Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Percentage Accurate: 99.6% → 99.6%

Time: 23.9s

Alternatives: 16

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[Log[z], $MachinePrecision] + N[(N[Log[N[(x + y), $MachinePrecision]], $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. +-commutative 99.6%

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

   3. associate--l+ 99.6%

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

   4. sub-neg 99.6%

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

   5. +-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-neg-in 99.6%

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

   8. fma-undefine 99.6%

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

   9. sub-neg 99.6%

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

   10. +-commutative 99.6%

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

   11. distribute-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. metadata-eval 99.6%

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

   14. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.6 \lor \neg \left(a - 0.5 \leq -0.48\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= (- a 0.5) -0.6) (not (<= (- a 0.5) -0.48)))
   (- (+ (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 - 0.5) <= -0.6) || !((a - 0.5) <= -0.48)) {
		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 - 0.5d0) <= (-0.6d0)) .or. (.not. ((a - 0.5d0) <= (-0.48d0)))) 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 - 0.5) <= -0.6) || !((a - 0.5) <= -0.48)) {
		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 - 0.5) <= -0.6) or not ((a - 0.5) <= -0.48):
		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 ((Float64(a - 0.5) <= -0.6) || !(Float64(a - 0.5) <= -0.48))
		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 - 0.5) <= -0.6) || ~(((a - 0.5) <= -0.48)))
		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[N[(a - 0.5), $MachinePrecision], -0.6], N[Not[LessEqual[N[(a - 0.5), $MachinePrecision], -0.48]], $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 (-.f64 a 1/2) < -0.599999999999999978 or -0.47999999999999998 < (-.f64 a 1/2)

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 75.3%

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

   4. Taylor expanded in a around inf 73.6%

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

      1. *-commutative 73.6%

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

   6. Simplified 73.6%

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

   if -0.599999999999999978 < (-.f64 a 1/2) < -0.47999999999999998

   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 63.7%

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

   4. Taylor expanded in a around 0 63.4%

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

      1. *-commutative 63.4%

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

   6. Simplified 63.4%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a - 0.5 \leq -0.6 \lor \neg \left(a - 0.5 \leq -0.48\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \left(\log z + \log t \cdot -0.5\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.065 \lor \neg \left(a \leq 0.00095\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.065) (not (<= a 0.00095)))
   (- (+ (log y) (* (log t) a)) t)
   (- (+ (log y) (log (* z (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.065) || !(a <= 0.00095)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = (log(y) + log((z * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.065d0)) .or. (.not. (a <= 0.00095d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = (log(y) + log((z * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.065) || !(a <= 0.00095)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = (Math.log(y) + Math.log((z * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.065) or not (a <= 0.00095):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = (math.log(y) + math.log((z * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.065) || !(a <= 0.00095))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(Float64(log(y) + log(Float64(z * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.065) || ~((a <= 0.00095)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = (log(y) + log((z * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.065], N[Not[LessEqual[a, 0.00095]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[y], $MachinePrecision] + N[Log[N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.065000000000000002 or 9.49999999999999998e-4 < 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 75.3%

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

   4. Taylor expanded in a around inf 73.6%

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

      1. *-commutative 73.6%

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

   6. Simplified 73.6%

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

   if -0.065000000000000002 < a < 9.49999999999999998e-4

   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 63.7%

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

      1. add-cube-cbrt 63.6%

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

      2. pow3 63.6%

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

      3. sub-neg 63.6%

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

      4. metadata-eval 63.6%

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

   5. Applied egg-rr 63.6%

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

   6. Taylor expanded in a around 0 63.4%

      \[\leadsto \left(\log y + \color{blue}{\left(\log z + -0.5 \cdot \left({1}^{0.3333333333333333} \cdot \log t\right)\right)}\right) - t \]
   7. Step-by-step derivation

      1. pow-base-1 63.4%

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

      2. *-lft-identity 63.4%

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

      3. log-pow 63.4%

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

      4. log-prod 56.4%

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

   8. Simplified 56.4%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.065 \lor \neg \left(a \leq 0.00095\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.038 \lor \neg \left(a \leq 0.0051\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.038) (not (<= a 0.0051)))
   (- (+ (log y) (* (log t) a)) t)
   (- (+ (log z) (log (* y (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.038) || !(a <= 0.0051)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = (log(z) + log((y * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-0.038d0)) .or. (.not. (a <= 0.0051d0))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = (log(z) + log((y * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.038) || !(a <= 0.0051)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = (Math.log(z) + Math.log((y * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -0.038) or not (a <= 0.0051):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = (math.log(z) + math.log((y * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.038) || !(a <= 0.0051))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(Float64(log(z) + log(Float64(y * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -0.038) || ~((a <= 0.0051)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = (log(z) + log((y * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.038], N[Not[LessEqual[a, 0.0051]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.0379999999999999991 or 0.0051000000000000004 < 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 75.3%

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

   4. Taylor expanded in a around inf 73.6%

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

      1. *-commutative 73.6%

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

   6. Simplified 73.6%

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

   if -0.0379999999999999991 < a < 0.0051000000000000004

   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 98.4%

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

      1. associate-+r+ 98.3%

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

      2. log-prod 68.6%

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

      3. *-commutative 68.6%

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

   5. Simplified 68.6%

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

   6. Taylor expanded in y around inf 63.5%

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

      1. +-commutative 63.5%

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

      2. mul-1-neg 63.5%

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

      3. unsub-neg 63.5%

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

      4. *-commutative 63.5%

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

      5. log-rec 63.5%

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

   8. Simplified 63.5%

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

      1. add-sqr-sqrt 20.1%

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

      2. sqrt-unprod 48.3%

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

      3. sqr-neg 48.3%

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

      4. sqrt-unprod 28.2%

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

      5. add-sqr-sqrt 44.0%

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

      6. unsub-neg 44.0%

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

      7. add-log-exp 44.0%

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

      8. neg-log 44.0%

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

      9. sum-log 40.9%

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

      10. pow-to-exp 40.9%

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

      11. add-exp-log 40.9%

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

      12. neg-log 40.9%

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

      13. add-sqr-sqrt 15.5%

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

      14. sqrt-unprod 54.4%

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

      15. sqr-neg 54.4%

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

      16. sqrt-unprod 38.9%

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

      17. add-sqr-sqrt 55.3%

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

      18. add-exp-log 55.5%

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

   10. Applied egg-rr 55.5%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.038 \lor \neg \left(a \leq 0.0051\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Final simplification 99.6%

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

Alternative 6: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = (log(y) + (log(z) + (log(t) * (a - 0.5)))) - 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[(a - 0.5), $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 69.5%

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

4. Final simplification 69.5%

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

Alternative 7: 58.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-7} \lor \neg \left(a \leq 2.3 \cdot 10^{-177} \lor \neg \left(a \leq 1.16 \cdot 10^{-108}\right) \land a \leq 9.5 \cdot 10^{-41}\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.5e-7)
         (not
          (or (<= a 2.3e-177) (and (not (<= a 1.16e-108)) (<= a 9.5e-41)))))
   (- (+ (log y) (* (log t) a)) t)
   (- (log (* z (* y (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.5e-7) || !((a <= 2.3e-177) || (!(a <= 1.16e-108) && (a <= 9.5e-41)))) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = log((z * (y * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-3.5d-7)) .or. (.not. (a <= 2.3d-177) .or. (.not. (a <= 1.16d-108)) .and. (a <= 9.5d-41))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = log((z * (y * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.5e-7) || !((a <= 2.3e-177) || (!(a <= 1.16e-108) && (a <= 9.5e-41)))) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = Math.log((z * (y * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -3.5e-7) or not ((a <= 2.3e-177) or (not (a <= 1.16e-108) and (a <= 9.5e-41))):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = math.log((z * (y * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.5e-7) || !((a <= 2.3e-177) || (!(a <= 1.16e-108) && (a <= 9.5e-41))))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(log(Float64(z * Float64(y * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -3.5e-7) || ~(((a <= 2.3e-177) || (~((a <= 1.16e-108)) && (a <= 9.5e-41)))))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = log((z * (y * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.5e-7], N[Not[Or[LessEqual[a, 2.3e-177], And[N[Not[LessEqual[a, 1.16e-108]], $MachinePrecision], LessEqual[a, 9.5e-41]]]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(z * N[(y * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.49999999999999984e-7 or 2.30000000000000022e-177 < a < 1.16e-108 or 9.4999999999999997e-41 < 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 72.8%

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

   4. Taylor expanded in a around inf 68.5%

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

      1. *-commutative 68.5%

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

   6. Simplified 68.5%

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

   if -3.49999999999999984e-7 < a < 2.30000000000000022e-177 or 1.16e-108 < a < 9.4999999999999997e-41

   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 99.5%

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

      1. associate-+r+ 99.5%

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

      2. log-prod 74.9%

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

      3. *-commutative 74.9%

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

   5. Simplified 74.9%

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

   6. Taylor expanded in y around inf 64.3%

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

      1. +-commutative 64.3%

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

      2. mul-1-neg 64.3%

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

      3. unsub-neg 64.3%

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

      4. *-commutative 64.3%

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

      5. log-rec 64.3%

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

   8. Simplified 64.3%

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

      1. add-log-exp 57.5%

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

      2. sum-log 49.3%

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

      3. sub-neg 49.3%

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

      4. exp-sum 49.3%

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

      5. pow-to-exp 49.4%

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

      6. remove-double-neg 49.4%

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

      7. add-exp-log 49.7%

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

   10. Applied egg-rr 49.7%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.5 \cdot 10^{-7} \lor \neg \left(a \leq 2.3 \cdot 10^{-177} \lor \neg \left(a \leq 1.16 \cdot 10^{-108}\right) \land a \leq 9.5 \cdot 10^{-41}\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(z \cdot \left(y \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-17} \lor \neg \left(a \leq 1.55 \cdot 10^{-40}\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(z \cdot y\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.02e-17) (not (<= a 1.55e-40)))
   (- (+ (log y) (* (log t) a)) t)
   (- (log (* (pow t -0.5) (* z y))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.02e-17) || !(a <= 1.55e-40)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = log((pow(t, -0.5) * (z * y))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.02d-17)) .or. (.not. (a <= 1.55d-40))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = log(((t ** (-0.5d0)) * (z * y))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.02e-17) || !(a <= 1.55e-40)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = Math.log((Math.pow(t, -0.5) * (z * y))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.02e-17) or not (a <= 1.55e-40):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = math.log((math.pow(t, -0.5) * (z * y))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.02e-17) || !(a <= 1.55e-40))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(log(Float64((t ^ -0.5) * Float64(z * y))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.02e-17) || ~((a <= 1.55e-40)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = log(((t ^ -0.5) * (z * y))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.02e-17], N[Not[LessEqual[a, 1.55e-40]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(N[Power[t, -0.5], $MachinePrecision] * N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.01999999999999997e-17 or 1.55000000000000005e-40 < a

   1. Initial program 99.7%

      \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   2. 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 70.9%

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

      1. *-commutative 70.9%

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

   6. Simplified 70.9%

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

   if -1.01999999999999997e-17 < a < 1.55000000000000005e-40

   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 64.2%

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

      1. add-cube-cbrt 64.0%

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

      2. pow3 64.0%

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

      3. sub-neg 64.0%

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

      4. metadata-eval 64.0%

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

   5. Applied egg-rr 64.0%

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

      1. rem-cube-cbrt 64.2%

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

      2. *-commutative 64.2%

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

      3. add-cube-cbrt 64.1%

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

      4. associate-*r* 64.1%

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

      5. pow2 64.1%

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

   7. Applied egg-rr 64.1%

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

   8. Taylor expanded in a around 0 64.2%

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

      1. associate-+r+ 64.1%

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

      2. log-prod 46.7%

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

      3. log-pow 46.7%

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

      4. log-prod 46.7%

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

   10. Simplified 46.7%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-17} \lor \neg \left(a \leq 1.55 \cdot 10^{-40}\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left({t}^{-0.5} \cdot \left(z \cdot y\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-45} \lor \neg \left(a \leq 1.32 \cdot 10^{-40}\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.02e-45) (not (<= a 1.32e-40)))
   (- (+ (log y) (* (log t) a)) t)
   (- (log (* y (* z (pow t -0.5)))) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.02e-45) || !(a <= 1.32e-40)) {
		tmp = (log(y) + (log(t) * a)) - t;
	} else {
		tmp = log((y * (z * pow(t, -0.5)))) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-1.02d-45)) .or. (.not. (a <= 1.32d-40))) then
        tmp = (log(y) + (log(t) * a)) - t
    else
        tmp = log((y * (z * (t ** (-0.5d0))))) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.02e-45) || !(a <= 1.32e-40)) {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	} else {
		tmp = Math.log((y * (z * Math.pow(t, -0.5)))) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -1.02e-45) or not (a <= 1.32e-40):
		tmp = (math.log(y) + (math.log(t) * a)) - t
	else:
		tmp = math.log((y * (z * math.pow(t, -0.5)))) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.02e-45) || !(a <= 1.32e-40))
		tmp = Float64(Float64(log(y) + Float64(log(t) * a)) - t);
	else
		tmp = Float64(log(Float64(y * Float64(z * (t ^ -0.5)))) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -1.02e-45) || ~((a <= 1.32e-40)))
		tmp = (log(y) + (log(t) * a)) - t;
	else
		tmp = log((y * (z * (t ^ -0.5)))) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.02e-45], N[Not[LessEqual[a, 1.32e-40]], $MachinePrecision]], N[(N[(N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision], N[(N[Log[N[(y * N[(z * N[Power[t, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.0199999999999999e-45 or 1.32000000000000009e-40 < a

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0 71.9%

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

   4. Taylor expanded in a around inf 68.3%

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

      1. *-commutative 68.3%

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

   6. Simplified 68.3%

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

   if -1.0199999999999999e-45 < a < 1.32000000000000009e-40

   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 66.1%

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

      1. add-cube-cbrt 66.0%

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

      2. pow3 66.0%

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

      3. sub-neg 66.0%

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

      4. metadata-eval 66.0%

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

   5. Applied egg-rr 66.0%

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

   6. Taylor expanded in a around 0 66.1%

      \[\leadsto \color{blue}{\left(\log y + \left(\log z + -0.5 \cdot \left({1}^{0.3333333333333333} \cdot \log t\right)\right)\right)} - t \]
   7. Step-by-step derivation

      1. associate-+r+ 66.1%

         \[\leadsto \color{blue}{\left(\left(\log y + \log z\right) + -0.5 \cdot \left({1}^{0.3333333333333333} \cdot \log t\right)\right)} - t \]

      2. pow-base-1 66.1%

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

      3. *-lft-identity 66.1%

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

      4. +-commutative 66.1%

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

      5. remove-double-neg 66.1%

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

      6. log-rec 66.1%

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

      7. mul-1-neg 66.1%

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

      8. associate-+r+ 66.2%

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

      9. mul-1-neg 66.2%

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

      10. log-rec 66.2%

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

      11. remove-double-neg 66.2%

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

      12. log-pow 66.2%

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

      13. log-prod 57.5%

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

      14. *-commutative 57.5%

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

      15. log-prod 49.9%

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

      16. associate-*r* 52.6%

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

   8. Simplified 52.6%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-45} \lor \neg \left(a \leq 1.32 \cdot 10^{-40}\right):\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(y \cdot \left(z \cdot {t}^{-0.5}\right)\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+18}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.7e+18)
   (+ (* (log t) (- a 0.5)) (- (log (* z (+ x y))) t))
   (- (+ (log y) (* (log t) a)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.7e+18) {
		tmp = (log(t) * (a - 0.5)) + (log((z * (x + y))) - t);
	} 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 <= 3.7d+18) then
        tmp = (log(t) * (a - 0.5d0)) + (log((z * (x + y))) - t)
    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 <= 3.7e+18) {
		tmp = (Math.log(t) * (a - 0.5)) + (Math.log((z * (x + y))) - t);
	} else {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.7e+18:
		tmp = (math.log(t) * (a - 0.5)) + (math.log((z * (x + y))) - t)
	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 <= 3.7e+18)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + Float64(log(Float64(z * Float64(x + y))) - t));
	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 <= 3.7e+18)
		tmp = (log(t) * (a - 0.5)) + (log((z * (x + y))) - t);
	else
		tmp = (log(y) + (log(t) * a)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.7e+18], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(z * N[(x + y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - t), $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 < 3.7e18

   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. Add Preprocessing

   3. Taylor expanded in z around 0 99.3%

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

      1. log-prod 74.7%

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

   5. Simplified 74.7%

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

   if 3.7e18 < 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 80.4%

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

   4. Taylor expanded in a around inf 80.4%

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

      1. *-commutative 80.4%

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

   6. Simplified 80.4%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+18}:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \left(\log \left(z \cdot \left(x + y\right)\right) - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.5:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 7.5)
   (+ (* (log t) (- a 0.5)) (log (* z (+ x y))))
   (- (+ (log y) (* (log t) a)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 7.5) {
		tmp = (log(t) * (a - 0.5)) + log((z * (x + y)));
	} 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 <= 7.5d0) then
        tmp = (log(t) * (a - 0.5d0)) + log((z * (x + y)))
    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 <= 7.5) {
		tmp = (Math.log(t) * (a - 0.5)) + Math.log((z * (x + y)));
	} else {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 7.5:
		tmp = (math.log(t) * (a - 0.5)) + math.log((z * (x + y)))
	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 <= 7.5)
		tmp = Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(z * Float64(x + y))));
	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 <= 7.5)
		tmp = (log(t) * (a - 0.5)) + log((z * (x + y)));
	else
		tmp = (log(y) + (log(t) * a)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 7.5], N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * N[(x + y), $MachinePrecision]), $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 < 7.5

   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. +-commutative 99.3%

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

      3. associate--l+ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. fma-undefine 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. metadata-eval 99.3%

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

      14. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in t around 0 97.6%

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

      1. log-prod 72.1%

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

   7. Simplified 72.1%

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

   if 7.5 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 80.1%

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

   4. Taylor expanded in a around inf 80.0%

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

      1. *-commutative 80.0%

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

   6. Simplified 80.0%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5:\\ \;\;\;\;\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot \left(x + y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+18}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 3.7e+18)
   (- (+ (* (log t) (- a 0.5)) (log (* z y))) t)
   (- (+ (log y) (* (log t) a)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 3.7e+18) {
		tmp = ((log(t) * (a - 0.5)) + log((z * y))) - t;
	} 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 <= 3.7d+18) then
        tmp = ((log(t) * (a - 0.5d0)) + log((z * y))) - t
    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 <= 3.7e+18) {
		tmp = ((Math.log(t) * (a - 0.5)) + Math.log((z * y))) - t;
	} else {
		tmp = (Math.log(y) + (Math.log(t) * a)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 3.7e+18:
		tmp = ((math.log(t) * (a - 0.5)) + math.log((z * y))) - t
	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 <= 3.7e+18)
		tmp = Float64(Float64(Float64(log(t) * Float64(a - 0.5)) + log(Float64(z * y))) - t);
	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 <= 3.7e+18)
		tmp = ((log(t) * (a - 0.5)) + log((z * y))) - t;
	else
		tmp = (log(y) + (log(t) * a)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 3.7e+18], N[(N[(N[(N[Log[t], $MachinePrecision] * N[(a - 0.5), $MachinePrecision]), $MachinePrecision] + N[Log[N[(z * y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t), $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 < 3.7e18

   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. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 63.2%

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

      2. pow2 63.2%

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

      3. +-commutative 63.2%

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

      4. *-commutative 63.2%

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

      5. fma-define 63.2%

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

      6. sub-neg 63.2%

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

      7. metadata-eval 63.2%

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

      8. +-commutative 63.2%

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

      9. sum-log 45.2%

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

   4. Applied egg-rr 45.2%

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

   5. Taylor expanded in x around 0 45.8%

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

   if 3.7e18 < 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 80.4%

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

   4. Taylor expanded in a around inf 80.4%

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

      1. *-commutative 80.4%

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

   6. Simplified 80.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.7 \cdot 10^{+18}:\\ \;\;\;\;\left(\log t \cdot \left(a - 0.5\right) + \log \left(z \cdot y\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\left(\log y + \log t \cdot a\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 57.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+52} \lor \neg \left(a \leq 1.45 \cdot 10^{+20}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -6.5e+52) (not (<= a 1.45e+20)))
   (* (log t) a)
   (- (+ (log z) (log y)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e+52) || !(a <= 1.45e+20)) {
		tmp = log(t) * a;
	} else {
		tmp = (log(z) + log(y)) - t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((a <= (-6.5d+52)) .or. (.not. (a <= 1.45d+20))) then
        tmp = log(t) * a
    else
        tmp = (log(z) + log(y)) - t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -6.5e+52) || !(a <= 1.45e+20)) {
		tmp = Math.log(t) * a;
	} else {
		tmp = (Math.log(z) + Math.log(y)) - t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -6.5e+52) or not (a <= 1.45e+20):
		tmp = math.log(t) * a
	else:
		tmp = (math.log(z) + math.log(y)) - t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -6.5e+52) || !(a <= 1.45e+20))
		tmp = Float64(log(t) * a);
	else
		tmp = Float64(Float64(log(z) + log(y)) - t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -6.5e+52) || ~((a <= 1.45e+20)))
		tmp = log(t) * a;
	else
		tmp = (log(z) + log(y)) - t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -6.5e+52], N[Not[LessEqual[a, 1.45e+20]], $MachinePrecision]], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], N[(N[(N[Log[z], $MachinePrecision] + N[Log[y], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -6.49999999999999996e52 or 1.45e20 < a

   1. Initial program 99.7%

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

      1. associate-+l- 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. sub-neg 99.7%

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

      5. +-commutative 99.7%

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

      6. *-commutative 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. fma-undefine 99.7%

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

      9. sub-neg 99.7%

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

      10. +-commutative 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. metadata-eval 99.7%

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

      13. metadata-eval 99.7%

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

      14. unsub-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in a around inf 76.9%

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

      1. *-commutative 76.9%

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

   7. Simplified 76.9%

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

   if -6.49999999999999996e52 < a < 1.45e20

   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 67.2%

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

      1. add-cube-cbrt 67.0%

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

      2. pow3 67.0%

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

      3. sub-neg 67.0%

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

      4. metadata-eval 67.0%

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

   5. Applied egg-rr 67.0%

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

   6. Taylor expanded in a around inf 46.4%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{+52} \lor \neg \left(a \leq 1.45 \cdot 10^{+20}\right):\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \log y\right) - t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.4% 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 69.5%

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

4. Taylor expanded in a around inf 58.5%

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

   1. *-commutative 58.5%

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

6. Simplified 58.5%

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

7. Final simplification 58.5%

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

Alternative 15: 62.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t 5.5e+16) (* (log t) a) (- t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= 5.5e+16) {
		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 <= 5.5d+16) 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 <= 5.5e+16) {
		tmp = Math.log(t) * a;
	} else {
		tmp = -t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= 5.5e+16:
		tmp = math.log(t) * a
	else:
		tmp = -t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= 5.5e+16)
		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 <= 5.5e+16)
		tmp = log(t) * a;
	else
		tmp = -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, 5.5e+16], N[(N[Log[t], $MachinePrecision] * a), $MachinePrecision], (-t)]

Derivation

1. Split input into 2 regimes

2. if t < 5.5e16

   1. Initial program 99.3%

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

      1. associate-+l- 99.3%

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

      2. +-commutative 99.3%

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

      3. associate--l+ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. *-commutative 99.3%

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

      7. distribute-rgt-neg-in 99.3%

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

      8. fma-undefine 99.3%

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

      9. sub-neg 99.3%

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

      10. +-commutative 99.3%

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

      11. distribute-neg-in 99.3%

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

      12. metadata-eval 99.3%

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

      13. metadata-eval 99.3%

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

      14. unsub-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in a around inf 50.9%

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

      1. *-commutative 50.9%

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

   7. Simplified 50.9%

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

   if 5.5e16 < t

   1. Initial program 99.9%

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

      1. associate-+l- 99.9%

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

      2. +-commutative 99.9%

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

      3. associate--l+ 99.9%

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

      4. sub-neg 99.9%

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

      5. +-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. distribute-rgt-neg-in 99.9%

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

      8. fma-undefine 99.9%

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

      9. sub-neg 99.9%

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

      10. +-commutative 99.9%

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

      11. distribute-neg-in 99.9%

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

      12. metadata-eval 99.9%

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

      13. metadata-eval 99.9%

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

      14. unsub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around inf 80.1%

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

      1. mul-1-neg 80.1%

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

   7. Simplified 80.1%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.5 \cdot 10^{+16}:\\ \;\;\;\;\log t \cdot a\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.8% accurate, 156.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-+l- 99.6%

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

   2. +-commutative 99.6%

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

   3. associate--l+ 99.6%

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

   4. sub-neg 99.6%

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

   5. +-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. distribute-rgt-neg-in 99.6%

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

   8. fma-undefine 99.6%

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

   9. sub-neg 99.6%

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

   10. +-commutative 99.6%

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

   11. distribute-neg-in 99.6%

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

   12. metadata-eval 99.6%

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

   13. metadata-eval 99.6%

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

   14. unsub-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in t around inf 40.6%

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

   1. mul-1-neg 40.6%

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

7. Simplified 40.6%

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

8. Final simplification 40.6%

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

Developer target: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024034 
(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))))