Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 96.4%

Time: 11.5s

Alternatives: 15

Speedup: 2.5×

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 96.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 10^{+139}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot t\_1 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 1e+139)
     (+
      (*
       (*
        J
        (*
         l
         (+
          2.0
          (*
           (* l l)
           (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))
       t_1)
      U)
     (+ U (* t_0 t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 1e+139) {
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * t_1) + U;
	} else {
		tmp = U + (t_0 * t_1);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = j * (exp(l) - exp(-l))
    t_1 = cos((k / 2.0d0))
    if (t_0 <= 1d+139) then
        tmp = ((j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))) * t_1) + u
    else
        tmp = u + (t_0 * t_1)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (Math.exp(l) - Math.exp(-l));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 1e+139) {
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * t_1) + U;
	} else {
		tmp = U + (t_0 * t_1);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (math.exp(l) - math.exp(-l))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 1e+139:
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * t_1) + U
	else:
		tmp = U + (t_0 * t_1)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 1e+139)
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))) * t_1) + U);
	else
		tmp = Float64(U + Float64(t_0 * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (exp(l) - exp(-l));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 1e+139)
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * t_1) + U;
	else
		tmp = U + (t_0 * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 1e+139], N[(N[(N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 1.00000000000000003e139

   1. Initial program 81.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 98.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{\ell}^{2} \cdot 0.016666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 98.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 98.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 98.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 98.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. Applied egg-rr 98.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 1.00000000000000003e139 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))))

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq 10^{+139}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -4.6:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{-14} \lor \neg \left(\ell \leq 5.4 \cdot 10^{+45}\right):\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot t\_1 + U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(e^{\ell} - t\_0\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -4.6)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (or (<= l 4.9e-14) (not (<= l 5.4e+45)))
       (+
        (*
         (*
          J
          (*
           l
           (+
            2.0
            (*
             (* l l)
             (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))
         t_1)
        U)
       (+ (* J (- (exp l) t_0)) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -4.6) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if ((l <= 4.9e-14) || !(l <= 5.4e+45)) {
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * t_1) + U;
	} else {
		tmp = (J * (exp(l) - t_0)) + U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-4.6d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if ((l <= 4.9d-14) .or. (.not. (l <= 5.4d+45))) then
        tmp = ((j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))) * t_1) + u
    else
        tmp = (j * (exp(l) - t_0)) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -4.6) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if ((l <= 4.9e-14) || !(l <= 5.4e+45)) {
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * t_1) + U;
	} else {
		tmp = (J * (Math.exp(l) - t_0)) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -4.6:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif (l <= 4.9e-14) or not (l <= 5.4e+45):
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * t_1) + U
	else:
		tmp = (J * (math.exp(l) - t_0)) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -4.6)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif ((l <= 4.9e-14) || !(l <= 5.4e+45))
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))) * t_1) + U);
	else
		tmp = Float64(Float64(J * Float64(exp(l) - t_0)) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -4.6)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif ((l <= 4.9e-14) || ~((l <= 5.4e+45)))
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * t_1) + U;
	else
		tmp = (J * (exp(l) - t_0)) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.6], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 4.9e-14], N[Not[LessEqual[l, 5.4e+45]], $MachinePrecision]], N[(N[(N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.5999999999999996

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 98.5%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -4.5999999999999996 < l < 4.89999999999999995e-14 or 5.39999999999999968e45 < l

   1. Initial program 80.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{\ell}^{2} \cdot 0.016666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 99.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 99.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 99.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. Applied egg-rr 99.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 4.89999999999999995e-14 < l < 5.39999999999999968e45

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 88.2%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 4.9 \cdot 10^{-14} \lor \neg \left(\ell \leq 5.4 \cdot 10^{+45}\right):\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.9 \cdot 10^{-14} \lor \neg \left(\ell \leq 5.6 \cdot 10^{+46}\right):\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l 4.9e-14) (not (<= l 5.6e+46)))
   (+
    (*
     (*
      J
      (*
       l
       (+
        2.0
        (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))
     (cos (/ K 2.0)))
    U)
   (+ (* J (- (exp l) (exp (- l)))) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 4.9e-14) || !(l <= 5.6e+46)) {
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * cos((K / 2.0))) + U;
	} else {
		tmp = (J * (exp(l) - exp(-l))) + U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= 4.9d-14) .or. (.not. (l <= 5.6d+46))) then
        tmp = ((j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))) * cos((k / 2.0d0))) + u
    else
        tmp = (j * (exp(l) - exp(-l))) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 4.9e-14) || !(l <= 5.6e+46)) {
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = (J * (Math.exp(l) - Math.exp(-l))) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= 4.9e-14) or not (l <= 5.6e+46):
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * math.cos((K / 2.0))) + U
	else:
		tmp = (J * (math.exp(l) - math.exp(-l))) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= 4.9e-14) || !(l <= 5.6e+46))
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= 4.9e-14) || ~((l <= 5.6e+46)))
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * cos((K / 2.0))) + U;
	else
		tmp = (J * (exp(l) - exp(-l))) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, 4.9e-14], N[Not[LessEqual[l, 5.6e+46]], $MachinePrecision]], N[(N[(N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.89999999999999995e-14 or 5.60000000000000037e46 < l

   1. Initial program 84.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 96.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 96.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{\ell}^{2} \cdot 0.016666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 96.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 96.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 96.8%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 96.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. Applied egg-rr 96.8%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 4.89999999999999995e-14 < l < 5.60000000000000037e46

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0 88.2%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} - e^{-\ell}\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.9 \cdot 10^{-14} \lor \neg \left(\ell \leq 5.6 \cdot 10^{+46}\right):\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(e^{\ell} - e^{-\ell}\right) + U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  (*
   (*
    J
    (*
     l
     (+
      2.0
      (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))
   (cos (/ K 2.0)))
  U))

C

double code(double J, double l, double K, double U) {
	return ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 85.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 92.5%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 92.5%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{\ell}^{2} \cdot 0.016666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Simplified 92.5%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
6. Step-by-step derivation

   1. unpow2 92.5%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

7. Applied egg-rr 92.5%

   \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
8. Step-by-step derivation

   1. unpow2 92.5%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

9. Applied egg-rr 92.5%

   \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
10. Add Preprocessing

Alternative 5: 56.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+170}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -920:\\ \;\;\;\;\frac{16 - U \cdot U}{U + -4}\\ \mathbf{elif}\;\ell \leq 10500:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+195}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* J (* l 2.0)) (+ 5.0 (* -8.0 (* K K)))))))
   (if (<= l -6e+170)
     t_0
     (if (<= l -920.0)
       (/ (- 16.0 (* U U)) (+ U -4.0))
       (if (<= l 10500.0)
         (+ U (* l (* J 2.0)))
         (if (<= l 2.05e+195) (pow U -4.0) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	double tmp;
	if (l <= -6e+170) {
		tmp = t_0;
	} else if (l <= -920.0) {
		tmp = (16.0 - (U * U)) / (U + -4.0);
	} else if (l <= 10500.0) {
		tmp = U + (l * (J * 2.0));
	} else if (l <= 2.05e+195) {
		tmp = pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((j * (l * 2.0d0)) * (5.0d0 + ((-8.0d0) * (k * k))))
    if (l <= (-6d+170)) then
        tmp = t_0
    else if (l <= (-920.0d0)) then
        tmp = (16.0d0 - (u * u)) / (u + (-4.0d0))
    else if (l <= 10500.0d0) then
        tmp = u + (l * (j * 2.0d0))
    else if (l <= 2.05d+195) then
        tmp = u ** (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	double tmp;
	if (l <= -6e+170) {
		tmp = t_0;
	} else if (l <= -920.0) {
		tmp = (16.0 - (U * U)) / (U + -4.0);
	} else if (l <= 10500.0) {
		tmp = U + (l * (J * 2.0));
	} else if (l <= 2.05e+195) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))))
	tmp = 0
	if l <= -6e+170:
		tmp = t_0
	elif l <= -920.0:
		tmp = (16.0 - (U * U)) / (U + -4.0)
	elif l <= 10500.0:
		tmp = U + (l * (J * 2.0))
	elif l <= 2.05e+195:
		tmp = math.pow(U, -4.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(5.0 + Float64(-8.0 * Float64(K * K)))))
	tmp = 0.0
	if (l <= -6e+170)
		tmp = t_0;
	elseif (l <= -920.0)
		tmp = Float64(Float64(16.0 - Float64(U * U)) / Float64(U + -4.0));
	elseif (l <= 10500.0)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	elseif (l <= 2.05e+195)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	tmp = 0.0;
	if (l <= -6e+170)
		tmp = t_0;
	elseif (l <= -920.0)
		tmp = (16.0 - (U * U)) / (U + -4.0);
	elseif (l <= 10500.0)
		tmp = U + (l * (J * 2.0));
	elseif (l <= 2.05e+195)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(5.0 + N[(-8.0 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6e+170], t$95$0, If[LessEqual[l, -920.0], N[(N[(16.0 - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 10500.0], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.05e+195], N[Power[U, -4.0], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.99999999999999994e170 or 2.05e195 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 53.1%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 53.1%

         \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 53.1%

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 53.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 33.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 33.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 33.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 33.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 33.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 33.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 33.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \left(K \cdot -4\right) + \color{blue}{4}\right) + U \]

   9. Simplified 33.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) + 4\right)} + U \]

   10. Taylor expanded in K around 0 52.2%

       \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(5 + -8 \cdot {K}^{2}\right)} + U \]
   11. Step-by-step derivation

       1. unpow2 52.2%

          \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   12. Applied egg-rr 52.2%

       \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   if -5.99999999999999994e170 < l < -920

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{-4 - U} \]
   5. Step-by-step derivation

      1. sub-neg 3.2%

         \[\leadsto \color{blue}{-4 + \left(-U\right)} \]

      2. flip-+ 34.2%

         \[\leadsto \color{blue}{\frac{-4 \cdot -4 - \left(-U\right) \cdot \left(-U\right)}{-4 - \left(-U\right)}} \]

      3. metadata-eval 34.2%

         \[\leadsto \frac{\color{blue}{16} - \left(-U\right) \cdot \left(-U\right)}{-4 - \left(-U\right)} \]

   6. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{\frac{16 - \left(-U\right) \cdot \left(-U\right)}{-4 - \left(-U\right)}} \]

   if -920 < l < 10500

   1. Initial program 75.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 98.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.3%

         \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 98.3%

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 98.3%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 84.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} + U \]
   7. Step-by-step derivation

      1. associate-*r* 84.2%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \ell} + U \]

      2. *-commutative 84.2%

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot J\right)} + U \]

   8. Simplified 84.2%

      \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot J\right)} + U \]

   if 10500 < l < 2.05e195

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 31.7%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+170}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\ell \leq -920:\\ \;\;\;\;\frac{16 - U \cdot U}{U + -4}\\ \mathbf{elif}\;\ell \leq 10500:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 2.05 \cdot 10^{+195}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-33}:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot t\_0\right) + \frac{U}{J}\right)\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+17}:\\ \;\;\;\;U + \ell \cdot \left(t\_0 \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+195}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (if (<= l -2.4e-33)
     (* J (+ (* 2.0 (* l t_0)) (/ U J)))
     (if (<= l 5.4e+17)
       (+ U (* l (* t_0 (* J 2.0))))
       (if (<= l 1.95e+195)
         (pow U -4.0)
         (+ U (* (* J (* l 2.0)) (+ 5.0 (* -8.0 (* K K))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double tmp;
	if (l <= -2.4e-33) {
		tmp = J * ((2.0 * (l * t_0)) + (U / J));
	} else if (l <= 5.4e+17) {
		tmp = U + (l * (t_0 * (J * 2.0)));
	} else if (l <= 1.95e+195) {
		tmp = pow(U, -4.0);
	} else {
		tmp = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    if (l <= (-2.4d-33)) then
        tmp = j * ((2.0d0 * (l * t_0)) + (u / j))
    else if (l <= 5.4d+17) then
        tmp = u + (l * (t_0 * (j * 2.0d0)))
    else if (l <= 1.95d+195) then
        tmp = u ** (-4.0d0)
    else
        tmp = u + ((j * (l * 2.0d0)) * (5.0d0 + ((-8.0d0) * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -2.4e-33) {
		tmp = J * ((2.0 * (l * t_0)) + (U / J));
	} else if (l <= 5.4e+17) {
		tmp = U + (l * (t_0 * (J * 2.0)));
	} else if (l <= 1.95e+195) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	tmp = 0
	if l <= -2.4e-33:
		tmp = J * ((2.0 * (l * t_0)) + (U / J))
	elif l <= 5.4e+17:
		tmp = U + (l * (t_0 * (J * 2.0)))
	elif l <= 1.95e+195:
		tmp = math.pow(U, -4.0)
	else:
		tmp = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -2.4e-33)
		tmp = Float64(J * Float64(Float64(2.0 * Float64(l * t_0)) + Float64(U / J)));
	elseif (l <= 5.4e+17)
		tmp = Float64(U + Float64(l * Float64(t_0 * Float64(J * 2.0))));
	elseif (l <= 1.95e+195)
		tmp = U ^ -4.0;
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(5.0 + Float64(-8.0 * Float64(K * K)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -2.4e-33)
		tmp = J * ((2.0 * (l * t_0)) + (U / J));
	elseif (l <= 5.4e+17)
		tmp = U + (l * (t_0 * (J * 2.0)));
	elseif (l <= 1.95e+195)
		tmp = U ^ -4.0;
	else
		tmp = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -2.4e-33], N[(J * N[(N[(2.0 * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.4e+17], N[(U + N[(l * N[(t$95$0 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.95e+195], N[Power[U, -4.0], $MachinePrecision], N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(5.0 + N[(-8.0 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.4e-33

   1. Initial program 95.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 36.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 36.7%

         \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 36.7%

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 36.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in J around inf 45.0%

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + \frac{U}{J}\right)} \]

   if -2.4e-33 < l < 5.4e17

   1. Initial program 76.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 96.0%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative 96.0%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)}\right) + U \]

      2. associate-*r* 96.0%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \ell\right)} + U \]

      3. associate-*l* 96.0%

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot \ell} + U \]

      4. *-commutative 96.0%

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

      5. *-commutative 96.0%

         \[\leadsto \ell \cdot \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot 2\right)} + U \]

      6. *-commutative 96.0%

         \[\leadsto \ell \cdot \left(\color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot J\right)} \cdot 2\right) + U \]

      7. associate-*l* 96.0%

         \[\leadsto \ell \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right)\right)} + U \]

   5. Simplified 96.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot 2\right)\right)} + U \]

   if 5.4e17 < l < 1.9499999999999999e195

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 36.3%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if 1.9499999999999999e195 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 55.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 55.8%

         \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 55.8%

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 55.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 34.4%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \left(K \cdot -4\right) + \color{blue}{4}\right) + U \]

   9. Simplified 34.4%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) + 4\right)} + U \]

   10. Taylor expanded in K around 0 59.5%

       \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(5 + -8 \cdot {K}^{2}\right)} + U \]
   11. Step-by-step derivation

       1. unpow2 59.5%

          \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   12. Applied egg-rr 59.5%

       \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{-33}:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) + \frac{U}{J}\right)\\ \mathbf{elif}\;\ell \leq 5.4 \cdot 10^{+17}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+195}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{+20}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+195}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 3.4e+20)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (if (<= l 5.5e+195)
     (pow U -4.0)
     (+ U (* (* J (* l 2.0)) (+ 5.0 (* -8.0 (* K K))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3.4e+20) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 5.5e+195) {
		tmp = pow(U, -4.0);
	} else {
		tmp = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 3.4d+20) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 5.5d+195) then
        tmp = u ** (-4.0d0)
    else
        tmp = u + ((j * (l * 2.0d0)) * (5.0d0 + ((-8.0d0) * (k * k))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3.4e+20) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 5.5e+195) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 3.4e+20:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 5.5e+195:
		tmp = math.pow(U, -4.0)
	else:
		tmp = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 3.4e+20)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 5.5e+195)
		tmp = U ^ -4.0;
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(5.0 + Float64(-8.0 * Float64(K * K)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 3.4e+20)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 5.5e+195)
		tmp = U ^ -4.0;
	else
		tmp = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 3.4e+20], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5e+195], N[Power[U, -4.0], $MachinePrecision], N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(5.0 + N[(-8.0 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 3.4e20

   1. Initial program 82.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 79.2%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if 3.4e20 < l < 5.49999999999999994e195

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 36.3%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if 5.49999999999999994e195 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 55.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 55.8%

         \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 55.8%

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 55.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 34.4%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 34.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \left(K \cdot -4\right) + \color{blue}{4}\right) + U \]

   9. Simplified 34.4%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) + 4\right)} + U \]

   10. Taylor expanded in K around 0 59.5%

       \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(5 + -8 \cdot {K}^{2}\right)} + U \]
   11. Step-by-step derivation

       1. unpow2 59.5%

          \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   12. Applied egg-rr 59.5%

       \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{+20}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+195}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right) + 1\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (* U (+ (* 2.0 (* J (/ (* l (cos (* K 0.5))) U))) 1.0)))

C

double code(double J, double l, double K, double U) {
	return U * ((2.0 * (J * ((l * cos((K * 0.5))) / U))) + 1.0);
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u * ((2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))) + 1.0d0)
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * ((2.0 * (J * ((l * Math.cos((K * 0.5))) / U))) + 1.0);
}

Python

def code(J, l, K, U):
	return U * ((2.0 * (J * ((l * math.cos((K * 0.5))) / U))) + 1.0)

Julia

function code(J, l, K, U)
	return Float64(U * Float64(Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U))) + 1.0))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * ((2.0 * (J * ((l * cos((K * 0.5))) / U))) + 1.0);
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 68.6%

   \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 68.6%

      \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. associate-*l* 68.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Simplified 68.6%

   \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

6. Taylor expanded in U around inf 71.5%

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
7. Step-by-step derivation

   1. associate-/l* 74.9%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

8. Simplified 74.9%

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

9. Final simplification 74.9%

   \[\leadsto U \cdot \left(2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right) + 1\right) \]
10. Add Preprocessing

Alternative 9: 57.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+170}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1350000:\\ \;\;\;\;\frac{16 - U \cdot U}{U + -4}\\ \mathbf{elif}\;\ell \leq 245:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* J (* l 2.0)) (+ 5.0 (* -8.0 (* K K)))))))
   (if (<= l -3.6e+170)
     t_0
     (if (<= l -1350000.0)
       (/ (- 16.0 (* U U)) (+ U -4.0))
       (if (<= l 245.0) (+ U (* l (* J 2.0))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	double tmp;
	if (l <= -3.6e+170) {
		tmp = t_0;
	} else if (l <= -1350000.0) {
		tmp = (16.0 - (U * U)) / (U + -4.0);
	} else if (l <= 245.0) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((j * (l * 2.0d0)) * (5.0d0 + ((-8.0d0) * (k * k))))
    if (l <= (-3.6d+170)) then
        tmp = t_0
    else if (l <= (-1350000.0d0)) then
        tmp = (16.0d0 - (u * u)) / (u + (-4.0d0))
    else if (l <= 245.0d0) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	double tmp;
	if (l <= -3.6e+170) {
		tmp = t_0;
	} else if (l <= -1350000.0) {
		tmp = (16.0 - (U * U)) / (U + -4.0);
	} else if (l <= 245.0) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))))
	tmp = 0
	if l <= -3.6e+170:
		tmp = t_0
	elif l <= -1350000.0:
		tmp = (16.0 - (U * U)) / (U + -4.0)
	elif l <= 245.0:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(5.0 + Float64(-8.0 * Float64(K * K)))))
	tmp = 0.0
	if (l <= -3.6e+170)
		tmp = t_0;
	elseif (l <= -1350000.0)
		tmp = Float64(Float64(16.0 - Float64(U * U)) / Float64(U + -4.0));
	elseif (l <= 245.0)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (l * 2.0)) * (5.0 + (-8.0 * (K * K))));
	tmp = 0.0;
	if (l <= -3.6e+170)
		tmp = t_0;
	elseif (l <= -1350000.0)
		tmp = (16.0 - (U * U)) / (U + -4.0);
	elseif (l <= 245.0)
		tmp = U + (l * (J * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(5.0 + N[(-8.0 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.6e+170], t$95$0, If[LessEqual[l, -1350000.0], N[(N[(16.0 - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 245.0], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.6e170 or 245 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 34.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 34.7%

         \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 34.7%

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 34.7%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 23.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 23.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 23.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 23.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 23.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 23.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 23.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \left(K \cdot -4\right) + \color{blue}{4}\right) + U \]

   9. Simplified 23.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) + 4\right)} + U \]

   10. Taylor expanded in K around 0 36.2%

       \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(5 + -8 \cdot {K}^{2}\right)} + U \]
   11. Step-by-step derivation

       1. unpow2 36.2%

          \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   12. Applied egg-rr 36.2%

       \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   if -3.6e170 < l < -1.35e6

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 3.2%

      \[\leadsto \color{blue}{-4 - U} \]
   5. Step-by-step derivation

      1. sub-neg 3.2%

         \[\leadsto \color{blue}{-4 + \left(-U\right)} \]

      2. flip-+ 34.2%

         \[\leadsto \color{blue}{\frac{-4 \cdot -4 - \left(-U\right) \cdot \left(-U\right)}{-4 - \left(-U\right)}} \]

      3. metadata-eval 34.2%

         \[\leadsto \frac{\color{blue}{16} - \left(-U\right) \cdot \left(-U\right)}{-4 - \left(-U\right)} \]

   6. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{\frac{16 - \left(-U\right) \cdot \left(-U\right)}{-4 - \left(-U\right)}} \]

   if -1.35e6 < l < 245

   1. Initial program 74.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 98.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*l* 98.9%

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 84.7%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} + U \]
   7. Step-by-step derivation

      1. associate-*r* 84.7%

         \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \ell} + U \]

      2. *-commutative 84.7%

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot J\right)} + U \]

   8. Simplified 84.7%

      \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot J\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{+170}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\ell \leq -1350000:\\ \;\;\;\;\frac{16 - U \cdot U}{U + -4}\\ \mathbf{elif}\;\ell \leq 245:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(5 + -8 \cdot \left(K \cdot K\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.2% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -980 \lor \neg \left(\ell \leq 265000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -980.0) (not (<= l 265000000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -980.0) || !(l <= 265000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-980.0d0)) .or. (.not. (l <= 265000000.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -980.0) || !(l <= 265000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -980.0) or not (l <= 265000000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -980.0) || !(l <= 265000000.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -980.0) || ~((l <= 265000000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -980.0], N[Not[LessEqual[l, 265000000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -980 or 2.65e8 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 15.1%

      \[\leadsto \color{blue}{U \cdot U} \]

   if -980 < l < 2.65e8

   1. Initial program 75.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 72.2%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -980 \lor \neg \left(\ell \leq 265000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.9% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{-24}:\\ \;\;\;\;J \cdot \left(-4 - \frac{U}{J}\right)\\ \mathbf{elif}\;\ell \leq 66000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.2e-24) (* J (- -4.0 (/ U J))) (if (<= l 66000000.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.2e-24) {
		tmp = J * (-4.0 - (U / J));
	} else if (l <= 66000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-4.2d-24)) then
        tmp = j * ((-4.0d0) - (u / j))
    else if (l <= 66000000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.2e-24) {
		tmp = J * (-4.0 - (U / J));
	} else if (l <= 66000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.2e-24:
		tmp = J * (-4.0 - (U / J))
	elif l <= 66000000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.2e-24)
		tmp = Float64(J * Float64(-4.0 - Float64(U / J)));
	elseif (l <= 66000000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -4.2e-24)
		tmp = J * (-4.0 - (U / J));
	elseif (l <= 66000000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.2e-24], N[(J * N[(-4.0 - N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 66000000.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.1999999999999999e-24

   1. Initial program 97.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 4.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, -4, -U\right)} \]
   5. Step-by-step derivation

      1. fmm-undef 4.0%

         \[\leadsto \color{blue}{J \cdot -4 - U} \]

   6. Simplified 4.0%

      \[\leadsto \color{blue}{J \cdot -4 - U} \]

   7. Taylor expanded in J around inf 12.4%

      \[\leadsto \color{blue}{J \cdot \left(-1 \cdot \frac{U}{J} - 4\right)} \]
   8. Step-by-step derivation

      1. sub-neg 12.4%

         \[\leadsto J \cdot \color{blue}{\left(-1 \cdot \frac{U}{J} + \left(-4\right)\right)} \]

      2. metadata-eval 12.4%

         \[\leadsto J \cdot \left(-1 \cdot \frac{U}{J} + \color{blue}{-4}\right) \]

      3. +-commutative 12.4%

         \[\leadsto J \cdot \color{blue}{\left(-4 + -1 \cdot \frac{U}{J}\right)} \]

      4. mul-1-neg 12.4%

         \[\leadsto J \cdot \left(-4 + \color{blue}{\left(-\frac{U}{J}\right)}\right) \]

      5. unsub-neg 12.4%

         \[\leadsto J \cdot \color{blue}{\left(-4 - \frac{U}{J}\right)} \]

   9. Simplified 12.4%

      \[\leadsto \color{blue}{J \cdot \left(-4 - \frac{U}{J}\right)} \]

   if -4.1999999999999999e-24 < l < 6.6e7

   1. Initial program 75.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0 74.2%

      \[\leadsto \color{blue}{U} \]

   if 6.6e7 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{U \cdot U} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.1% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + \ell \cdot \left(J \cdot 2\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* l (* J 2.0))))

C

double code(double J, double l, double K, double U) {
	return U + (l * (J * 2.0));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (l * (j * 2.0d0))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (l * (J * 2.0));
}

Python

def code(J, l, K, U):
	return U + (l * (J * 2.0))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(l * Float64(J * 2.0)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (l * (J * 2.0));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 68.6%

   \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 68.6%

      \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. associate-*l* 68.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Simplified 68.6%

   \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

6. Taylor expanded in K around 0 56.5%

   \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} + U \]
7. Step-by-step derivation

   1. associate-*r* 56.5%

      \[\leadsto \color{blue}{\left(2 \cdot J\right) \cdot \ell} + U \]

   2. *-commutative 56.5%

      \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot J\right)} + U \]

8. Simplified 56.5%

   \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot J\right)} + U \]

9. Final simplification 56.5%

   \[\leadsto U + \ell \cdot \left(J \cdot 2\right) \]
10. Add Preprocessing

Alternative 13: 47.8% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + 10 \cdot \left(J \cdot \ell\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* 10.0 (* J l))))

C

double code(double J, double l, double K, double U) {
	return U + (10.0 * (J * l));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (10.0d0 * (j * l))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (10.0 * (J * l));
}

Python

def code(J, l, K, U):
	return U + (10.0 * (J * l))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(10.0 * Float64(J * l)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (10.0 * (J * l));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(10.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 68.6%

   \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 68.6%

      \[\leadsto \color{blue}{\left(\left(J \cdot \ell\right) \cdot 2\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. associate-*l* 68.6%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Simplified 68.6%

   \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

7. Applied egg-rr 51.1%

   \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
8. Step-by-step derivation

   1. log1p-undefine 51.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

   2. rem-exp-log 51.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

   3. +-commutative 51.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

   4. associate--l+ 51.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

   5. *-commutative 51.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

   6. metadata-eval 51.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \left(K \cdot -4\right) + \color{blue}{4}\right) + U \]

9. Simplified 51.1%

   \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) + 4\right)} + U \]

10. Taylor expanded in K around 0 51.1%

    \[\leadsto \color{blue}{10 \cdot \left(J \cdot \ell\right)} + U \]
11. Step-by-step derivation

    1. *-commutative 51.1%

       \[\leadsto 10 \cdot \color{blue}{\left(\ell \cdot J\right)} + U \]

12. Simplified 51.1%

    \[\leadsto \color{blue}{10 \cdot \left(\ell \cdot J\right)} + U \]

13. Final simplification 51.1%

    \[\leadsto U + 10 \cdot \left(J \cdot \ell\right) \]
14. Add Preprocessing

Alternative 14: 36.4% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 85.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0 42.0%

   \[\leadsto \color{blue}{U} \]
4. Add Preprocessing

Alternative 15: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -4.0)

C

double code(double J, double l, double K, double U) {
	return -4.0;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = -4.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -4.0;
}

Python

def code(J, l, K, U):
	return -4.0

Julia

function code(J, l, K, U)
	return -4.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -4.0;
end

Wolfram

code[J_, l_, K_, U_] := -4.0

Derivation

1. Initial program 85.9%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

4. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-4 - U} \]

5. Taylor expanded in U around 0 3.0%

   \[\leadsto \color{blue}{-4} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024186 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))