Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 17.9s

Alternatives: 14

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

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

FPCore

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

C

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

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

Python

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

Julia

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

MATLAB

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

3. Final simplification 99.6%

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

Alternative 2: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.00029:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.00029)
   (+ (log z) (+ (log (+ x y)) (* (log t) (- a 0.5))))
   (+ (+ (log z) (log y)) (- (* a (log t)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00029) {
		tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5)));
	} else {
		tmp = (log(z) + log(y)) + ((a * log(t)) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 0.00029d0) then
        tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5d0)))
    else
        tmp = (log(z) + log(y)) + ((a * log(t)) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00029) {
		tmp = Math.log(z) + (Math.log((x + y)) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = (Math.log(z) + Math.log(y)) + ((a * Math.log(t)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.00029:
		tmp = math.log(z) + (math.log((x + y)) + (math.log(t) * (a - 0.5)))
	else:
		tmp = (math.log(z) + math.log(y)) + ((a * math.log(t)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.00029)
		tmp = Float64(log(z) + Float64(log(Float64(x + y)) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(Float64(log(z) + log(y)) + Float64(Float64(a * log(t)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.00029)
		tmp = log(z) + (log((x + y)) + (log(t) * (a - 0.5)));
	else
		tmp = (log(z) + log(y)) + ((a * log(t)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.00029], N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.9e-4

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 99.1%

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

      1. associate--l+ 99.1%

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

      2. +-commutative 99.1%

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

   7. Simplified 99.1%

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

   if 2.9e-4 < 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%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.2%

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

   6. Taylor expanded in a around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. distribute-rgt-neg-in 74.6%

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

   8. Simplified 74.6%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00029:\\ \;\;\;\;\log z + \left(\log \left(x + y\right) + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{+38}:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.2e+38) (+ (log z) (+ (log y) (* (log t) (- a 0.5)))) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.2e+38) {
		tmp = log(z) + (log(y) + (log(t) * (a - 0.5)));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 3.2d+38) then
        tmp = log(z) + (log(y) + (log(t) * (a - 0.5d0)))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.2e+38) {
		tmp = Math.log(z) + (Math.log(y) + (Math.log(t) * (a - 0.5)));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.2e+38:
		tmp = math.log(z) + (math.log(y) + (math.log(t) * (a - 0.5)))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.2e+38)
		tmp = Float64(log(z) + Float64(log(y) + Float64(log(t) * Float64(a - 0.5))));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 3.2e+38)
		tmp = log(z) + (log(y) + (log(t) * (a - 0.5)));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.2e+38], N[(N[Log[z], $MachinePrecision] + N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 3.19999999999999985e38

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 97.2%

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

      1. associate--l+ 97.1%

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

      2. +-commutative 97.1%

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

   7. Simplified 97.1%

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

   8. Taylor expanded in y around inf 67.6%

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

      1. mul-1-neg 67.6%

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

      2. log-rec 67.6%

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

      3. remove-double-neg 67.6%

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

      4. *-commutative 67.6%

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

   10. Simplified 67.6%

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

   if 3.19999999999999985e38 < 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%

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

      2. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-undefine 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 77.5%

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

      1. neg-mul-1 77.5%

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

   7. Simplified 77.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.2 \cdot 10^{+38}:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{+38}:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7e+38) (+ (+ (log z) (log y)) (* (log t) (- a 0.5))) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7e+38) {
		tmp = (log(z) + log(y)) + (log(t) * (a - 0.5));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 7d+38) then
        tmp = (log(z) + log(y)) + (log(t) * (a - 0.5d0))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7e+38) {
		tmp = (Math.log(z) + Math.log(y)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 7e+38:
		tmp = (math.log(z) + math.log(y)) + (math.log(t) * (a - 0.5))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 7e+38)
		tmp = Float64(Float64(log(z) + log(y)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 7e+38)
		tmp = (log(z) + log(y)) + (log(t) * (a - 0.5));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7e+38], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 7.00000000000000003e38

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 69.8%

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

   6. Taylor expanded in t around 0 67.6%

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

      1. +-commutative 67.6%

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

      2. *-commutative 67.6%

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

   8. Simplified 67.6%

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

   if 7.00000000000000003e38 < 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%

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

      2. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-undefine 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 77.5%

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

      1. neg-mul-1 77.5%

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

   7. Simplified 77.5%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7 \cdot 10^{+38}:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log z + \log y\\ \mathbf{if}\;t \leq 0.00029:\\ \;\;\;\;t\_1 + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(a \cdot \log t - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log z) (log y))))
   (if (<= t 0.00029)
     (+ t_1 (* (log t) (- a 0.5)))
     (+ t_1 (- (* a (log t)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(z) + log(y);
	double tmp;
	if (t <= 0.00029) {
		tmp = t_1 + (log(t) * (a - 0.5));
	} else {
		tmp = t_1 + ((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) :: t_1
    real(8) :: tmp
    t_1 = log(z) + log(y)
    if (t <= 0.00029d0) then
        tmp = t_1 + (log(t) * (a - 0.5d0))
    else
        tmp = t_1 + ((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 t_1 = Math.log(z) + Math.log(y);
	double tmp;
	if (t <= 0.00029) {
		tmp = t_1 + (Math.log(t) * (a - 0.5));
	} else {
		tmp = t_1 + ((a * Math.log(t)) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(z) + math.log(y)
	tmp = 0
	if t <= 0.00029:
		tmp = t_1 + (math.log(t) * (a - 0.5))
	else:
		tmp = t_1 + ((a * math.log(t)) - t)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(z) + log(y))
	tmp = 0.0
	if (t <= 0.00029)
		tmp = Float64(t_1 + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(t_1 + Float64(Float64(a * log(t)) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(z) + log(y);
	tmp = 0.0;
	if (t <= 0.00029)
		tmp = t_1 + (log(t) * (a - 0.5));
	else
		tmp = t_1 + ((a * log(t)) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 0.00029], N[(t$95$1 + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.9e-4

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 67.6%

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

   6. Taylor expanded in t around 0 67.3%

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

      1. +-commutative 67.3%

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

      2. *-commutative 67.3%

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

   8. Simplified 67.3%

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

   if 2.9e-4 < 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%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 75.2%

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

   6. Taylor expanded in a around inf 74.6%

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

      1. mul-1-neg 74.6%

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

      2. distribute-rgt-neg-in 74.6%

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

   8. Simplified 74.6%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00029:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) + \left(a \cdot \log t - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in x around 0 71.6%

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

6. Final simplification 71.6%

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

Alternative 7: 68.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+98}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-125}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{elif}\;a \leq 0.00052:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \left(t \cdot \frac{\log t}{t}\right) \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.5e+98)
   (* a (log t))
   (if (<= a 9.5e-125)
     (+ (log (* (+ x y) z)) (- (* (log t) (+ a -0.5)) t))
     (if (<= a 0.00052)
       (- (+ (log z) (log y)) t)
       (- (+ (log (* y z)) (* (* t (/ (log t) t)) (- a 0.5))) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.5e+98) {
		tmp = a * log(t);
	} else if (a <= 9.5e-125) {
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	} else if (a <= 0.00052) {
		tmp = (log(z) + log(y)) - t;
	} else {
		tmp = (log((y * z)) + ((t * (log(t) / t)) * (a - 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 <= (-5.5d+98)) then
        tmp = a * log(t)
    else if (a <= 9.5d-125) then
        tmp = log(((x + y) * z)) + ((log(t) * (a + (-0.5d0))) - t)
    else if (a <= 0.00052d0) then
        tmp = (log(z) + log(y)) - t
    else
        tmp = (log((y * z)) + ((t * (log(t) / t)) * (a - 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 <= -5.5e+98) {
		tmp = a * Math.log(t);
	} else if (a <= 9.5e-125) {
		tmp = Math.log(((x + y) * z)) + ((Math.log(t) * (a + -0.5)) - t);
	} else if (a <= 0.00052) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else {
		tmp = (Math.log((y * z)) + ((t * (Math.log(t) / t)) * (a - 0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -5.5e+98:
		tmp = a * math.log(t)
	elif a <= 9.5e-125:
		tmp = math.log(((x + y) * z)) + ((math.log(t) * (a + -0.5)) - t)
	elif a <= 0.00052:
		tmp = (math.log(z) + math.log(y)) - t
	else:
		tmp = (math.log((y * z)) + ((t * (math.log(t) / t)) * (a - 0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.5e+98)
		tmp = Float64(a * log(t));
	elseif (a <= 9.5e-125)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + Float64(Float64(log(t) * Float64(a + -0.5)) - t));
	elseif (a <= 0.00052)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	else
		tmp = Float64(Float64(log(Float64(y * z)) + Float64(Float64(t * Float64(log(t) / t)) * Float64(a - 0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -5.5e+98)
		tmp = a * log(t);
	elseif (a <= 9.5e-125)
		tmp = log(((x + y) * z)) + ((log(t) * (a + -0.5)) - t);
	elseif (a <= 0.00052)
		tmp = (log(z) + log(y)) - t;
	else
		tmp = (log((y * z)) + ((t * (log(t) / t)) * (a - 0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.5e+98], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.5e-125], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 0.00052], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(t * N[(N[Log[t], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.49999999999999946e98

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around inf 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if -5.49999999999999946e98 < a < 9.50000000000000031e-125

   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+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-undefine 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. associate-+r+ 99.5%

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

      6. associate--l+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+l- 99.6%

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

      9. sum-log 80.3%

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

      10. sub-neg 80.3%

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

      11. metadata-eval 80.3%

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

      12. *-commutative 80.3%

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

   6. Applied egg-rr 80.3%

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

   if 9.50000000000000031e-125 < a < 5.19999999999999954e-4

   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%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 74.9%

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

   6. Taylor expanded in t around inf 58.5%

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

   if 5.19999999999999954e-4 < a

   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%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 76.0%

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

   6. Taylor expanded in t around inf 56.5%

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

      1. associate-*r/ 56.5%

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

      2. mul-1-neg 56.5%

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

      3. log-rec 56.5%

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

      4. distribute-lft-neg-in 56.5%

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

      5. remove-double-neg 56.5%

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

      6. *-commutative 56.5%

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

   8. Simplified 56.5%

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

      1. +-commutative 56.5%

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

      2. distribute-rgt-in 56.5%

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

      3. associate-/l* 56.5%

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

      4. associate-*l* 76.0%

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

      5. *-un-lft-identity 76.0%

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

   10. Applied egg-rr 76.0%

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

       1. associate--r+ 76.0%

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

       2. sub-neg 76.0%

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

       3. sum-log 64.1%

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

       4. *-commutative 64.1%

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

       5. *-commutative 64.1%

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

   12. Applied egg-rr 64.1%

       \[\leadsto \color{blue}{\left(\log \left(y \cdot z\right) - \left(t \cdot \frac{\log t}{t}\right) \cdot \left(0.5 - a\right)\right) + \left(-t\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+98}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-125}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{elif}\;a \leq 0.00052:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(y \cdot z\right) + \left(t \cdot \frac{\log t}{t}\right) \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+94}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-125} \lor \neg \left(a \leq 0.00021\right):\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.5e+94)
   (* a (log t))
   (if (or (<= a 9.5e-125) (not (<= a 0.00021)))
     (+ (log (* y z)) (- (* (log t) (+ a -0.5)) t))
     (- (+ (log z) (log y)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.5e+94) {
		tmp = a * log(t);
	} else if ((a <= 9.5e-125) || !(a <= 0.00021)) {
		tmp = log((y * z)) + ((log(t) * (a + -0.5)) - t);
	} else {
		tmp = (log(z) + log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-1.5d+94)) then
        tmp = a * log(t)
    else if ((a <= 9.5d-125) .or. (.not. (a <= 0.00021d0))) then
        tmp = log((y * z)) + ((log(t) * (a + (-0.5d0))) - t)
    else
        tmp = (log(z) + log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.5e+94) {
		tmp = a * Math.log(t);
	} else if ((a <= 9.5e-125) || !(a <= 0.00021)) {
		tmp = Math.log((y * z)) + ((Math.log(t) * (a + -0.5)) - t);
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.5e+94:
		tmp = a * math.log(t)
	elif (a <= 9.5e-125) or not (a <= 0.00021):
		tmp = math.log((y * z)) + ((math.log(t) * (a + -0.5)) - t)
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.5e+94)
		tmp = Float64(a * log(t));
	elseif ((a <= 9.5e-125) || !(a <= 0.00021))
		tmp = Float64(log(Float64(y * z)) + Float64(Float64(log(t) * Float64(a + -0.5)) - t));
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.5e+94)
		tmp = a * log(t);
	elseif ((a <= 9.5e-125) || ~((a <= 0.00021)))
		tmp = log((y * z)) + ((log(t) * (a + -0.5)) - t);
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.5e+94], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a, 9.5e-125], N[Not[LessEqual[a, 0.00021]], $MachinePrecision]], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.5e94

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around inf 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if -1.5e94 < a < 9.50000000000000031e-125 or 2.1000000000000001e-4 < a

   1. Initial program 99.6%

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

      1. associate--l+ 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. +-commutative 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. add-cube-cbrt 98.1%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)} \cdot \sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}\right) \cdot \sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}} \]

      2. pow3 98.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}\right)}^{3}} \]

   6. Applied egg-rr 79.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\log t, a + -0.5, \log \left(\left(x + y\right) \cdot z\right) - t\right)}\right)}^{3}} \]

   7. Taylor expanded in x around 0 57.0%

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

      1. associate--l+ 57.0%

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

      2. *-commutative 57.0%

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

      3. sub-neg 57.0%

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

      4. metadata-eval 57.0%

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

   9. Simplified 57.0%

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

   if 9.50000000000000031e-125 < a < 2.1000000000000001e-4

   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%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 74.9%

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

   6. Taylor expanded in t around inf 58.5%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.5 \cdot 10^{+94}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-125} \lor \neg \left(a \leq 0.00021\right):\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot \left(a + -0.5\right) - t\\ \mathbf{if}\;a \leq -1.15 \cdot 10^{+101}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-125}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + t\_1\\ \mathbf{elif}\;a \leq 0.00021:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (* (log t) (+ a -0.5)) t)))
   (if (<= a -1.15e+101)
     (* a (log t))
     (if (<= a 9.5e-125)
       (+ (log (* (+ x y) z)) t_1)
       (if (<= a 0.00021) (- (+ (log z) (log y)) t) (+ (log (* y z)) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (log(t) * (a + -0.5)) - t;
	double tmp;
	if (a <= -1.15e+101) {
		tmp = a * log(t);
	} else if (a <= 9.5e-125) {
		tmp = log(((x + y) * z)) + t_1;
	} else if (a <= 0.00021) {
		tmp = (log(z) + log(y)) - t;
	} else {
		tmp = log((y * z)) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (log(t) * (a + (-0.5d0))) - t
    if (a <= (-1.15d+101)) then
        tmp = a * log(t)
    else if (a <= 9.5d-125) then
        tmp = log(((x + y) * z)) + t_1
    else if (a <= 0.00021d0) then
        tmp = (log(z) + log(y)) - t
    else
        tmp = log((y * z)) + t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (Math.log(t) * (a + -0.5)) - t;
	double tmp;
	if (a <= -1.15e+101) {
		tmp = a * Math.log(t);
	} else if (a <= 9.5e-125) {
		tmp = Math.log(((x + y) * z)) + t_1;
	} else if (a <= 0.00021) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else {
		tmp = Math.log((y * z)) + t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (math.log(t) * (a + -0.5)) - t
	tmp = 0
	if a <= -1.15e+101:
		tmp = a * math.log(t)
	elif a <= 9.5e-125:
		tmp = math.log(((x + y) * z)) + t_1
	elif a <= 0.00021:
		tmp = (math.log(z) + math.log(y)) - t
	else:
		tmp = math.log((y * z)) + t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(log(t) * Float64(a + -0.5)) - t)
	tmp = 0.0
	if (a <= -1.15e+101)
		tmp = Float64(a * log(t));
	elseif (a <= 9.5e-125)
		tmp = Float64(log(Float64(Float64(x + y) * z)) + t_1);
	elseif (a <= 0.00021)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	else
		tmp = Float64(log(Float64(y * z)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (log(t) * (a + -0.5)) - t;
	tmp = 0.0;
	if (a <= -1.15e+101)
		tmp = a * log(t);
	elseif (a <= 9.5e-125)
		tmp = log(((x + y) * z)) + t_1;
	elseif (a <= 0.00021)
		tmp = (log(z) + log(y)) - t;
	else
		tmp = log((y * z)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[Log[t], $MachinePrecision] * N[(a + -0.5), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]}, If[LessEqual[a, -1.15e+101], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.5e-125], N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[a, 0.00021], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -1.1500000000000001e101

   1. Initial program 99.5%

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

      1. associate-+l- 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in a around inf 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

   if -1.1500000000000001e101 < a < 9.50000000000000031e-125

   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+ 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. +-commutative 99.5%

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

      2. fma-undefine 99.5%

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

      3. metadata-eval 99.5%

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

      4. sub-neg 99.5%

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

      5. associate-+r+ 99.5%

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

      6. associate--l+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+l- 99.6%

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

      9. sum-log 80.3%

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

      10. sub-neg 80.3%

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

      11. metadata-eval 80.3%

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

      12. *-commutative 80.3%

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

   6. Applied egg-rr 80.3%

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

   if 9.50000000000000031e-125 < a < 2.1000000000000001e-4

   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%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. +-commutative 99.7%

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

      5. *-commutative 99.7%

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

      6. distribute-rgt-neg-in 99.7%

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

      7. fma-undefine 99.7%

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

      8. sub-neg 99.7%

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

      9. +-commutative 99.7%

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

      10. distribute-neg-in 99.7%

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

      11. metadata-eval 99.7%

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

      12. metadata-eval 99.7%

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

      13. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 74.9%

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

   6. Taylor expanded in t around inf 58.5%

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

   if 2.1000000000000001e-4 < a

   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. +-commutative 99.8%

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

      3. associate-+l+ 99.8%

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

      4. +-commutative 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. add-cube-cbrt 97.8%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)} \cdot \sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}\right) \cdot \sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}} \]

      2. pow3 98.0%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}\right)}^{3}} \]

   6. Applied egg-rr 81.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\log t, a + -0.5, \log \left(\left(x + y\right) \cdot z\right) - t\right)}\right)}^{3}} \]

   7. Taylor expanded in x around 0 64.1%

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

      1. associate--l+ 64.1%

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

      2. *-commutative 64.1%

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

      3. sub-neg 64.1%

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

      4. metadata-eval 64.1%

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

   9. Simplified 64.1%

      \[\leadsto \color{blue}{\log \left(z \cdot y\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.15 \cdot 10^{+101}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-125}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \mathbf{elif}\;a \leq 0.00021:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot \left(a + -0.5\right) - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -7.4 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-126}:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot -0.5 - t\right)\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+64}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+130} \lor \neg \left(a \leq 6 \cdot 10^{+165}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -7.4e+17)
     t_1
     (if (<= a 8e-126)
       (+ (log (* y z)) (- (* (log t) -0.5) t))
       (if (<= a 7e+64)
         (- (+ (log z) (log y)) t)
         (if (or (<= a 2.2e+130) (not (<= a 6e+165))) 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 (a <= -7.4e+17) {
		tmp = t_1;
	} else if (a <= 8e-126) {
		tmp = log((y * z)) + ((log(t) * -0.5) - t);
	} else if (a <= 7e+64) {
		tmp = (log(z) + log(y)) - t;
	} else if ((a <= 2.2e+130) || !(a <= 6e+165)) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-7.4d+17)) then
        tmp = t_1
    else if (a <= 8d-126) then
        tmp = log((y * z)) + ((log(t) * (-0.5d0)) - t)
    else if (a <= 7d+64) then
        tmp = (log(z) + log(y)) - t
    else if ((a <= 2.2d+130) .or. (.not. (a <= 6d+165))) then
        tmp = t_1
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -7.4e+17) {
		tmp = t_1;
	} else if (a <= 8e-126) {
		tmp = Math.log((y * z)) + ((Math.log(t) * -0.5) - t);
	} else if (a <= 7e+64) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else if ((a <= 2.2e+130) || !(a <= 6e+165)) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -7.4e+17:
		tmp = t_1
	elif a <= 8e-126:
		tmp = math.log((y * z)) + ((math.log(t) * -0.5) - t)
	elif a <= 7e+64:
		tmp = (math.log(z) + math.log(y)) - t
	elif (a <= 2.2e+130) or not (a <= 6e+165):
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -7.4e+17)
		tmp = t_1;
	elseif (a <= 8e-126)
		tmp = Float64(log(Float64(y * z)) + Float64(Float64(log(t) * -0.5) - t));
	elseif (a <= 7e+64)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	elseif ((a <= 2.2e+130) || !(a <= 6e+165))
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -7.4e+17)
		tmp = t_1;
	elseif (a <= 8e-126)
		tmp = log((y * z)) + ((log(t) * -0.5) - t);
	elseif (a <= 7e+64)
		tmp = (log(z) + log(y)) - t;
	elseif ((a <= 2.2e+130) || ~((a <= 6e+165)))
		tmp = t_1;
	else
		tmp = -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[a, -7.4e+17], t$95$1, If[LessEqual[a, 8e-126], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 7e+64], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[a, 2.2e+130], N[Not[LessEqual[a, 6e+165]], $MachinePrecision]], t$95$1, (-t)]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -7.4e17 or 6.9999999999999997e64 < a < 2.19999999999999993e130 or 5.99999999999999981e165 < a

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around inf 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

   if -7.4e17 < a < 7.9999999999999996e-126

   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. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. add-cube-cbrt 98.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)} \cdot \sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}\right) \cdot \sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}} \]

      2. pow3 98.1%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\log z - t\right) + \mathsf{fma}\left(a + -0.5, \log t, \log \left(x + y\right)\right)}\right)}^{3}} \]

   6. Applied egg-rr 77.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(\log t, a + -0.5, \log \left(\left(x + y\right) \cdot z\right) - t\right)}\right)}^{3}} \]

   7. Taylor expanded in x around 0 52.4%

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

      1. associate--l+ 52.5%

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

      2. *-commutative 52.5%

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

      3. sub-neg 52.5%

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

      4. metadata-eval 52.5%

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

   9. Simplified 52.5%

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

   10. Taylor expanded in a around 0 50.6%

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

       1. *-commutative 50.6%

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

   12. Simplified 50.6%

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

   if 7.9999999999999996e-126 < a < 6.9999999999999997e64

   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%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. +-commutative 99.8%

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

      5. *-commutative 99.8%

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

      6. distribute-rgt-neg-in 99.8%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 77.7%

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

   6. Taylor expanded in t around inf 57.1%

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

   if 2.19999999999999993e130 < a < 5.99999999999999981e165

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l+ 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-undefine 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 75.7%

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

      1. neg-mul-1 75.7%

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

   7. Simplified 75.7%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.4 \cdot 10^{+17}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-126}:\\ \;\;\;\;\log \left(y \cdot z\right) + \left(\log t \cdot -0.5 - t\right)\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+64}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+130} \lor \neg \left(a \leq 6 \cdot 10^{+165}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \log t\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+64}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+130} \lor \neg \left(a \leq 6 \cdot 10^{+165}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (log t))))
   (if (<= a -6.5e+17)
     t_1
     (if (<= a 7e+64)
       (- (+ (log z) (log y)) t)
       (if (or (<= a 1.8e+130) (not (<= a 6e+165))) 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 (a <= -6.5e+17) {
		tmp = t_1;
	} else if (a <= 7e+64) {
		tmp = (log(z) + log(y)) - t;
	} else if ((a <= 1.8e+130) || !(a <= 6e+165)) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = a * log(t)
    if (a <= (-6.5d+17)) then
        tmp = t_1
    else if (a <= 7d+64) then
        tmp = (log(z) + log(y)) - t
    else if ((a <= 1.8d+130) .or. (.not. (a <= 6d+165))) then
        tmp = t_1
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * Math.log(t);
	double tmp;
	if (a <= -6.5e+17) {
		tmp = t_1;
	} else if (a <= 7e+64) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else if ((a <= 1.8e+130) || !(a <= 6e+165)) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * math.log(t)
	tmp = 0
	if a <= -6.5e+17:
		tmp = t_1
	elif a <= 7e+64:
		tmp = (math.log(z) + math.log(y)) - t
	elif (a <= 1.8e+130) or not (a <= 6e+165):
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * log(t))
	tmp = 0.0
	if (a <= -6.5e+17)
		tmp = t_1;
	elseif (a <= 7e+64)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	elseif ((a <= 1.8e+130) || !(a <= 6e+165))
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * log(t);
	tmp = 0.0;
	if (a <= -6.5e+17)
		tmp = t_1;
	elseif (a <= 7e+64)
		tmp = (log(z) + log(y)) - t;
	elseif ((a <= 1.8e+130) || ~((a <= 6e+165)))
		tmp = t_1;
	else
		tmp = -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[a, -6.5e+17], t$95$1, If[LessEqual[a, 7e+64], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[a, 1.8e+130], N[Not[LessEqual[a, 6e+165]], $MachinePrecision]], t$95$1, (-t)]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.5e17 or 6.9999999999999997e64 < a < 1.8000000000000001e130 or 5.99999999999999981e165 < a

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around inf 83.3%

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

      1. *-commutative 83.3%

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

   7. Simplified 83.3%

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

   if -6.5e17 < a < 6.9999999999999997e64

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 69.5%

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

   6. Taylor expanded in t around inf 45.7%

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

   if 1.8000000000000001e130 < a < 5.99999999999999981e165

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. associate--l+ 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. *-commutative 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. fma-undefine 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. metadata-eval 100.0%

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

      13. unsub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 75.7%

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

      1. neg-mul-1 75.7%

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

   7. Simplified 75.7%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+17}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{elif}\;a \leq 7 \cdot 10^{+64}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+130} \lor \neg \left(a \leq 6 \cdot 10^{+165}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.05 \cdot 10^{+38}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+120} \lor \neg \left(t \leq 1.25 \cdot 10^{+144}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.05e+38)
   (+ (log (* y z)) (* (log t) (- a 0.5)))
   (if (or (<= t 3.8e+120) (not (<= t 1.25e+144))) (- t) (* a (log t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.05e+38) {
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	} else if ((t <= 3.8e+120) || !(t <= 1.25e+144)) {
		tmp = -t;
	} else {
		tmp = a * log(t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 3.05d+38) then
        tmp = log((y * z)) + (log(t) * (a - 0.5d0))
    else if ((t <= 3.8d+120) .or. (.not. (t <= 1.25d+144))) then
        tmp = -t
    else
        tmp = a * log(t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.05e+38) {
		tmp = Math.log((y * z)) + (Math.log(t) * (a - 0.5));
	} else if ((t <= 3.8e+120) || !(t <= 1.25e+144)) {
		tmp = -t;
	} else {
		tmp = a * Math.log(t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.05e+38:
		tmp = math.log((y * z)) + (math.log(t) * (a - 0.5))
	elif (t <= 3.8e+120) or not (t <= 1.25e+144):
		tmp = -t
	else:
		tmp = a * math.log(t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 3.05e+38)
		tmp = Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5)));
	elseif ((t <= 3.8e+120) || !(t <= 1.25e+144))
		tmp = Float64(-t);
	else
		tmp = Float64(a * log(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 3.05e+38)
		tmp = log((y * z)) + (log(t) * (a - 0.5));
	elseif ((t <= 3.8e+120) || ~((t <= 1.25e+144)))
		tmp = -t;
	else
		tmp = a * log(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.05e+38], N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, 3.8e+120], N[Not[LessEqual[t, 1.25e+144]], $MachinePrecision]], (-t), N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 3.05e38

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. fma-undefine 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 69.8%

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

   6. Taylor expanded in t around 0 67.6%

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

      1. +-commutative 67.6%

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

      2. *-commutative 67.6%

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

   8. Simplified 67.6%

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

      1. sum-log 55.4%

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

   10. Applied egg-rr 55.4%

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

   if 3.05e38 < t < 3.7999999999999998e120 or 1.25e144 < 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%

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

      2. associate--l+ 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. distribute-rgt-neg-in 99.9%

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

      7. fma-undefine 99.9%

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

      8. sub-neg 99.9%

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

      9. +-commutative 99.9%

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

      10. distribute-neg-in 99.9%

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

      11. metadata-eval 99.9%

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

      12. metadata-eval 99.9%

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

      13. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 82.8%

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

      1. neg-mul-1 82.8%

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

   7. Simplified 82.8%

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

   if 3.7999999999999998e120 < t < 1.25e144

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.8%

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

      8. sub-neg 99.8%

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

      9. +-commutative 99.8%

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

      10. distribute-neg-in 99.8%

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

      11. metadata-eval 99.8%

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

      12. metadata-eval 99.8%

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

      13. unsub-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in a around inf 66.7%

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

      1. *-commutative 66.7%

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

   7. Simplified 66.7%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.05 \cdot 10^{+38}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+120} \lor \neg \left(t \leq 1.25 \cdot 10^{+144}\right):\\ \;\;\;\;-t\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+17} \lor \neg \left(a \leq 7.8 \cdot 10^{+64}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.5e+17) (not (<= a 7.8e+64))) (* a (log t)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e+17) || !(a <= 7.8e+64)) {
		tmp = a * log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-6.5d+17)) .or. (.not. (a <= 7.8d+64))) then
        tmp = a * log(t)
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e+17) || !(a <= 7.8e+64)) {
		tmp = a * Math.log(t);
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6.5e+17) or not (a <= 7.8e+64):
		tmp = a * math.log(t)
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.5e+17) || !(a <= 7.8e+64))
		tmp = Float64(a * log(t));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -6.5e+17) || ~((a <= 7.8e+64)))
		tmp = a * log(t);
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.5e+17], N[Not[LessEqual[a, 7.8e+64]], $MachinePrecision]], N[(a * N[Log[t], $MachinePrecision]), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if a < -6.5e17 or 7.7999999999999996e64 < a

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. fma-undefine 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in a around inf 79.3%

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

      1. *-commutative 79.3%

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

   7. Simplified 79.3%

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

   if -6.5e17 < a < 7.7999999999999996e64

   1. Initial program 99.6%

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

      1. associate-+l- 99.6%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. fma-undefine 99.5%

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

      8. sub-neg 99.5%

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

      9. +-commutative 99.5%

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

      10. distribute-neg-in 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. unsub-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in t around inf 52.1%

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

      1. neg-mul-1 52.1%

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

   7. Simplified 52.1%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+17} \lor \neg \left(a \leq 7.8 \cdot 10^{+64}\right):\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.9% 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%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-undefine 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 37.9%

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

   1. neg-mul-1 37.9%

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

7. Simplified 37.9%

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

8. Final simplification 37.9%

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

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024066 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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