Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 28.2s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. +-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-rgt-neg-in 99.6%

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

   7. fma-udef 99.6%

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

   8. sub-neg 99.6%

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

   9. +-commutative 99.6%

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

   10. distribute-neg-in 99.6%

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

   11. metadata-eval 99.6%

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

   12. metadata-eval 99.6%

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

   13. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -680000000000 \lor \neg \left(a \leq 1.4\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -680000000000.0) (not (<= a 1.4)))
   (- (+ (log y) (* (log t) a)) t)
   (- (+ (log y) (+ (log z) (* (log t) -0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -680000000000.0) || !(a <= 1.4)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-680000000000.0d0)) .or. (.not. (a <= 1.4d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = (log(y) + (log(z) + (log(t) * (-0.5d0)))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -680000000000.0) || !(a <= 1.4)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = (Math.log(y) + (Math.log(z) + (Math.log(t) * -0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -680000000000.0) or not (a <= 1.4):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = (math.log(y) + (math.log(z) + (math.log(t) * -0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -680000000000.0) || !(a <= 1.4))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(Float64(log(y) + Float64(log(z) + Float64(log(t) * -0.5))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -680000000000.0) || ~((a <= 1.4)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = (log(y) + (log(z) + (log(t) * -0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -680000000000.0], N[Not[LessEqual[a, 1.4]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[z], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.8e11 or 1.3999999999999999 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 74.2%

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

   4. Taylor expanded in a around inf 74.2%

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

      1. *-commutative 74.2%

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

   6. Simplified 74.2%

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

   if -6.8e11 < a < 1.3999999999999999

   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. Add Preprocessing

   3. Taylor expanded in x around 0 63.6%

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

   4. Taylor expanded in a around 0 62.3%

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

      1. *-commutative 62.3%

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

   6. Simplified 62.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -680000000000 \lor \neg \left(a \leq 1.4\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.00016:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 0.00016)
   (+ (+ (log z) (log y)) (* (log t) (- a 0.5)))
   (- (+ (log y) (* (log t) a)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 0.00016) {
		tmp = (log(z) + log(y)) + (log(t) * (a - 0.5));
	} else {
		tmp = (log(y) + (log(t) * a)) - 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 <= 0.00016d0) then
        tmp = (log(z) + log(y)) + (log(t) * (a - 0.5d0))
    else
        tmp = (log(y) + (log(t) * a)) - 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 <= 0.00016) {
		tmp = (Math.log(z) + Math.log(y)) + (Math.log(t) * (a - 0.5));
	} else {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 0.00016:
		tmp = (math.log(z) + math.log(y)) + (math.log(t) * (a - 0.5))
	else:
		tmp = (math.log(y) + (math.log(t) * a)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 0.00016)
		tmp = Float64(Float64(log(z) + log(y)) + Float64(log(t) * Float64(a - 0.5)));
	else
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 0.00016)
		tmp = (log(z) + log(y)) + (log(t) * (a - 0.5));
	else
		tmp = (log(y) + (log(t) * a)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 0.00016], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.60000000000000013e-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+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-udef 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in 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. log-prod 76.0%

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

      2. +-commutative 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in y around inf 67.8%

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

      1. mul-1-neg 67.8%

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

      2. unsub-neg 67.8%

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

      3. log-rec 67.8%

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

   10. Simplified 67.8%

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

   if 1.60000000000000013e-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. Add Preprocessing

   3. Taylor expanded in x around 0 70.2%

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

   4. Taylor expanded in a around inf 70.2%

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

      1. *-commutative 70.2%

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

   6. Simplified 70.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 0.00016:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 5: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0 69.1%

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

4. Final simplification 69.1%

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

Alternative 6: 56.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -2900000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;\log y - t\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-119}:\\ \;\;\;\;\log \left(\frac{\left(x + y\right) \cdot z}{\sqrt{t}}\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{+48} \lor \neg \left(a \leq 1.65 \cdot 10^{+74}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= a -2900000000000.0)
     t_1
     (if (<= a -1.2e-71)
       (- (log y) t)
       (if (<= a -3.7e-119)
         (log (/ (* (+ x y) z) (sqrt t)))
         (if (<= a 1.65e+31)
           (- (+ (log z) (log y)) t)
           (if (or (<= a 4.9e+48) (not (<= a 1.65e+74))) t_1 (- t))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if (a <= -2900000000000.0) {
		tmp = t_1;
	} else if (a <= -1.2e-71) {
		tmp = log(y) - t;
	} else if (a <= -3.7e-119) {
		tmp = log((((x + y) * z) / sqrt(t)));
	} else if (a <= 1.65e+31) {
		tmp = (log(z) + log(y)) - t;
	} else if ((a <= 4.9e+48) || !(a <= 1.65e+74)) {
		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 = log(t) * a
    if (a <= (-2900000000000.0d0)) then
        tmp = t_1
    else if (a <= (-1.2d-71)) then
        tmp = log(y) - t
    else if (a <= (-3.7d-119)) then
        tmp = log((((x + y) * z) / sqrt(t)))
    else if (a <= 1.65d+31) then
        tmp = (log(z) + log(y)) - t
    else if ((a <= 4.9d+48) .or. (.not. (a <= 1.65d+74))) 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 = Math.log(t) * a;
	double tmp;
	if (a <= -2900000000000.0) {
		tmp = t_1;
	} else if (a <= -1.2e-71) {
		tmp = Math.log(y) - t;
	} else if (a <= -3.7e-119) {
		tmp = Math.log((((x + y) * z) / Math.sqrt(t)));
	} else if (a <= 1.65e+31) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else if ((a <= 4.9e+48) || !(a <= 1.65e+74)) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if a <= -2900000000000.0:
		tmp = t_1
	elif a <= -1.2e-71:
		tmp = math.log(y) - t
	elif a <= -3.7e-119:
		tmp = math.log((((x + y) * z) / math.sqrt(t)))
	elif a <= 1.65e+31:
		tmp = (math.log(z) + math.log(y)) - t
	elif (a <= 4.9e+48) or not (a <= 1.65e+74):
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (a <= -2900000000000.0)
		tmp = t_1;
	elseif (a <= -1.2e-71)
		tmp = Float64(log(y) - t);
	elseif (a <= -3.7e-119)
		tmp = log(Float64(Float64(Float64(x + y) * z) / sqrt(t)));
	elseif (a <= 1.65e+31)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	elseif ((a <= 4.9e+48) || !(a <= 1.65e+74))
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if (a <= -2900000000000.0)
		tmp = t_1;
	elseif (a <= -1.2e-71)
		tmp = log(y) - t;
	elseif (a <= -3.7e-119)
		tmp = log((((x + y) * z) / sqrt(t)));
	elseif (a <= 1.65e+31)
		tmp = (log(z) + log(y)) - t;
	elseif ((a <= 4.9e+48) || ~((a <= 1.65e+74)))
		tmp = t_1;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -2900000000000.0], t$95$1, If[LessEqual[a, -1.2e-71], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], If[LessEqual[a, -3.7e-119], N[Log[N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[a, 1.65e+31], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[a, 4.9e+48], N[Not[LessEqual[a, 1.65e+74]], $MachinePrecision]], t$95$1, (-t)]]]]]]

Derivation

1. Split input into 5 regimes

2. if a < -2.9e12 or 1.64999999999999996e31 < a < 4.9000000000000003e48 or 1.6500000000000001e74 < 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-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around 0 83.3%

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

      1. log-prod 68.3%

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

      2. +-commutative 68.3%

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

   7. Simplified 68.3%

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

      1. add-sqr-sqrt 39.9%

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

      2. pow2 39.8%

         \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]

   9. Applied egg-rr 39.8%

      \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 39.9%

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

       2. add-sqr-sqrt 68.3%

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

       3. add-sqr-sqrt 34.8%

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

       4. associate-*r* 34.8%

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

   11. Applied egg-rr 34.8%

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

   12. Taylor expanded in a around inf 83.3%

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

       1. *-commutative 83.3%

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

   14. Simplified 83.3%

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

   if -2.9e12 < a < -1.2e-71

   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-udef 99.3%

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

      8. sub-neg 99.3%

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

      9. +-commutative 99.3%

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

      10. distribute-neg-in 99.3%

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

      11. metadata-eval 99.3%

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

      12. metadata-eval 99.3%

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

      13. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around inf 60.9%

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

      1. neg-mul-1 60.9%

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

   7. Simplified 60.9%

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

   8. Taylor expanded in x around 0 14.5%

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

   if -1.2e-71 < a < -3.7000000000000001e-119

   1. Initial program 99.2%

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

      1. associate-+l- 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. +-commutative 99.2%

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

      5. *-commutative 99.2%

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

      6. distribute-rgt-neg-in 99.2%

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

      7. fma-udef 99.2%

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

      8. sub-neg 99.2%

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

      9. +-commutative 99.2%

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

      10. distribute-neg-in 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. unsub-neg 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in t around 0 89.3%

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

      1. log-prod 80.0%

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

      2. +-commutative 80.0%

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

   7. Simplified 80.0%

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

      1. add-log-exp 80.0%

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

      2. diff-log 70.7%

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

      3. exp-to-pow 70.7%

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

   9. Applied egg-rr 70.7%

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

   10. Taylor expanded in a around 0 70.7%

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

   if -3.7000000000000001e-119 < a < 1.64999999999999996e31

   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. Add Preprocessing

   3. Taylor expanded in x around 0 69.9%

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

      1. +-commutative 69.9%

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

      2. add-cube-cbrt 69.8%

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

      3. fma-def 69.8%

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

      4. pow2 69.8%

         \[\leadsto \left(\log y + \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\log t \cdot \left(a - 0.5\right)}\right)}^{2}}, \sqrt[3]{\log t \cdot \left(a - 0.5\right)}, \log z\right)\right) - t \]

      5. sub-neg 69.8%

         \[\leadsto \left(\log y + \mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \color{blue}{\left(a + \left(-0.5\right)\right)}}\right)}^{2}, \sqrt[3]{\log t \cdot \left(a - 0.5\right)}, \log z\right)\right) - t \]

      6. metadata-eval 69.8%

         \[\leadsto \left(\log y + \mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \left(a + \color{blue}{-0.5}\right)}\right)}^{2}, \sqrt[3]{\log t \cdot \left(a - 0.5\right)}, \log z\right)\right) - t \]

      7. sub-neg 69.8%

         \[\leadsto \left(\log y + \mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \left(a + -0.5\right)}\right)}^{2}, \sqrt[3]{\log t \cdot \color{blue}{\left(a + \left(-0.5\right)\right)}}, \log z\right)\right) - t \]

      8. metadata-eval 69.8%

         \[\leadsto \left(\log y + \mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \left(a + -0.5\right)}\right)}^{2}, \sqrt[3]{\log t \cdot \left(a + \color{blue}{-0.5}\right)}, \log z\right)\right) - t \]

   5. Applied egg-rr 69.8%

      \[\leadsto \left(\log y + \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \left(a + -0.5\right)}\right)}^{2}, \sqrt[3]{\log t \cdot \left(a + -0.5\right)}, \log z\right)}\right) - t \]

   6. Taylor expanded in a around inf 43.6%

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

   if 4.9000000000000003e48 < a < 1.6500000000000001e74

   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. Add Preprocessing

   3. Taylor expanded in x around 0 60.0%

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

   4. Taylor expanded in t around inf 80.6%

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

      1. mul-1-neg 80.6%

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

   6. Simplified 80.6%

      \[\leadsto \color{blue}{-t} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2900000000000:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a \leq -1.2 \cdot 10^{-71}:\\ \;\;\;\;\log y - t\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-119}:\\ \;\;\;\;\log \left(\frac{\left(x + y\right) \cdot z}{\sqrt{t}}\right)\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 4.9 \cdot 10^{+48} \lor \neg \left(a \leq 1.65 \cdot 10^{+74}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ t_2 := \log t \cdot a\\ \mathbf{if}\;t \leq 1.36 \cdot 10^{-151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;\log \left(x + y\right) + t_2\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + t_2\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (* y z)) (* (log t) (- a 0.5)))) (t_2 (* (log t) a)))
   (if (<= t 1.36e-151)
     t_1
     (if (<= t 8.5e-125)
       (+ (log (+ x y)) t_2)
       (if (<= t 3.1e-5) t_1 (- (+ (log y) t_2) t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((y * z)) + (log(t) * (a - 0.5));
	double t_2 = log(t) * a;
	double tmp;
	if (t <= 1.36e-151) {
		tmp = t_1;
	} else if (t <= 8.5e-125) {
		tmp = log((x + y)) + t_2;
	} else if (t <= 3.1e-5) {
		tmp = t_1;
	} else {
		tmp = (log(y) + t_2) - 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) :: t_2
    real(8) :: tmp
    t_1 = log((y * z)) + (log(t) * (a - 0.5d0))
    t_2 = log(t) * a
    if (t <= 1.36d-151) then
        tmp = t_1
    else if (t <= 8.5d-125) then
        tmp = log((x + y)) + t_2
    else if (t <= 3.1d-5) then
        tmp = t_1
    else
        tmp = (log(y) + t_2) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = Math.log((y * z)) + (Math.log(t) * (a - 0.5));
	double t_2 = Math.log(t) * a;
	double tmp;
	if (t <= 1.36e-151) {
		tmp = t_1;
	} else if (t <= 8.5e-125) {
		tmp = Math.log((x + y)) + t_2;
	} else if (t <= 3.1e-5) {
		tmp = t_1;
	} else {
		tmp = (Math.log(y) + t_2) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((y * z)) + (math.log(t) * (a - 0.5))
	t_2 = math.log(t) * a
	tmp = 0
	if t <= 1.36e-151:
		tmp = t_1
	elif t <= 8.5e-125:
		tmp = math.log((x + y)) + t_2
	elif t <= 3.1e-5:
		tmp = t_1
	else:
		tmp = (math.log(y) + t_2) - t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(y * z)) + Float64(log(t) * Float64(a - 0.5)))
	t_2 = Float64(log(t) * a)
	tmp = 0.0
	if (t <= 1.36e-151)
		tmp = t_1;
	elseif (t <= 8.5e-125)
		tmp = Float64(log(Float64(x + y)) + t_2);
	elseif (t <= 3.1e-5)
		tmp = t_1;
	else
		tmp = Float64(Float64(log(y) + t_2) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((y * z)) + (log(t) * (a - 0.5));
	t_2 = log(t) * a;
	tmp = 0.0;
	if (t <= 1.36e-151)
		tmp = t_1;
	elseif (t <= 8.5e-125)
		tmp = log((x + y)) + t_2;
	elseif (t <= 3.1e-5)
		tmp = t_1;
	else
		tmp = (log(y) + t_2) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[t, 1.36e-151], t$95$1, If[LessEqual[t, 8.5e-125], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[t, 3.1e-5], t$95$1, N[(N[(N[Log[y], $MachinePrecision] + t$95$2), $MachinePrecision] - t), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.35999999999999994e-151 or 8.5000000000000002e-125 < t < 3.10000000000000014e-5

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-udef 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in 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. log-prod 79.3%

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

      2. +-commutative 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in x around 0 50.5%

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

      1. *-commutative 50.5%

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

   10. Simplified 50.5%

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

   if 1.35999999999999994e-151 < t < 8.5000000000000002e-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- 99.6%

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

      2. associate--l+ 99.7%

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

      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-udef 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 a around inf 77.7%

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

      1. *-commutative 77.7%

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

   7. Simplified 77.7%

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

   if 3.10000000000000014e-5 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 70.2%

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

   4. Taylor expanded in a around inf 70.2%

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

      1. *-commutative 70.2%

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

   6. Simplified 70.2%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.36 \cdot 10^{-151}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;\log \left(x + y\right) + \log t \cdot a\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;\log \left(y \cdot z\right) + \log t \cdot \left(a - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+38} \lor \neg \left(a \leq 920\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5e+38) (not (<= a 920.0)))
   (- (+ (log y) (* (log t) a)) t)
   (- (+ (log (* (+ x y) z)) (* (log t) (- a 0.5))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5e+38) || !(a <= 920.0)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = (log(((x + y) * z)) + (log(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 <= (-5d+38)) .or. (.not. (a <= 920.0d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = (log(((x + y) * z)) + (log(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 <= -5e+38) || !(a <= 920.0)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = (Math.log(((x + y) * z)) + (Math.log(t) * (a - 0.5))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -5e+38) or not (a <= 920.0):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = (math.log(((x + y) * z)) + (math.log(t) * (a - 0.5))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5e+38) || !(a <= 920.0))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(Float64(log(Float64(Float64(x + y) * z)) + Float64(log(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 <= -5e+38) || ~((a <= 920.0)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = (log(((x + y) * z)) + (log(t) * (a - 0.5))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5e+38], N[Not[LessEqual[a, 920.0]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.9999999999999997e38 or 920 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 73.8%

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

   4. Taylor expanded in a around inf 73.8%

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

      1. *-commutative 73.8%

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

   6. Simplified 73.8%

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

   if -4.9999999999999997e38 < a < 920

   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-udef 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. Step-by-step derivation

      1. associate-+r- 99.5%

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

      2. fma-udef 99.5%

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

      3. associate--r+ 99.5%

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

      4. +-commutative 99.5%

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

      5. sum-log 77.8%

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

   6. Applied egg-rr 77.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5 \cdot 10^{+38} \lor \neg \left(a \leq 920\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log \left(\left(x + y\right) \cdot z\right) + \log t \cdot \left(a - 0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(x + y\right) + \log t \cdot a\\ \mathbf{if}\;t \leq 4.5 \cdot 10^{-80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-69}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot \frac{z}{\sqrt{t}}\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+49}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (log (+ x y)) (* (log t) a))))
   (if (<= t 4.5e-80)
     t_1
     (if (<= t 8.5e-69)
       (log (* (+ x y) (/ z (sqrt t))))
       (if (<= t 4.3e+49) t_1 (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log((x + y)) + (log(t) * a);
	double tmp;
	if (t <= 4.5e-80) {
		tmp = t_1;
	} else if (t <= 8.5e-69) {
		tmp = log(((x + y) * (z / sqrt(t))));
	} else if (t <= 4.3e+49) {
		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 = log((x + y)) + (log(t) * a)
    if (t <= 4.5d-80) then
        tmp = t_1
    else if (t <= 8.5d-69) then
        tmp = log(((x + y) * (z / sqrt(t))))
    else if (t <= 4.3d+49) 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 = Math.log((x + y)) + (Math.log(t) * a);
	double tmp;
	if (t <= 4.5e-80) {
		tmp = t_1;
	} else if (t <= 8.5e-69) {
		tmp = Math.log(((x + y) * (z / Math.sqrt(t))));
	} else if (t <= 4.3e+49) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log((x + y)) + (math.log(t) * a)
	tmp = 0
	if t <= 4.5e-80:
		tmp = t_1
	elif t <= 8.5e-69:
		tmp = math.log(((x + y) * (z / math.sqrt(t))))
	elif t <= 4.3e+49:
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(Float64(x + y)) + Float64(log(t) * a))
	tmp = 0.0
	if (t <= 4.5e-80)
		tmp = t_1;
	elseif (t <= 8.5e-69)
		tmp = log(Float64(Float64(x + y) * Float64(z / sqrt(t))));
	elseif (t <= 4.3e+49)
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log((x + y)) + (log(t) * a);
	tmp = 0.0;
	if (t <= 4.5e-80)
		tmp = t_1;
	elseif (t <= 8.5e-69)
		tmp = log(((x + y) * (z / sqrt(t))));
	elseif (t <= 4.3e+49)
		tmp = t_1;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 4.5e-80], t$95$1, If[LessEqual[t, 8.5e-69], N[Log[N[(N[(x + y), $MachinePrecision] * N[(z / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 4.3e+49], t$95$1, (-t)]]]]

Derivation

1. Split input into 3 regimes

2. if t < 4.5000000000000003e-80 or 8.50000000000000046e-69 < t < 4.2999999999999999e49

   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+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-udef 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in a around inf 61.1%

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

      1. *-commutative 61.1%

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

   7. Simplified 61.1%

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

   if 4.5000000000000003e-80 < t < 8.50000000000000046e-69

   1. Initial program 99.0%

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

      1. associate-+l- 99.0%

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

      2. associate--l+ 98.9%

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

      3. sub-neg 98.9%

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

      4. +-commutative 98.9%

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

      5. *-commutative 98.9%

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

      6. distribute-rgt-neg-in 98.9%

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

      7. fma-udef 98.9%

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

      8. sub-neg 98.9%

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

      9. +-commutative 98.9%

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

      10. distribute-neg-in 98.9%

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

      11. metadata-eval 98.9%

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

      12. metadata-eval 98.9%

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

      13. unsub-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in t around 0 99.0%

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

      1. log-prod 85.7%

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

      2. +-commutative 85.7%

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

   7. Simplified 85.7%

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

      1. add-log-exp 85.7%

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

      2. diff-log 85.9%

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

      3. exp-to-pow 86.2%

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

   9. Applied egg-rr 86.2%

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

       1. associate-/l* 86.2%

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

       2. log-div 98.7%

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

   11. Applied egg-rr 98.7%

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

   12. Taylor expanded in a around 0 87.6%

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

       1. associate-*r/ 87.6%

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

       2. *-rgt-identity 87.6%

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

       3. +-commutative 87.6%

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

       4. log-div 74.9%

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

       5. associate-/r/ 74.9%

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

       6. *-commutative 74.9%

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

       7. +-commutative 74.9%

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

   14. Simplified 74.9%

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

   if 4.2999999999999999e49 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 71.6%

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

   4. Taylor expanded in t around inf 79.4%

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

      1. mul-1-neg 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.5 \cdot 10^{-80}:\\ \;\;\;\;\log \left(x + y\right) + \log t \cdot a\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-69}:\\ \;\;\;\;\log \left(\left(x + y\right) \cdot \frac{z}{\sqrt{t}}\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{+49}:\\ \;\;\;\;\log \left(x + y\right) + \log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-40} \lor \neg \left(a \leq 0.8\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.5e-40) (not (<= a 0.8)))
   (- (+ (log y) (* (log t) a)) t)
   (- (+ (* (log t) -0.5) (log (* (+ x y) z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e-40) || !(a <= 0.8)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = ((log(t) * -0.5) + log(((x + y) * z))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-6.5d-40)) .or. (.not. (a <= 0.8d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = ((log(t) * (-0.5d0)) + log(((x + y) * z))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e-40) || !(a <= 0.8)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = ((Math.log(t) * -0.5) + Math.log(((x + y) * z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6.5e-40) or not (a <= 0.8):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = ((math.log(t) * -0.5) + math.log(((x + y) * z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.5e-40) || !(a <= 0.8))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(Float64(Float64(log(t) * -0.5) + log(Float64(Float64(x + y) * z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -6.5e-40) || ~((a <= 0.8)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = ((log(t) * -0.5) + log(((x + y) * z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.5e-40], N[Not[LessEqual[a, 0.8]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] + N[Log[N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4999999999999999e-40 or 0.80000000000000004 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 73.1%

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

   4. Taylor expanded in a around inf 71.9%

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

      1. *-commutative 71.9%

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

   6. Simplified 71.9%

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

   if -6.4999999999999999e-40 < a < 0.80000000000000004

   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. Add Preprocessing

   3. Taylor expanded in a around 0 98.1%

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

      1. associate-+r+ 98.2%

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

      2. log-prod 76.6%

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

      3. +-commutative 76.6%

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

      4. *-commutative 76.6%

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

   5. Simplified 76.6%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-40} \lor \neg \left(a \leq 0.8\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(\left(x + y\right) \cdot z\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-40} \lor \neg \left(a \leq 0.85\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.5e-40) (not (<= a 0.85)))
   (- (+ (log y) (* (log t) a)) t)
   (- (+ (* (log t) -0.5) (log (* y z))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e-40) || !(a <= 0.85)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = ((log(t) * -0.5) + log((y * z))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-6.5d-40)) .or. (.not. (a <= 0.85d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = ((log(t) * (-0.5d0)) + log((y * z))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e-40) || !(a <= 0.85)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = ((Math.log(t) * -0.5) + Math.log((y * z))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6.5e-40) or not (a <= 0.85):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = ((math.log(t) * -0.5) + math.log((y * z))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.5e-40) || !(a <= 0.85))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(Float64(Float64(log(t) * -0.5) + log(Float64(y * z))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -6.5e-40) || ~((a <= 0.85)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = ((log(t) * -0.5) + log((y * z))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.5e-40], N[Not[LessEqual[a, 0.85]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[(N[Log[t], $MachinePrecision] * -0.5), $MachinePrecision] + N[Log[N[(y * z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4999999999999999e-40 or 0.849999999999999978 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 73.1%

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

   4. Taylor expanded in a around inf 71.9%

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

      1. *-commutative 71.9%

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

   6. Simplified 71.9%

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

   if -6.4999999999999999e-40 < a < 0.849999999999999978

   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. Add Preprocessing

   3. Taylor expanded in x around 0 64.3%

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

   4. Taylor expanded in a around 0 63.5%

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

      1. associate-+r+ 63.4%

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

      2. log-prod 50.1%

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

      3. *-commutative 50.1%

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

      4. *-commutative 50.1%

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

   6. Simplified 50.1%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-40} \lor \neg \left(a \leq 0.85\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log t \cdot -0.5 + \log \left(y \cdot z\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+48} \lor \neg \left(a \leq 3.2 \cdot 10^{+77}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= a -1.45e+16)
     t_1
     (if (<= a 1.65e+31)
       (- (+ (log z) (log y)) t)
       (if (or (<= a 2.2e+48) (not (<= a 3.2e+77))) t_1 (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if (a <= -1.45e+16) {
		tmp = t_1;
	} else if (a <= 1.65e+31) {
		tmp = (log(z) + log(y)) - t;
	} else if ((a <= 2.2e+48) || !(a <= 3.2e+77)) {
		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 = log(t) * a
    if (a <= (-1.45d+16)) then
        tmp = t_1
    else if (a <= 1.65d+31) then
        tmp = (log(z) + log(y)) - t
    else if ((a <= 2.2d+48) .or. (.not. (a <= 3.2d+77))) 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 = Math.log(t) * a;
	double tmp;
	if (a <= -1.45e+16) {
		tmp = t_1;
	} else if (a <= 1.65e+31) {
		tmp = (Math.log(z) + Math.log(y)) - t;
	} else if ((a <= 2.2e+48) || !(a <= 3.2e+77)) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if a <= -1.45e+16:
		tmp = t_1
	elif a <= 1.65e+31:
		tmp = (math.log(z) + math.log(y)) - t
	elif (a <= 2.2e+48) or not (a <= 3.2e+77):
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (a <= -1.45e+16)
		tmp = t_1;
	elseif (a <= 1.65e+31)
		tmp = Float64(Float64(log(z) + log(y)) - t);
	elseif ((a <= 2.2e+48) || !(a <= 3.2e+77))
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if (a <= -1.45e+16)
		tmp = t_1;
	elseif (a <= 1.65e+31)
		tmp = (log(z) + log(y)) - t;
	elseif ((a <= 2.2e+48) || ~((a <= 3.2e+77)))
		tmp = t_1;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -1.45e+16], t$95$1, If[LessEqual[a, 1.65e+31], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[a, 2.2e+48], N[Not[LessEqual[a, 3.2e+77]], $MachinePrecision]], t$95$1, (-t)]]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.45e16 or 1.64999999999999996e31 < a < 2.1999999999999999e48 or 3.2000000000000002e77 < 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-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around 0 83.3%

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

      1. log-prod 68.3%

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

      2. +-commutative 68.3%

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

   7. Simplified 68.3%

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

      1. add-sqr-sqrt 39.9%

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

      2. pow2 39.8%

         \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]

   9. Applied egg-rr 39.8%

      \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 39.9%

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

       2. add-sqr-sqrt 68.3%

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

       3. add-sqr-sqrt 34.8%

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

       4. associate-*r* 34.8%

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

   11. Applied egg-rr 34.8%

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

   12. Taylor expanded in a around inf 83.3%

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

       1. *-commutative 83.3%

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

   14. Simplified 83.3%

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

   if -1.45e16 < a < 1.64999999999999996e31

   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. Add Preprocessing

   3. Taylor expanded in x around 0 65.3%

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

      1. +-commutative 65.3%

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

      2. add-cube-cbrt 65.1%

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

      3. fma-def 65.1%

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

      4. pow2 65.1%

         \[\leadsto \left(\log y + \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\log t \cdot \left(a - 0.5\right)}\right)}^{2}}, \sqrt[3]{\log t \cdot \left(a - 0.5\right)}, \log z\right)\right) - t \]

      5. sub-neg 65.1%

         \[\leadsto \left(\log y + \mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \color{blue}{\left(a + \left(-0.5\right)\right)}}\right)}^{2}, \sqrt[3]{\log t \cdot \left(a - 0.5\right)}, \log z\right)\right) - t \]

      6. metadata-eval 65.1%

         \[\leadsto \left(\log y + \mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \left(a + \color{blue}{-0.5}\right)}\right)}^{2}, \sqrt[3]{\log t \cdot \left(a - 0.5\right)}, \log z\right)\right) - t \]

      7. sub-neg 65.1%

         \[\leadsto \left(\log y + \mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \left(a + -0.5\right)}\right)}^{2}, \sqrt[3]{\log t \cdot \color{blue}{\left(a + \left(-0.5\right)\right)}}, \log z\right)\right) - t \]

      8. metadata-eval 65.1%

         \[\leadsto \left(\log y + \mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \left(a + -0.5\right)}\right)}^{2}, \sqrt[3]{\log t \cdot \left(a + \color{blue}{-0.5}\right)}, \log z\right)\right) - t \]

   5. Applied egg-rr 65.1%

      \[\leadsto \left(\log y + \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\log t \cdot \left(a + -0.5\right)}\right)}^{2}, \sqrt[3]{\log t \cdot \left(a + -0.5\right)}, \log z\right)}\right) - t \]

   6. Taylor expanded in a around inf 38.2%

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

   if 2.1999999999999999e48 < a < 3.2000000000000002e77

   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. Add Preprocessing

   3. Taylor expanded in x around 0 60.0%

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

   4. Taylor expanded in t around inf 80.6%

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

      1. mul-1-neg 80.6%

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

   6. Simplified 80.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+31}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{+48} \lor \neg \left(a \leq 3.2 \cdot 10^{+77}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 56.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0 69.1%

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

4. Taylor expanded in a around inf 55.7%

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

   1. *-commutative 55.7%

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

6. Simplified 55.7%

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

7. Final simplification 55.7%

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

Alternative 14: 56.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -5200000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+31}:\\ \;\;\;\;\log y - t\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+47} \lor \neg \left(a \leq 2 \cdot 10^{+75}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= a -5200000000000.0)
     t_1
     (if (<= a 1.8e+31)
       (- (log y) t)
       (if (or (<= a 6.4e+47) (not (<= a 2e+75))) t_1 (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if (a <= -5200000000000.0) {
		tmp = t_1;
	} else if (a <= 1.8e+31) {
		tmp = log(y) - t;
	} else if ((a <= 6.4e+47) || !(a <= 2e+75)) {
		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 = log(t) * a
    if (a <= (-5200000000000.0d0)) then
        tmp = t_1
    else if (a <= 1.8d+31) then
        tmp = log(y) - t
    else if ((a <= 6.4d+47) .or. (.not. (a <= 2d+75))) 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 = Math.log(t) * a;
	double tmp;
	if (a <= -5200000000000.0) {
		tmp = t_1;
	} else if (a <= 1.8e+31) {
		tmp = Math.log(y) - t;
	} else if ((a <= 6.4e+47) || !(a <= 2e+75)) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if a <= -5200000000000.0:
		tmp = t_1
	elif a <= 1.8e+31:
		tmp = math.log(y) - t
	elif (a <= 6.4e+47) or not (a <= 2e+75):
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (a <= -5200000000000.0)
		tmp = t_1;
	elseif (a <= 1.8e+31)
		tmp = Float64(log(y) - t);
	elseif ((a <= 6.4e+47) || !(a <= 2e+75))
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if (a <= -5200000000000.0)
		tmp = t_1;
	elseif (a <= 1.8e+31)
		tmp = log(y) - t;
	elseif ((a <= 6.4e+47) || ~((a <= 2e+75)))
		tmp = t_1;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -5200000000000.0], t$95$1, If[LessEqual[a, 1.8e+31], N[(N[Log[y], $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[a, 6.4e+47], N[Not[LessEqual[a, 2e+75]], $MachinePrecision]], t$95$1, (-t)]]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.2e12 or 1.79999999999999998e31 < a < 6.4e47 or 1.99999999999999985e75 < 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-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around 0 83.3%

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

      1. log-prod 68.3%

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

      2. +-commutative 68.3%

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

   7. Simplified 68.3%

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

      1. add-sqr-sqrt 39.9%

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

      2. pow2 39.8%

         \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]

   9. Applied egg-rr 39.8%

      \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 39.9%

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

       2. add-sqr-sqrt 68.3%

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

       3. add-sqr-sqrt 34.8%

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

       4. associate-*r* 34.8%

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

   11. Applied egg-rr 34.8%

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

   12. Taylor expanded in a around inf 83.3%

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

       1. *-commutative 83.3%

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

   14. Simplified 83.3%

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

   if -5.2e12 < a < 1.79999999999999998e31

   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-udef 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 53.0%

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

      1. neg-mul-1 53.0%

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

   7. Simplified 53.0%

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

   8. Taylor expanded in x around 0 37.3%

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

   if 6.4e47 < a < 1.99999999999999985e75

   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. Add Preprocessing

   3. Taylor expanded in x around 0 60.0%

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

   4. Taylor expanded in t around inf 80.6%

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

      1. mul-1-neg 80.6%

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

   6. Simplified 80.6%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5200000000000:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+31}:\\ \;\;\;\;\log y - t\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+47} \lor \neg \left(a \leq 2 \cdot 10^{+75}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log t \cdot a\\ \mathbf{if}\;a \leq -950000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.72 \cdot 10^{+31}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 10^{+48} \lor \neg \left(a \leq 1.3 \cdot 10^{+73}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (log t) a)))
   (if (<= a -950000000000.0)
     t_1
     (if (<= a 1.72e+31)
       (- (log (+ x y)) t)
       (if (or (<= a 1e+48) (not (<= a 1.3e+73))) t_1 (- t))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = log(t) * a;
	double tmp;
	if (a <= -950000000000.0) {
		tmp = t_1;
	} else if (a <= 1.72e+31) {
		tmp = log((x + y)) - t;
	} else if ((a <= 1e+48) || !(a <= 1.3e+73)) {
		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 = log(t) * a
    if (a <= (-950000000000.0d0)) then
        tmp = t_1
    else if (a <= 1.72d+31) then
        tmp = log((x + y)) - t
    else if ((a <= 1d+48) .or. (.not. (a <= 1.3d+73))) 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 = Math.log(t) * a;
	double tmp;
	if (a <= -950000000000.0) {
		tmp = t_1;
	} else if (a <= 1.72e+31) {
		tmp = Math.log((x + y)) - t;
	} else if ((a <= 1e+48) || !(a <= 1.3e+73)) {
		tmp = t_1;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = math.log(t) * a
	tmp = 0
	if a <= -950000000000.0:
		tmp = t_1
	elif a <= 1.72e+31:
		tmp = math.log((x + y)) - t
	elif (a <= 1e+48) or not (a <= 1.3e+73):
		tmp = t_1
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(log(t) * a)
	tmp = 0.0
	if (a <= -950000000000.0)
		tmp = t_1;
	elseif (a <= 1.72e+31)
		tmp = Float64(log(Float64(x + y)) - t);
	elseif ((a <= 1e+48) || !(a <= 1.3e+73))
		tmp = t_1;
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = log(t) * a;
	tmp = 0.0;
	if (a <= -950000000000.0)
		tmp = t_1;
	elseif (a <= 1.72e+31)
		tmp = log((x + y)) - t;
	elseif ((a <= 1e+48) || ~((a <= 1.3e+73)))
		tmp = t_1;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]}, If[LessEqual[a, -950000000000.0], t$95$1, If[LessEqual[a, 1.72e+31], N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision], If[Or[LessEqual[a, 1e+48], N[Not[LessEqual[a, 1.3e+73]], $MachinePrecision]], t$95$1, (-t)]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.5e11 or 1.72e31 < a < 1.00000000000000004e48 or 1.3e73 < 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-udef 99.6%

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

      8. sub-neg 99.6%

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

      9. +-commutative 99.6%

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

      10. distribute-neg-in 99.6%

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

      11. metadata-eval 99.6%

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

      12. metadata-eval 99.6%

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

      13. unsub-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in t around 0 83.3%

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

      1. log-prod 68.3%

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

      2. +-commutative 68.3%

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

   7. Simplified 68.3%

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

      1. add-sqr-sqrt 39.9%

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

      2. pow2 39.8%

         \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]

   9. Applied egg-rr 39.8%

      \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 39.9%

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

       2. add-sqr-sqrt 68.3%

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

       3. add-sqr-sqrt 34.8%

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

       4. associate-*r* 34.8%

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

   11. Applied egg-rr 34.8%

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

   12. Taylor expanded in a around inf 83.3%

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

       1. *-commutative 83.3%

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

   14. Simplified 83.3%

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

   if -9.5e11 < a < 1.72e31

   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-udef 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 53.0%

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

      1. neg-mul-1 53.0%

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

   7. Simplified 53.0%

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

   if 1.00000000000000004e48 < a < 1.3e73

   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. Add Preprocessing

   3. Taylor expanded in x around 0 60.0%

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

   4. Taylor expanded in t around inf 80.6%

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

      1. mul-1-neg 80.6%

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

   6. Simplified 80.6%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -950000000000:\\ \;\;\;\;\log t \cdot a\\ \mathbf{elif}\;a \leq 1.72 \cdot 10^{+31}:\\ \;\;\;\;\log \left(x + y\right) - t\\ \mathbf{elif}\;a \leq 10^{+48} \lor \neg \left(a \leq 1.3 \cdot 10^{+73}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 40.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 135:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= t 135.0) (log (+ x y)) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 135.0) {
		tmp = log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 135.0d0) then
        tmp = log((x + y))
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 135.0) {
		tmp = Math.log((x + y));
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 135.0:
		tmp = math.log((x + y))
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 135.0)
		tmp = log(Float64(x + y));
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 135.0)
		tmp = log((x + y));
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 135.0], N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 135

   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+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-udef 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around inf 9.1%

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

      1. neg-mul-1 9.1%

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

   7. Simplified 9.1%

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

   8. Taylor expanded in t around 0 9.1%

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

      1. +-commutative 9.1%

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

   10. Simplified 9.1%

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

   if 135 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 70.0%

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

   4. Taylor expanded in t around inf 72.1%

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

      1. mul-1-neg 72.1%

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

   6. Simplified 72.1%

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

4. Final simplification 37.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 135:\\ \;\;\;\;\log \left(x + y\right)\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+50}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 1.3e+50) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.3e+50) {
		tmp = log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= 1.3d+50) then
        tmp = log(t) * a
    else
        tmp = -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 1.3e+50) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 1.3e+50:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 1.3e+50)
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(-t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= 1.3e+50)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 1.3e+50], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 1.3000000000000001e50

   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+ 99.4%

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

      3. sub-neg 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. fma-udef 99.4%

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

      8. sub-neg 99.4%

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

      9. +-commutative 99.4%

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

      10. distribute-neg-in 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. unsub-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in t around 0 95.3%

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

      1. log-prod 73.7%

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

      2. +-commutative 73.7%

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

   7. Simplified 73.7%

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

      1. add-sqr-sqrt 24.8%

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

      2. pow2 24.8%

         \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]

   9. Applied egg-rr 24.8%

      \[\leadsto \log \left(z \cdot \left(y + x\right)\right) - \color{blue}{{\left(\sqrt{\log t \cdot \left(0.5 - a\right)}\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 24.8%

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

       2. add-sqr-sqrt 73.7%

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

       3. add-sqr-sqrt 51.3%

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

       4. associate-*r* 51.3%

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

   11. Applied egg-rr 51.3%

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

   12. Taylor expanded in a around inf 53.6%

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

       1. *-commutative 53.6%

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

   14. Simplified 53.6%

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

   if 1.3000000000000001e50 < 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. Add Preprocessing

   3. Taylor expanded in x around 0 71.6%

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

   4. Taylor expanded in t around inf 79.4%

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

      1. mul-1-neg 79.4%

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

   6. Simplified 79.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.3 \cdot 10^{+50}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.3% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around 0 69.1%

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

4. Taylor expanded in t around inf 34.4%

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

   1. mul-1-neg 34.4%

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

6. Simplified 34.4%

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

7. Final simplification 34.4%

   \[\leadsto -t \]
8. 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 2024020 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

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

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