.5)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = ((log(z) + log(y)) + (log(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 - 0.5d0) <= (-20000000.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (a * log(t)) - t
    else
        tmp = ((log(z) + log(y)) + (log(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 - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = ((Math.log(z) + Math.log(y)) + (Math.log(t) * -0.5)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -20000000.0) or not ((a - 0.5) <= -0.4):
		tmp = (a * math.log(t)) - t
	else:
		tmp = ((math.log(z) + math.log(y)) + (math.log(t) * -0.5)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -20000000.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(Float64(log(z) + log(y)) + Float64(log(t) * -0.5)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -20000000.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (a * log(t)) - t;
	else
		tmp = ((log(z) + log(y)) + (log(t) * -0.5)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -20000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -2e7 or -0.40000000000000002 < (-.f64 a 1/2)

   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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.5%

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

   5. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   if -2e7 < (-.f64 a 1/2) < -0.40000000000000002

   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. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 67.4%

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

   5. Taylor expanded in a around 0 66.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log z + \log y\right) + \log t \cdot -0.5\right) - t\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -20000000.0) (not (<= (- a 0.5) -0.4)))
   (- (* a (log t)) t)
   (- (+ (log z) (log (* y (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((a - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log(z) + log((y * 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 - 0.5d0) <= (-20000000.0d0)) .or. (.not. ((a - 0.5d0) <= (-0.4d0)))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log(z) + log((y * (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 - 0.5) <= -20000000.0) || !((a - 0.5) <= -0.4)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log(z) + Math.log((y * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((a - 0.5) <= -20000000.0) or not ((a - 0.5) <= -0.4):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log(z) + math.log((y * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((Float64(a - 0.5) <= -20000000.0) || !(Float64(a - 0.5) <= -0.4))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(z) + log(Float64(y * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((a - 0.5) <= -20000000.0) || ~(((a - 0.5) <= -0.4)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log(z) + log((y * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[N[(a - 0.5), $MachinePrecision], -20000000.0], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.4]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 a 1/2) < -2e7 or -0.40000000000000002 < (-.f64 a 1/2)

   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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.5%

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

   5. Taylor expanded in a around inf 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   if -2e7 < (-.f64 a 1/2) < -0.40000000000000002

   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. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 67.4%

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

   5. Taylor expanded in a around 0 66.6%

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

   6. Taylor expanded in y around inf 66.6%

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

      1. mul-1-neg 66.6%

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

      2. log-rec 66.6%

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

      3. remove-double-neg 66.6%

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

      4. log-pow 66.6%

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

      5. log-prod 58.6%

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

   8. Simplified 58.6%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -20000000 \lor \neg \left(a - 0.5 \leq -0.4\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]

Alternative 7: 69.8% accurate, 1.0× speedup

Math

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

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(Float64(a - 0.5) * log(t)) + 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[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 72.0%

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

5. Final simplification 72.0%

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

Alternative 8: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.95 \cdot 10^{-10} \lor \neg \left(a \leq 0.235\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + 0.5 \cdot \log \left(\frac{1}{t}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.95e-10) (not (<= a 0.235)))
   (- (* a (log t)) t)
   (- (+ (log (* y z)) (* 0.5 (log (/ 1.0 t)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.95e-10) || !(a <= 0.235)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = (log((y * z)) + (0.5 * log((1.0 / t)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.95d-10)) .or. (.not. (a <= 0.235d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = (log((y * z)) + (0.5d0 * log((1.0d0 / t)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.95e-10) || !(a <= 0.235)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = (Math.log((y * z)) + (0.5 * Math.log((1.0 / t)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.95e-10) or not (a <= 0.235):
		tmp = (a * math.log(t)) - t
	else:
		tmp = (math.log((y * z)) + (0.5 * math.log((1.0 / t)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.95e-10) || !(a <= 0.235))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(0.5 * log(Float64(1.0 / t)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.95e-10) || ~((a <= 0.235)))
		tmp = (a * log(t)) - t;
	else
		tmp = (log((y * z)) + (0.5 * log((1.0 / t)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.95e-10], N[Not[LessEqual[a, 0.235]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[Log[N[(1.0 / t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.95e-10 or 0.23499999999999999 < 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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.4%

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

   5. Taylor expanded in a around inf 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if -1.95e-10 < a < 0.23499999999999999

   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. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 67.3%

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

   5. Taylor expanded in a around 0 66.4%

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

      1. associate-+r+ 66.6%

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

      2. log-prod 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in t around inf 48.0%

      \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) + 0.5 \cdot \log \left(\frac{1}{t}\right)\right)} - t \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.95 \cdot 10^{-10} \lor \neg \left(a \leq 0.235\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + 0.5 \cdot \log \left(\frac{1}{t}\right)\right) - t\\ \end{array} \]

Alternative 9: 86.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-11} \lor \neg \left(a \leq 0.0075\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3e-11) (not (<= a 0.0075)))
   (- (* a (log t)) t)
   (- (+ (* (log t) -0.5) (log (* (+ x y) z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3e-11) || !(a <= 0.0075)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = ((log(t) * -0.5) + log(((x + y) * z))) - 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 <= (-3d-11)) .or. (.not. (a <= 0.0075d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = ((log(t) * (-0.5d0)) + log(((x + y) * z))) - 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 <= -3e-11) || !(a <= 0.0075)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = ((Math.log(t) * -0.5) + Math.log(((x + y) * z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3e-11) or not (a <= 0.0075):
		tmp = (a * math.log(t)) - t
	else:
		tmp = ((math.log(t) * -0.5) + math.log(((x + y) * z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3e-11) || !(a <= 0.0075))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(Float64(log(t) * -0.5) + log(Float64(Float64(x + y) * z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3e-11) || ~((a <= 0.0075)))
		tmp = (a * log(t)) - t;
	else
		tmp = ((log(t) * -0.5) + log(((x + y) * z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3e-11], N[Not[LessEqual[a, 0.0075]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3e-11 or 0.0074999999999999997 < 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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.4%

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

   5. Taylor expanded in a around inf 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if -3e-11 < a < 0.0074999999999999997

   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. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a around 0 98.8%

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

      1. associate-+r+ 98.9%

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

      2. +-commutative 98.9%

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

      3. associate-+r+ 98.7%

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

      4. +-commutative 98.7%

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

      5. associate-+r+ 98.9%

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

      6. log-prod 76.2%

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

      7. +-commutative 76.2%

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

   6. Simplified 76.2%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-11} \lor \neg \left(a \leq 0.0075\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \end{array} \]

Alternative 10: 87.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 28500000000:\\ \;\;\;\;\left(\log t \cdot \left(a + -0.5\right) + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 28500000000.0)
   (- (+ (* (log t) (+ a -0.5)) (log (* (+ x y) z))) t)
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 28500000000.0) {
		tmp = ((log(t) * (a + -0.5)) + log(((x + y) * z))) - t;
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 28500000000.0d0) then
        tmp = ((log(t) * (a + (-0.5d0))) + log(((x + y) * z))) - t
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 28500000000.0) {
		tmp = ((Math.log(t) * (a + -0.5)) + Math.log(((x + y) * z))) - t;
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 28500000000.0:
		tmp = ((math.log(t) * (a + -0.5)) + math.log(((x + y) * z))) - t
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 28500000000.0)
		tmp = Float64(Float64(Float64(log(t) * Float64(a + -0.5)) + log(Float64(Float64(x + y) * z))) - t);
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 28500000000.0)
		tmp = ((log(t) * (a + -0.5)) + log(((x + y) * z))) - t;
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 28500000000.0], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.85e10

   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. +-commutative 99.4%

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

      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. Step-by-step derivation

      1. +-commutative 99.3%

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

      2. fma-udef 99.3%

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

      3. metadata-eval 99.3%

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

      4. sub-neg 99.3%

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

      5. associate-+r+ 99.4%

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

      6. associate--l+ 99.4%

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

      7. associate-+r- 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

      10. +-commutative 99.4%

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

      11. sum-log 78.8%

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

   5. Applied egg-rr 78.8%

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

   if 2.85e10 < 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. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 76.8%

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

   5. Taylor expanded in a around inf 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 28500000000:\\ \;\;\;\;\left(\log t \cdot \left(a + -0.5\right) + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 11: 73.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-12} \lor \neg \left(a \leq 0.024\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.5e-12) (not (<= a 0.024)))
   (- (* a (log t)) t)
   (- (+ (* (log t) -0.5) (log (* y z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.5e-12) || !(a <= 0.024)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = ((log(t) * -0.5) + log((y * z))) - 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 <= (-4.5d-12)) .or. (.not. (a <= 0.024d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = ((log(t) * (-0.5d0)) + log((y * z))) - 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 <= -4.5e-12) || !(a <= 0.024)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = ((Math.log(t) * -0.5) + Math.log((y * z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -4.5e-12) or not (a <= 0.024):
		tmp = (a * math.log(t)) - t
	else:
		tmp = ((math.log(t) * -0.5) + math.log((y * z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.5e-12) || !(a <= 0.024))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(Float64(Float64(log(t) * -0.5) + log(Float64(y * z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -4.5e-12) || ~((a <= 0.024)))
		tmp = (a * log(t)) - t;
	else
		tmp = ((log(t) * -0.5) + log((y * z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.5e-12], N[Not[LessEqual[a, 0.024]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.49999999999999981e-12 or 0.024 < 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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.4%

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

   5. Taylor expanded in a around inf 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if -4.49999999999999981e-12 < a < 0.024

   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. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 67.3%

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

   5. Taylor expanded in a around 0 66.4%

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

      1. associate-+r+ 66.6%

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

      2. log-prod 48.0%

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

   7. Simplified 48.0%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-12} \lor \neg \left(a \leq 0.024\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \]

Alternative 12: 73.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 46000000000:\\ \;\;\;\;\left(\left(a - 0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 46000000000.0)
   (- (+ (* (- a 0.5) (log t)) (log (* y z))) t)
   (- (* a (log t)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 46000000000.0) {
		tmp = (((a - 0.5) * log(t)) + log((y * z))) - t;
	} else {
		tmp = (a * log(t)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 46000000000.0d0) then
        tmp = (((a - 0.5d0) * log(t)) + log((y * z))) - t
    else
        tmp = (a * log(t)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 46000000000.0) {
		tmp = (((a - 0.5) * Math.log(t)) + Math.log((y * z))) - t;
	} else {
		tmp = (a * Math.log(t)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 46000000000.0:
		tmp = (((a - 0.5) * math.log(t)) + math.log((y * z))) - t
	else:
		tmp = (a * math.log(t)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 46000000000.0)
		tmp = Float64(Float64(Float64(Float64(a - 0.5) * log(t)) + log(Float64(y * z))) - t);
	else
		tmp = Float64(Float64(a * log(t)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 46000000000.0)
		tmp = (((a - 0.5) * log(t)) + log((y * z))) - t;
	else
		tmp = (a * log(t)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 46000000000.0], N[(N[(N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.6e10

   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. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. remove-double-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 67.7%

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

      1. sum-log 56.5%

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

   6. Applied egg-rr 56.5%

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

   if 4.6e10 < 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. associate-+l+ 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-def 99.9%

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

      5. remove-double-neg 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 76.8%

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

   5. Taylor expanded in a around inf 99.9%

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

      1. *-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 46000000000:\\ \;\;\;\;\left(\left(a - 0.5\right) \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 13: 71.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-11} \lor \neg \left(a \leq 0.007\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(y \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3e-11) (not (<= a 0.007)))
   (- (* a (log t)) t)
   (- (log (* (pow t -0.5) (* y z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3e-11) || !(a <= 0.007)) {
		tmp = (a * log(t)) - t;
	} else {
		tmp = log((pow(t, -0.5) * (y * z))) - 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 <= (-3d-11)) .or. (.not. (a <= 0.007d0))) then
        tmp = (a * log(t)) - t
    else
        tmp = log(((t ** (-0.5d0)) * (y * z))) - 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 <= -3e-11) || !(a <= 0.007)) {
		tmp = (a * Math.log(t)) - t;
	} else {
		tmp = Math.log((Math.pow(t, -0.5) * (y * z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3e-11) or not (a <= 0.007):
		tmp = (a * math.log(t)) - t
	else:
		tmp = math.log((math.pow(t, -0.5) * (y * z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3e-11) || !(a <= 0.007))
		tmp = Float64(Float64(a * log(t)) - t);
	else
		tmp = Float64(log(Float64((t ^ -0.5) * Float64(y * z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3e-11) || ~((a <= 0.007)))
		tmp = (a * log(t)) - t;
	else
		tmp = log(((t ^ -0.5) * (y * z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3e-11], N[Not[LessEqual[a, 0.007]], $MachinePrecision]], N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(N[Power[t, -0.5], $MachinePrecision] * N[(y * z), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3e-11 or 0.00700000000000000015 < 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. associate-+l+ 99.7%

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

      3. +-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. remove-double-neg 99.7%

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

      6. remove-double-neg 99.7%

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

      7. sub-neg 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 76.4%

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

   5. Taylor expanded in a around inf 98.6%

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

      1. *-commutative 98.6%

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

   7. Simplified 98.6%

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

   if -3e-11 < a < 0.00700000000000000015

   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. associate-+l+ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

      5. remove-double-neg 99.5%

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

      6. remove-double-neg 99.5%

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

      7. sub-neg 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 67.3%

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

   5. Taylor expanded in a around 0 66.4%

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

      1. associate-+r+ 66.6%

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

      2. log-prod 48.0%

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

   7. Simplified 48.0%

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

   8. Taylor expanded in z around 0 66.4%

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

      1. log-pow 66.4%

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

      2. associate-+r+ 66.6%

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

      3. log-prod 48.0%

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

      4. log-prod 43.5%

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

      5. *-commutative 43.5%

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

   10. Simplified 43.5%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{-11} \lor \neg \left(a \leq 0.007\right):\\ \;\;\;\;a \cdot \log t - t\\ \mathbf{else}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(y \cdot z\right)\right) - t\\ \end{array} \]

Alternative 14: 77.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;t \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;t_1 + \log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= t 2.6e-17) (+ t_1 (log (+ x y))) (- t_1 t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * log(t);
	double tmp;
	if (t <= 2.6e-17) {
		tmp = t_1 + log((x + y));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (t <= 2.6d-17) then
        tmp = t_1 + log((x + y))
    else
        tmp = t_1 - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (t <= 2.6e-17) {
		tmp = t_1 + Math.log((x + y));
	} else {
		tmp = t_1 - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if t <= 2.6e-17:
		tmp = t_1 + math.log((x + y))
	else:
		tmp = t_1 - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (t <= 2.6e-17)
		tmp = Float64(t_1 + log(Float64(x + y)));
	else
		tmp = Float64(t_1 - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (t <= 2.6e-17)
		tmp = t_1 + log((x + y));
	else
		tmp = t_1 - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 2.6e-17], N[(t$95$1 + N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.60000000000000003e-17

   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. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. remove-double-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in a around inf 57.0%

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

      1. *-commutative 57.0%

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

   6. Simplified 57.0%

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

   if 2.60000000000000003e-17 < 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. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. remove-double-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 76.5%

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

   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%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;a \cdot \log t + \log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 15: 41.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 2.6e-17) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.6e-17) {
		tmp = log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 2.6d-17) then
        tmp = log((x + y))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 2.6e-17) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 2.6e-17:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 2.6e-17)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 2.6e-17)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 2.6e-17], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 2.60000000000000003e-17

   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. associate-+l+ 99.4%

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

      3. +-commutative 99.4%

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

      4. fma-def 99.4%

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

      5. remove-double-neg 99.4%

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

      6. remove-double-neg 99.4%

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

      7. sub-neg 99.4%

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

      8. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in t around inf 9.4%

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

      1. neg-mul-1 9.4%

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

   6. Simplified 9.4%

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

   7. Taylor expanded in t around 0 9.4%

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

   if 2.60000000000000003e-17 < 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. associate-+l+ 99.8%

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

      3. +-commutative 99.8%

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

      4. fma-def 99.8%

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

      5. remove-double-neg 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 76.5%

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

   5. Taylor expanded in t around inf 72.8%

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

      1. neg-mul-1 72.8%

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

   7. Simplified 72.8%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2.6 \cdot 10^{-17}:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 16: 74.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ a \cdot \log t - t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 72.0%

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

5. Taylor expanded in a around inf 75.4%

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

   1. *-commutative 75.4%

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

7. Simplified 75.4%

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

8. Final simplification 75.4%

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

Alternative 17: 31.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in t around inf 42.0%

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

   1. neg-mul-1 42.0%

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

6. Simplified 42.0%

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

7. Taylor expanded in x around 0 31.7%

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

8. Final simplification 31.7%

   \[\leadsto \log y - t \]

Alternative 18: 38.7% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate--l+ 99.6%

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

   2. associate-+l+ 99.6%

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

   3. +-commutative 99.6%

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

   4. fma-def 99.6%

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

   5. remove-double-neg 99.6%

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

   6. remove-double-neg 99.6%

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

   7. sub-neg 99.6%

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

   8. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 72.0%

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

5. Taylor expanded in t around inf 38.7%

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

   1. neg-mul-1 38.7%

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

7. Simplified 38.7%

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

8. Final simplification 38.7%

   \[\leadsto -t \]

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 2023192 
(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))))