Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.9%

Time: 15.4s

Alternatives: 21

Speedup: 2.5×

• 49.3% 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: 86.2% 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: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return ((cos((K / 2.0)) * sinh(l)) * (2.0 * J)) + 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 = ((cos((k / 2.0d0)) * sinh(l)) * (2.0d0 * j)) + u
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

   1. sinh-undef N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(J \cdot 2\right) \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot 2\right) \cdot \left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right)\right), U\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(J \cdot 2\right)\right), U\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right), \left(J \cdot 2\right)\right), U\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{K}{2}\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

   10. sinh-lowering-sinh.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \left(J \cdot 2\right)\right), U\right) \]

   11. *-lowering-*.f64 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 87.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (*
    U
    (+
     1.0
     (/
      (* J (* (* l (* l (* l 0.3333333333333333))) (+ 1.0 (* -0.125 (* K K)))))
      U)))
   (+ U (* (sinh l) (* 2.0 J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.04) {
		tmp = U * (1.0 + ((J * ((l * (l * (l * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K))))) / U));
	} else {
		tmp = U + (sinh(l) * (2.0 * J));
	}
	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 (cos((k / 2.0d0)) <= (-0.04d0)) then
        tmp = u * (1.0d0 + ((j * ((l * (l * (l * 0.3333333333333333d0))) * (1.0d0 + ((-0.125d0) * (k * k))))) / u))
    else
        tmp = u + (sinh(l) * (2.0d0 * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.04) {
		tmp = U * (1.0 + ((J * ((l * (l * (l * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K))))) / U));
	} else {
		tmp = U + (Math.sinh(l) * (2.0 * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.04:
		tmp = U * (1.0 + ((J * ((l * (l * (l * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K))))) / U))
	else:
		tmp = U + (math.sinh(l) * (2.0 * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.04)
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(Float64(l * Float64(l * Float64(l * 0.3333333333333333))) * Float64(1.0 + Float64(-0.125 * Float64(K * K))))) / U)));
	else
		tmp = Float64(U + Float64(sinh(l) * Float64(2.0 * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.04)
		tmp = U * (1.0 + ((J * ((l * (l * (l * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K))))) / U));
	else
		tmp = U + (sinh(l) * (2.0 * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.04], N[(U * N[(1.0 + N[(N[(J * N[(N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0400000000000000008

   1. Initial program 88.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 87.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 55.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 55.5%

      \[\leadsto \ell \cdot \left(\color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) + U \]

   9. Taylor expanded in l around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right)\right)}\right)\right), U\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(\left(\frac{1}{3} \cdot J\right) \cdot {\ell}^{2}\right)\right)\right), U\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(\left(J \cdot \frac{1}{3}\right) \cdot {\ell}^{2}\right)\right)\right), U\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

       7. *-lowering-*.f64 60.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   11. Simplified 60.0%

       \[\leadsto \ell \cdot \left(\left(1 + -0.125 \cdot \left(K \cdot K\right)\right) \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)}\right) + U \]

   12. Taylor expanded in U around inf

       \[\leadsto \color{blue}{U \cdot \left(1 + \frac{1}{3} \cdot \frac{J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)}{U}\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \color{blue}{\left(1 + \frac{1}{3} \cdot \frac{J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)}{U}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \frac{J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)}{U}\right)}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)}{\color{blue}{U}}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \left(J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)}{U}\right)\right)\right) \]

       5. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{-1}{3} \cdot \left(J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)\right)}{U}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\frac{-1}{3} \cdot J\right) \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)}{U}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(J \cdot \frac{-1}{3}\right) \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)}{U}\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(J \cdot \left(\frac{-1}{3} \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)\right)}{U}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(J \cdot \left(\frac{-1}{3} \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)\right)\right), \color{blue}{U}\right)\right)\right) \]

   14. Simplified 64.5%

       \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)}{U}\right)} \]

   if -0.0400000000000000008 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.0%

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

      1. sinh-undef N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(J \cdot 2\right) \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot 2\right) \cdot \left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right)\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(J \cdot 2\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{K}{2}\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      10. sinh-lowering-sinh.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]
   6. Step-by-step derivation

   7. Simplified 94.2%

      \[\leadsto \left(\color{blue}{1} \cdot \sinh \ell\right) \cdot \left(J \cdot 2\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-7)
   (+ U (* (sinh l) (* 2.0 J)))
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (+
        2.0
        (*
         (* l l)
         (+
          0.3333333333333333
          (*
           l
           (*
            l
            (+
             0.016666666666666666
             (* (* l l) 0.0003968253968253968)))))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-7) {
		tmp = U + (sinh(l) * (2.0 * J));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
	}
	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 ((k / 2.0d0) <= 5d-7) then
        tmp = u + (sinh(l) * (2.0d0 * j))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-7) {
		tmp = U + (Math.sinh(l) * (2.0 * J));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-7:
		tmp = U + (math.sinh(l) * (2.0 * J))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-7)
		tmp = Float64(U + Float64(sinh(l) * Float64(2.0 * J)));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-7)
		tmp = U + (sinh(l) * (2.0 * J));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-7], N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.99999999999999977e-7

   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. Step-by-step derivation

      1. sinh-undef N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(J \cdot 2\right) \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot 2\right) \cdot \left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right)\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(J \cdot 2\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{K}{2}\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      10. sinh-lowering-sinh.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]
   6. Step-by-step derivation

   7. Simplified 85.0%

      \[\leadsto \left(\color{blue}{1} \cdot \sinh \ell\right) \cdot \left(J \cdot 2\right) + U \]

   if 4.99999999999999977e-7 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 89.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \left(\frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \left({\ell}^{2} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      15. *-lowering-*.f64 94.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 94.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-7)
   (+ U (* (sinh l) (* 2.0 J)))
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (+
        2.0
        (*
         (* l l)
         (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-7) {
		tmp = U + (sinh(l) * (2.0 * J));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	}
	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 ((k / 2.0d0) <= 5d-7) then
        tmp = u + (sinh(l) * (2.0d0 * j))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-7) {
		tmp = U + (Math.sinh(l) * (2.0 * J));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-7:
		tmp = U + (math.sinh(l) * (2.0 * J))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-7)
		tmp = Float64(U + Float64(sinh(l) * Float64(2.0 * J)));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-7)
		tmp = U + (sinh(l) * (2.0 * J));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-7], N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * 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]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.99999999999999977e-7

   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. Step-by-step derivation

      1. sinh-undef N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(J \cdot 2\right) \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot 2\right) \cdot \left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right)\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(J \cdot 2\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{K}{2}\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      10. sinh-lowering-sinh.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]
   6. Step-by-step derivation

   7. Simplified 85.0%

      \[\leadsto \left(\color{blue}{1} \cdot \sinh \ell\right) \cdot \left(J \cdot 2\right) + U \]

   if 4.99999999999999977e-7 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 89.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 92.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 92.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-7)
   (+ U (* (sinh l) (* 2.0 J)))
   (+
    U
    (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-7) {
		tmp = U + (sinh(l) * (2.0 * J));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	}
	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 ((k / 2.0d0) <= 5d-7) then
        tmp = u + (sinh(l) * (2.0d0 * j))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-7) {
		tmp = U + (Math.sinh(l) * (2.0 * J));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-7:
		tmp = U + (math.sinh(l) * (2.0 * J))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-7)
		tmp = Float64(U + Float64(sinh(l) * Float64(2.0 * J)));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-7)
		tmp = U + (sinh(l) * (2.0 * J));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-7], N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.99999999999999977e-7

   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. Step-by-step derivation

      1. sinh-undef N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(J \cdot 2\right) \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot 2\right) \cdot \left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right)\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(J \cdot 2\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{K}{2}\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      10. sinh-lowering-sinh.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]
   6. Step-by-step derivation

   7. Simplified 85.0%

      \[\leadsto \left(\color{blue}{1} \cdot \sinh \ell\right) \cdot \left(J \cdot 2\right) + U \]

   if 4.99999999999999977e-7 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 89.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 87.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 87.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-7)
   (+ U (* (sinh l) (* 2.0 J)))
   (+
    U
    (* l (* (cos (* K 0.5)) (* J (+ 2.0 (* (* l l) 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-7) {
		tmp = U + (sinh(l) * (2.0 * J));
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * (2.0 + ((l * l) * 0.3333333333333333)))));
	}
	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 ((k / 2.0d0) <= 5d-7) then
        tmp = u + (sinh(l) * (2.0d0 * j))
    else
        tmp = u + (l * (cos((k * 0.5d0)) * (j * (2.0d0 + ((l * l) * 0.3333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-7) {
		tmp = U + (Math.sinh(l) * (2.0 * J));
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * (2.0 + ((l * l) * 0.3333333333333333)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-7:
		tmp = U + (math.sinh(l) * (2.0 * J))
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * (2.0 + ((l * l) * 0.3333333333333333)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-7)
		tmp = Float64(U + Float64(sinh(l) * Float64(2.0 * J)));
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-7)
		tmp = U + (sinh(l) * (2.0 * J));
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * (2.0 + ((l * l) * 0.3333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-7], N[(U + N[(N[Sinh[l], $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.99999999999999977e-7

   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. Step-by-step derivation

      1. sinh-undef N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \left(2 \cdot \sinh \ell\right)\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(J \cdot 2\right) \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right)\right), U\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot 2\right) \cdot \left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right)\right), U\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \left(J \cdot 2\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\sinh \ell \cdot \cos \left(\frac{K}{2}\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\cos \left(\frac{K}{2}\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \sinh \ell\right), \left(J \cdot 2\right)\right), U\right) \]

      10. sinh-lowering-sinh.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \left(J \cdot 2\right)\right), U\right) \]

      11. *-lowering-*.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right), \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sinh.f64}\left(\ell\right)\right), \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]
   6. Step-by-step derivation

   7. Simplified 85.0%

      \[\leadsto \left(\color{blue}{1} \cdot \sinh \ell\right) \cdot \left(J \cdot 2\right) + U \]

   if 4.99999999999999977e-7 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 89.5%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 83.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-7}:\\ \;\;\;\;U + \sinh \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ U + \frac{J}{\frac{0.5}{\sinh \ell}} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (/ J (/ 0.5 (sinh l)))))

C

double code(double J, double l, double K, double U) {
	return U + (J / (0.5 / sinh(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 + (j / (0.5d0 / sinh(l)))
end function

Java

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

Python

def code(J, l, K, U):
	return U + (J / (0.5 / math.sinh(l)))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J / Float64(0.5 / sinh(l))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J / (0.5 / sinh(l)));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J / N[(0.5 / N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.8%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right), U\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   3. sinh-undef N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\right), U\right) \]

   4. sinh-def N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(2 \cdot \frac{e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}}{2}\right)\right), U\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \frac{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}{2}\right), U\right) \]

   6. clear-num N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \frac{1}{\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}}\right), U\right) \]

   7. un-div-inv N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{\cos \left(\frac{K}{2}\right) \cdot J}{\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}}\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \cos \left(\frac{K}{2}\right)\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

   11. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

   13. clear-num N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{1}{\frac{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}{2}}\right)\right), U\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{1}{2 \cdot \frac{e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}}{2}}\right)\right), U\right) \]

   15. sinh-def N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{1}{2 \cdot \sinh \ell}\right)\right), U\right) \]

   16. associate-/r* N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{\frac{1}{2}}{\sinh \ell}\right)\right), U\right) \]

   17. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2}\right), \sinh \ell\right)\right), U\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \sinh \ell\right)\right), U\right) \]

   19. sinh-lowering-sinh.f64 99.9%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sinh.f64}\left(\ell\right)\right)\right), U\right) \]

4. Applied egg-rr 99.9%

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

5. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{J}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sinh.f64}\left(\ell\right)\right)\right), U\right) \]
6. Step-by-step derivation

7. Simplified 81.3%

   \[\leadsto \frac{\color{blue}{J}}{\frac{0.5}{\sinh \ell}} + U \]

8. Final simplification 81.3%

   \[\leadsto U + \frac{J}{\frac{0.5}{\sinh \ell}} \]
9. Add Preprocessing

Alternative 8: 77.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+219}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -106000000:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -6e+219)
   (* J (* l (* (* l l) 0.3333333333333333)))
   (if (<= l -106000000.0)
     (+
      U
      (*
       (*
        J
        (*
         l
         (+
          2.0
          (*
           (* l l)
           (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))
       (+ 1.0 (* -0.125 (* K K)))))
     (+
      U
      (*
       (* l J)
       (+
        2.0
        (*
         (* l l)
         (+
          0.3333333333333333
          (*
           (* l l)
           (+ 0.016666666666666666 (* l (* l 0.0003968253968253968))))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6e+219) {
		tmp = J * (l * ((l * l) * 0.3333333333333333));
	} else if (l <= -106000000.0) {
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	} else {
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))));
	}
	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 <= (-6d+219)) then
        tmp = j * (l * ((l * l) * 0.3333333333333333d0))
    else if (l <= (-106000000.0d0)) then
        tmp = u + ((j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))) * (1.0d0 + ((-0.125d0) * (k * k))))
    else
        tmp = u + ((l * j) * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * (0.016666666666666666d0 + (l * (l * 0.0003968253968253968d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6e+219) {
		tmp = J * (l * ((l * l) * 0.3333333333333333));
	} else if (l <= -106000000.0) {
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	} else {
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -6e+219:
		tmp = J * (l * ((l * l) * 0.3333333333333333))
	elif l <= -106000000.0:
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))))
	else:
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -6e+219)
		tmp = Float64(J * Float64(l * Float64(Float64(l * l) * 0.3333333333333333)));
	elseif (l <= -106000000.0)
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))) * Float64(1.0 + Float64(-0.125 * Float64(K * K)))));
	else
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(l * Float64(l * 0.0003968253968253968)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -6e+219)
		tmp = J * (l * ((l * l) * 0.3333333333333333));
	elseif (l <= -106000000.0)
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	else
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -6e+219], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -106000000.0], N[(U + 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[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(l * N[(l * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.9999999999999995e219

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 88.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 88.9%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{3} \cdot J\right) \cdot \color{blue}{{\ell}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \left(J \cdot \frac{1}{3}\right) \cdot {\color{blue}{\ell}}^{3} \]

       3. associate-*l* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       4. unpow3 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\ell}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 88.9%

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   11. Simplified 88.9%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -5.9999999999999995e219 < l < -1.06e8

   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

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 72.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 72.2%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 69.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right)\right), U\right) \]

   8. Simplified 69.6%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} + U \]

   if -1.06e8 < l

   1. Initial program 80.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \left(\frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \left({\ell}^{2} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      15. *-lowering-*.f64 95.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 95.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(J \cdot \ell\right), \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \left(\frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \left({\ell}^{2} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \left(\ell \cdot \left(\ell \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

      17. *-lowering-*.f64 81.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 81.3%

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right)\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+219}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -106000000:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.0% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+218}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -250000000:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5e+218)
   (* J (* l (* (* l l) 0.3333333333333333)))
   (if (<= l -250000000.0)
     (*
      U
      (+
       1.0
       (/
        (*
         J
         (* (* l (* l (* l 0.3333333333333333))) (+ 1.0 (* -0.125 (* K K)))))
        U)))
     (+
      U
      (*
       J
       (*
        l
        (+
         2.0
         (*
          (* l l)
          (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5e+218) {
		tmp = J * (l * ((l * l) * 0.3333333333333333));
	} else if (l <= -250000000.0) {
		tmp = U * (1.0 + ((J * ((l * (l * (l * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K))))) / U));
	} else {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	}
	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 <= (-5d+218)) then
        tmp = j * (l * ((l * l) * 0.3333333333333333d0))
    else if (l <= (-250000000.0d0)) then
        tmp = u * (1.0d0 + ((j * ((l * (l * (l * 0.3333333333333333d0))) * (1.0d0 + ((-0.125d0) * (k * k))))) / u))
    else
        tmp = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5e+218) {
		tmp = J * (l * ((l * l) * 0.3333333333333333));
	} else if (l <= -250000000.0) {
		tmp = U * (1.0 + ((J * ((l * (l * (l * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K))))) / U));
	} else {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5e+218:
		tmp = J * (l * ((l * l) * 0.3333333333333333))
	elif l <= -250000000.0:
		tmp = U * (1.0 + ((J * ((l * (l * (l * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K))))) / U))
	else:
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5e+218)
		tmp = Float64(J * Float64(l * Float64(Float64(l * l) * 0.3333333333333333)));
	elseif (l <= -250000000.0)
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(Float64(l * Float64(l * Float64(l * 0.3333333333333333))) * Float64(1.0 + Float64(-0.125 * Float64(K * K))))) / U)));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5e+218)
		tmp = J * (l * ((l * l) * 0.3333333333333333));
	elseif (l <= -250000000.0)
		tmp = U * (1.0 + ((J * ((l * (l * (l * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K))))) / U));
	else
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5e+218], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -250000000.0], N[(U * N[(1.0 + N[(N[(J * N[(N[(l * N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + 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]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.99999999999999983e218

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 88.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 88.9%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{3} \cdot J\right) \cdot \color{blue}{{\ell}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \left(J \cdot \frac{1}{3}\right) \cdot {\color{blue}{\ell}}^{3} \]

       3. associate-*l* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       4. unpow3 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\ell}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 88.9%

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   11. Simplified 88.9%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -4.99999999999999983e218 < l < -2.5e8

   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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 54.2%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 53.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 53.6%

      \[\leadsto \ell \cdot \left(\color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) + U \]

   9. Taylor expanded in l around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right)\right)}\right)\right), U\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(\left(\frac{1}{3} \cdot J\right) \cdot {\ell}^{2}\right)\right)\right), U\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(\left(J \cdot \frac{1}{3}\right) \cdot {\ell}^{2}\right)\right)\right), U\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

       7. *-lowering-*.f64 53.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   11. Simplified 53.6%

       \[\leadsto \ell \cdot \left(\left(1 + -0.125 \cdot \left(K \cdot K\right)\right) \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)}\right) + U \]

   12. Taylor expanded in U around inf

       \[\leadsto \color{blue}{U \cdot \left(1 + \frac{1}{3} \cdot \frac{J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)}{U}\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \color{blue}{\left(1 + \frac{1}{3} \cdot \frac{J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)}{U}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{3} \cdot \frac{J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)}{U}\right)}\right)\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)}{\color{blue}{U}}\right)\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(\frac{-1}{3}\right)\right) \cdot \left(J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)}{U}\right)\right)\right) \]

       5. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{-1}{3} \cdot \left(J \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)\right)}{U}\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(\frac{-1}{3} \cdot J\right) \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)}{U}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(\left(J \cdot \frac{-1}{3}\right) \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)}{U}\right)\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\mathsf{neg}\left(J \cdot \left(\frac{-1}{3} \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)\right)}{U}\right)\right)\right) \]

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\mathsf{neg}\left(J \cdot \left(\frac{-1}{3} \cdot \left({\ell}^{3} \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)\right)\right), \color{blue}{U}\right)\right)\right) \]

   14. Simplified 65.4%

       \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)}{U}\right)} \]

   if -2.5e8 < l

   1. Initial program 80.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 94.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 94.8%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \color{blue}{1}\right), U\right) \]
   7. Step-by-step derivation

   8. Simplified 80.3%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \color{blue}{1} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+218}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -250000000:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\left(\ell \cdot \left(\ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 8 \cdot 10^{+140}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J 8e+140)
   (+
    U
    (*
     (* l J)
     (+
      2.0
      (*
       (* l l)
       (+
        0.3333333333333333
        (*
         (* l l)
         (+ 0.016666666666666666 (* l (* l 0.0003968253968253968)))))))))
   (+
    U
    (*
     l
     (*
      (* J (+ 2.0 (* (* l l) 0.3333333333333333)))
      (+ 1.0 (* -0.125 (* K K))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 8e+140) {
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))));
	} else {
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (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 (j <= 8d+140) then
        tmp = u + ((l * j) * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * (0.016666666666666666d0 + (l * (l * 0.0003968253968253968d0))))))))
    else
        tmp = u + (l * ((j * (2.0d0 + ((l * l) * 0.3333333333333333d0))) * (1.0d0 + ((-0.125d0) * (k * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 8e+140) {
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))));
	} else {
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= 8e+140:
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))))
	else:
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= 8e+140)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(l * Float64(l * 0.0003968253968253968)))))))));
	else
		tmp = Float64(U + Float64(l * Float64(Float64(J * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))) * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= 8e+140)
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))));
	else
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, 8e+140], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(l * N[(l * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[(J * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 8.00000000000000047e140

   1. Initial program 88.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \left(\frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \left({\ell}^{2} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      15. *-lowering-*.f64 92.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 92.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(J \cdot \ell\right), \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \left(\frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \left({\ell}^{2} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \left(\ell \cdot \left(\ell \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

      17. *-lowering-*.f64 78.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 78.4%

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right)\right)\right)} + U \]

   if 8.00000000000000047e140 < J

   1. Initial program 70.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 93.3%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 74.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 74.0%

      \[\leadsto \ell \cdot \left(\color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 8 \cdot 10^{+140}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.7% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(U \cdot \left(\frac{1}{J} + \frac{\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)}{U}\right)\right)\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1200:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (*
          J
          (*
           U
           (+ (/ 1.0 J) (/ (* l (+ 2.0 (* l (* l 0.3333333333333333)))) U))))))
   (if (<= l -6.2e+18) t_0 (if (<= l 1200.0) (+ U (* l (* 2.0 J))) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (U * ((1.0 / J) + ((l * (2.0 + (l * (l * 0.3333333333333333)))) / U)));
	double tmp;
	if (l <= -6.2e+18) {
		tmp = t_0;
	} else if (l <= 1200.0) {
		tmp = U + (l * (2.0 * J));
	} 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 = j * (u * ((1.0d0 / j) + ((l * (2.0d0 + (l * (l * 0.3333333333333333d0)))) / u)))
    if (l <= (-6.2d+18)) then
        tmp = t_0
    else if (l <= 1200.0d0) then
        tmp = u + (l * (2.0d0 * j))
    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 = J * (U * ((1.0 / J) + ((l * (2.0 + (l * (l * 0.3333333333333333)))) / U)));
	double tmp;
	if (l <= -6.2e+18) {
		tmp = t_0;
	} else if (l <= 1200.0) {
		tmp = U + (l * (2.0 * J));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (U * ((1.0 / J) + ((l * (2.0 + (l * (l * 0.3333333333333333)))) / U)))
	tmp = 0
	if l <= -6.2e+18:
		tmp = t_0
	elif l <= 1200.0:
		tmp = U + (l * (2.0 * J))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(U * Float64(Float64(1.0 / J) + Float64(Float64(l * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333)))) / U))))
	tmp = 0.0
	if (l <= -6.2e+18)
		tmp = t_0;
	elseif (l <= 1200.0)
		tmp = Float64(U + Float64(l * Float64(2.0 * J)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (U * ((1.0 / J) + ((l * (2.0 + (l * (l * 0.3333333333333333)))) / U)));
	tmp = 0.0;
	if (l <= -6.2e+18)
		tmp = t_0;
	elseif (l <= 1200.0)
		tmp = U + (l * (2.0 * J));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(U * N[(N[(1.0 / J), $MachinePrecision] + N[(N[(l * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.2e+18], t$95$0, If[LessEqual[l, 1200.0], N[(U + N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -6.2e18 or 1200 < 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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 69.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 51.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 51.3%

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

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{U}{J}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{U}{J}\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\frac{U}{J} + \color{blue}{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(\frac{U}{J}\right), \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\color{blue}{\ell} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{*.f64}\left(\ell, \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 54.2%

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 54.2%

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

   12. Taylor expanded in U around inf

       \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(U \cdot \left(\frac{1}{J} + \frac{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}{U}\right)\right)}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \color{blue}{\left(\frac{1}{J} + \frac{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}{U}\right)}\right)\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\left(\frac{1}{J}\right), \color{blue}{\left(\frac{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}{U}\right)}\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \left(\frac{\color{blue}{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}}{U}\right)\right)\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \mathsf{/.f64}\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right), \color{blue}{U}\right)\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right), U\right)\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right), U\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right), U\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right), U\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right), U\right)\right)\right)\right) \]

       11. *-lowering-*.f64 66.4%

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, J\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right), U\right)\right)\right)\right) \]

   14. Simplified 66.4%

       \[\leadsto J \cdot \color{blue}{\left(U \cdot \left(\frac{1}{J} + \frac{\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)}{U}\right)\right)} \]

   if -6.2e18 < l < 1200

   1. Initial program 73.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 95.6%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 81.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 81.9%

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

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(2 \cdot J\right)}\right), U\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot 2\right)\right), U\right) \]

       2. *-lowering-*.f64 81.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   11. Simplified 81.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+18}:\\ \;\;\;\;J \cdot \left(U \cdot \left(\frac{1}{J} + \frac{\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 1200:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(U \cdot \left(\frac{1}{J} + \frac{\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 74.3% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 8.2 \cdot 10^{+140}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J 8.2e+140)
   (+
    U
    (*
     J
     (*
      l
      (+
       2.0
       (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
   (+
    U
    (*
     l
     (*
      (* J (+ 2.0 (* (* l l) 0.3333333333333333)))
      (+ 1.0 (* -0.125 (* K K))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 8.2e+140) {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else {
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (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 (j <= 8.2d+140) then
        tmp = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))
    else
        tmp = u + (l * ((j * (2.0d0 + ((l * l) * 0.3333333333333333d0))) * (1.0d0 + ((-0.125d0) * (k * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 8.2e+140) {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else {
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= 8.2e+140:
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))
	else:
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= 8.2e+140)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))));
	else
		tmp = Float64(U + Float64(l * Float64(Float64(J * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))) * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= 8.2e+140)
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	else
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, 8.2e+140], N[(U + 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]), $MachinePrecision], N[(U + N[(l * N[(N[(J * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 8.1999999999999998e140

   1. Initial program 88.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 90.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 90.0%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \color{blue}{1}\right), U\right) \]
   7. Step-by-step derivation

   8. Simplified 76.6%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \color{blue}{1} + U \]

   if 8.1999999999999998e140 < J

   1. Initial program 70.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 93.3%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 74.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 74.0%

      \[\leadsto \ell \cdot \left(\color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 8.2 \cdot 10^{+140}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.5% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 2.05 \cdot 10^{+139}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J 2.05e+139)
   (+
    U
    (*
     (* l J)
     (+
      2.0
      (* l (* l (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
   (+
    U
    (*
     l
     (*
      (* J (+ 2.0 (* (* l l) 0.3333333333333333)))
      (+ 1.0 (* -0.125 (* K K))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 2.05e+139) {
		tmp = U + ((l * J) * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else {
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (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 (j <= 2.05d+139) then
        tmp = u + ((l * j) * (2.0d0 + (l * (l * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))
    else
        tmp = u + (l * ((j * (2.0d0 + ((l * l) * 0.3333333333333333d0))) * (1.0d0 + ((-0.125d0) * (k * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 2.05e+139) {
		tmp = U + ((l * J) * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else {
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= 2.05e+139:
		tmp = U + ((l * J) * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))
	else:
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= 2.05e+139)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(l * Float64(l * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))));
	else
		tmp = Float64(U + Float64(l * Float64(Float64(J * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))) * Float64(1.0 + Float64(-0.125 * Float64(K * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= 2.05e+139)
		tmp = U + ((l * J) * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	else
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * (1.0 + (-0.125 * (K * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, 2.05e+139], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(l * N[(l * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[(J * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 2.0500000000000001e139

   1. Initial program 88.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 90.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 90.0%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \left(J \cdot \ell\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      14. *-lowering-*.f64 75.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(J, \ell\right)\right), U\right) \]

   8. Simplified 75.7%

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

   if 2.0500000000000001e139 < J

   1. Initial program 70.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 93.3%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 74.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 74.0%

      \[\leadsto \ell \cdot \left(\color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 2.05 \cdot 10^{+139}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.1% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{if}\;\ell \leq -7.7 \cdot 10^{+68}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -400:\\ \;\;\;\;\left(\ell \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(\left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ \mathbf{elif}\;\ell \leq 60000:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l (* (* l l) 0.3333333333333333)))))
   (if (<= l -7.7e+68)
     t_0
     (if (<= l -400.0)
       (* (* l (* J (* l l))) (* (* K K) -0.041666666666666664))
       (if (<= l 60000.0) (+ U (* l (* 2.0 J))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * ((l * l) * 0.3333333333333333));
	double tmp;
	if (l <= -7.7e+68) {
		tmp = t_0;
	} else if (l <= -400.0) {
		tmp = (l * (J * (l * l))) * ((K * K) * -0.041666666666666664);
	} else if (l <= 60000.0) {
		tmp = U + (l * (2.0 * J));
	} 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 = j * (l * ((l * l) * 0.3333333333333333d0))
    if (l <= (-7.7d+68)) then
        tmp = t_0
    else if (l <= (-400.0d0)) then
        tmp = (l * (j * (l * l))) * ((k * k) * (-0.041666666666666664d0))
    else if (l <= 60000.0d0) then
        tmp = u + (l * (2.0d0 * j))
    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 = J * (l * ((l * l) * 0.3333333333333333));
	double tmp;
	if (l <= -7.7e+68) {
		tmp = t_0;
	} else if (l <= -400.0) {
		tmp = (l * (J * (l * l))) * ((K * K) * -0.041666666666666664);
	} else if (l <= 60000.0) {
		tmp = U + (l * (2.0 * J));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l * ((l * l) * 0.3333333333333333))
	tmp = 0
	if l <= -7.7e+68:
		tmp = t_0
	elif l <= -400.0:
		tmp = (l * (J * (l * l))) * ((K * K) * -0.041666666666666664)
	elif l <= 60000.0:
		tmp = U + (l * (2.0 * J))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * Float64(Float64(l * l) * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -7.7e+68)
		tmp = t_0;
	elseif (l <= -400.0)
		tmp = Float64(Float64(l * Float64(J * Float64(l * l))) * Float64(Float64(K * K) * -0.041666666666666664));
	elseif (l <= 60000.0)
		tmp = Float64(U + Float64(l * Float64(2.0 * J)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l * ((l * l) * 0.3333333333333333));
	tmp = 0.0;
	if (l <= -7.7e+68)
		tmp = t_0;
	elseif (l <= -400.0)
		tmp = (l * (J * (l * l))) * ((K * K) * -0.041666666666666664);
	elseif (l <= 60000.0)
		tmp = U + (l * (2.0 * J));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.7e+68], t$95$0, If[LessEqual[l, -400.0], N[(N[(l * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(K * K), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 60000.0], N[(U + N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -7.6999999999999998e68 or 6e4 < 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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 76.2%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 57.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 57.8%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{3} \cdot J\right) \cdot \color{blue}{{\ell}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \left(J \cdot \frac{1}{3}\right) \cdot {\color{blue}{\ell}}^{3} \]

       3. associate-*l* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       4. unpow3 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\ell}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 64.2%

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   11. Simplified 64.2%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -7.6999999999999998e68 < l < -400

   1. Initial program 99.9%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 16.8%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 33.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 33.9%

      \[\leadsto \ell \cdot \left(\color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) + U \]

   9. Taylor expanded in l around inf

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right)\right)}\right)\right), U\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(\left(\frac{1}{3} \cdot J\right) \cdot {\ell}^{2}\right)\right)\right), U\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(\left(J \cdot \frac{1}{3}\right) \cdot {\ell}^{2}\right)\right)\right), U\right) \]

       3. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

       7. *-lowering-*.f64 33.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   11. Simplified 33.9%

       \[\leadsto \ell \cdot \left(\left(1 + -0.125 \cdot \left(K \cdot K\right)\right) \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)}\right) + U \]

   12. Taylor expanded in K around inf

       \[\leadsto \color{blue}{\frac{-1}{24} \cdot \left(J \cdot \left({K}^{2} \cdot {\ell}^{3}\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{-1}{24} \cdot \left(J \cdot \left({\ell}^{3} \cdot \color{blue}{{K}^{2}}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \frac{-1}{24} \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \color{blue}{{K}^{2}}\right) \]

       3. associate-*l* N/A

          \[\leadsto \left(\frac{-1}{24} \cdot \left(J \cdot {\ell}^{3}\right)\right) \cdot \color{blue}{{K}^{2}} \]

       4. *-commutative N/A

          \[\leadsto \left(\left(J \cdot {\ell}^{3}\right) \cdot \frac{-1}{24}\right) \cdot {\color{blue}{K}}^{2} \]

       5. associate-*l* N/A

          \[\leadsto \left(J \cdot {\ell}^{3}\right) \cdot \color{blue}{\left(\frac{-1}{24} \cdot {K}^{2}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(J \cdot {\ell}^{3}\right), \color{blue}{\left(\frac{-1}{24} \cdot {K}^{2}\right)}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\ell}^{3} \cdot J\right), \left(\color{blue}{\frac{-1}{24}} \cdot {K}^{2}\right)\right) \]

       8. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot J\right), \left(\frac{-1}{24} \cdot {K}^{2}\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\left(\ell \cdot {\ell}^{2}\right) \cdot J\right), \left(\frac{-1}{24} \cdot {K}^{2}\right)\right) \]

       10. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \left({\ell}^{2} \cdot J\right)\right), \left(\color{blue}{\frac{-1}{24}} \cdot {K}^{2}\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \left(J \cdot {\ell}^{2}\right)\right), \left(\frac{-1}{24} \cdot {K}^{2}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot {\ell}^{2}\right)\right), \left(\color{blue}{\frac{-1}{24}} \cdot {K}^{2}\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left({\ell}^{2}\right)\right)\right), \left(\frac{-1}{24} \cdot {K}^{2}\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(\ell \cdot \ell\right)\right)\right), \left(\frac{-1}{24} \cdot {K}^{2}\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left(\frac{-1}{24} \cdot {K}^{2}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(\frac{-1}{24}, \color{blue}{\left({K}^{2}\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(\frac{-1}{24}, \left(K \cdot \color{blue}{K}\right)\right)\right) \]

       18. *-lowering-*.f64 33.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{*.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(K, \color{blue}{K}\right)\right)\right) \]

   14. Simplified 33.0%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(-0.041666666666666664 \cdot \left(K \cdot K\right)\right)} \]

   if -400 < l < 6e4

   1. Initial program 72.1%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 85.6%

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

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(2 \cdot J\right)}\right), U\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot 2\right)\right), U\right) \]

       2. *-lowering-*.f64 85.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   11. Simplified 85.6%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.7 \cdot 10^{+68}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -400:\\ \;\;\;\;\left(\ell \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(\left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ \mathbf{elif}\;\ell \leq 60000:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 72.9% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 1.4 \cdot 10^{+143}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J + \left(J \cdot \left(K \cdot K\right)\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J 1.4e+143)
   (+
    U
    (*
     (* l J)
     (+
      2.0
      (* l (* l (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
   (+ U (* l (+ (* 2.0 J) (* (* J (* K K)) -0.25))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 1.4e+143) {
		tmp = U + ((l * J) * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else {
		tmp = U + (l * ((2.0 * J) + ((J * (K * K)) * -0.25)));
	}
	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 (j <= 1.4d+143) then
        tmp = u + ((l * j) * (2.0d0 + (l * (l * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))
    else
        tmp = u + (l * ((2.0d0 * j) + ((j * (k * k)) * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 1.4e+143) {
		tmp = U + ((l * J) * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else {
		tmp = U + (l * ((2.0 * J) + ((J * (K * K)) * -0.25)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= 1.4e+143:
		tmp = U + ((l * J) * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))
	else:
		tmp = U + (l * ((2.0 * J) + ((J * (K * K)) * -0.25)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= 1.4e+143)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(l * Float64(l * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))));
	else
		tmp = Float64(U + Float64(l * Float64(Float64(2.0 * J) + Float64(Float64(J * Float64(K * K)) * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= 1.4e+143)
		tmp = U + ((l * J) * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	else
		tmp = U + (l * ((2.0 * J) + ((J * (K * K)) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, 1.4e+143], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(l * N[(l * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[(2.0 * J), $MachinePrecision] + N[(N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.39999999999999999e143

   1. Initial program 89.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 90.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 90.1%

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

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \left(J \cdot \ell\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      14. *-lowering-*.f64 75.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(J, \ell\right)\right), U\right) \]

   8. Simplified 75.9%

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

   if 1.39999999999999999e143 < J

   1. Initial program 68.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 93.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{-1}{8} \cdot \left(J \cdot \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{-1}{8} \cdot \left(\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right), U\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot J\right) \cdot \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 42.7%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) + \left(\left(K \cdot K\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(J \cdot -0.125\right)\right)} + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right) + 2 \cdot J\right)}\right), U\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(2 \cdot J + \frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right), U\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(2 \cdot J\right), \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(J \cdot 2\right), \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \left(\left(J \cdot {K}^{2}\right) \cdot \frac{-1}{4}\right)\right)\right), U\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \mathsf{*.f64}\left(\left(J \cdot {K}^{2}\right), \frac{-1}{4}\right)\right)\right), U\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \left({K}^{2}\right)\right), \frac{-1}{4}\right)\right)\right), U\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \left(K \cdot K\right)\right), \frac{-1}{4}\right)\right)\right), U\right) \]

       9. *-lowering-*.f64 72.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(K, K\right)\right), \frac{-1}{4}\right)\right)\right), U\right) \]

   11. Simplified 72.7%

       \[\leadsto \ell \cdot \color{blue}{\left(J \cdot 2 + \left(J \cdot \left(K \cdot K\right)\right) \cdot -0.25\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.4 \cdot 10^{+143}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J + \left(J \cdot \left(K \cdot K\right)\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 69.9% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 1.4 \cdot 10^{+143}:\\ \;\;\;\;U + \frac{J}{\frac{0.5}{\ell \cdot \left(1 + \left(\ell \cdot \ell\right) \cdot 0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J + \left(J \cdot \left(K \cdot K\right)\right) \cdot -0.25\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J 1.4e+143)
   (+ U (/ J (/ 0.5 (* l (+ 1.0 (* (* l l) 0.16666666666666666))))))
   (+ U (* l (+ (* 2.0 J) (* (* J (* K K)) -0.25))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 1.4e+143) {
		tmp = U + (J / (0.5 / (l * (1.0 + ((l * l) * 0.16666666666666666)))));
	} else {
		tmp = U + (l * ((2.0 * J) + ((J * (K * K)) * -0.25)));
	}
	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 (j <= 1.4d+143) then
        tmp = u + (j / (0.5d0 / (l * (1.0d0 + ((l * l) * 0.16666666666666666d0)))))
    else
        tmp = u + (l * ((2.0d0 * j) + ((j * (k * k)) * (-0.25d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= 1.4e+143) {
		tmp = U + (J / (0.5 / (l * (1.0 + ((l * l) * 0.16666666666666666)))));
	} else {
		tmp = U + (l * ((2.0 * J) + ((J * (K * K)) * -0.25)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= 1.4e+143:
		tmp = U + (J / (0.5 / (l * (1.0 + ((l * l) * 0.16666666666666666)))))
	else:
		tmp = U + (l * ((2.0 * J) + ((J * (K * K)) * -0.25)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= 1.4e+143)
		tmp = Float64(U + Float64(J / Float64(0.5 / Float64(l * Float64(1.0 + Float64(Float64(l * l) * 0.16666666666666666))))));
	else
		tmp = Float64(U + Float64(l * Float64(Float64(2.0 * J) + Float64(Float64(J * Float64(K * K)) * -0.25))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= 1.4e+143)
		tmp = U + (J / (0.5 / (l * (1.0 + ((l * l) * 0.16666666666666666)))));
	else
		tmp = U + (l * ((2.0 * J) + ((J * (K * K)) * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, 1.4e+143], N[(U + N[(J / N[(0.5 / N[(l * N[(1.0 + N[(N[(l * l), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[(2.0 * J), $MachinePrecision] + N[(N[(J * N[(K * K), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.39999999999999999e143

   1. Initial program 89.0%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      3. sinh-undef N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(2 \cdot \sinh \ell\right)\right), U\right) \]

      4. sinh-def N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \left(2 \cdot \frac{e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}}{2}\right)\right), U\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \frac{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}{2}\right), U\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right) \cdot \frac{1}{\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}}\right), U\right) \]

      7. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\cos \left(\frac{K}{2}\right) \cdot J}{\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}}\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(\cos \left(\frac{K}{2}\right) \cdot J\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left(J \cdot \cos \left(\frac{K}{2}\right)\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \cos \left(\frac{K}{2}\right)\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\left(\frac{K}{2}\right)\right)\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{2}{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}\right)\right), U\right) \]

      13. clear-num N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{1}{\frac{2 \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)}{2}}\right)\right), U\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{1}{2 \cdot \frac{e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}}{2}}\right)\right), U\right) \]

      15. sinh-def N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{1}{2 \cdot \sinh \ell}\right)\right), U\right) \]

      16. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \left(\frac{\frac{1}{2}}{\sinh \ell}\right)\right), U\right) \]

      17. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{2}\right), \sinh \ell\right)\right), U\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \sinh \ell\right)\right), U\right) \]

      19. sinh-lowering-sinh.f64 99.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sinh.f64}\left(\ell\right)\right)\right), U\right) \]

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\color{blue}{J}, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{sinh.f64}\left(\ell\right)\right)\right), U\right) \]
   6. Step-by-step derivation

   7. Simplified 84.4%

      \[\leadsto \frac{\color{blue}{J}}{\frac{0.5}{\sinh \ell}} + U \]

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(\frac{1}{2}, \color{blue}{\left(\ell \cdot \left(1 + \frac{1}{6} \cdot {\ell}^{2}\right)\right)}\right)\right), U\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \left(1 + \frac{1}{6} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \left(\frac{1}{6} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \left({\ell}^{2} \cdot \frac{1}{6}\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{6}\right)\right)\right)\right)\right), U\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{6}\right)\right)\right)\right)\right), U\right) \]

      6. *-lowering-*.f64 71.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(J, \mathsf{/.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{6}\right)\right)\right)\right)\right), U\right) \]

   10. Simplified 71.5%

       \[\leadsto \frac{J}{\frac{0.5}{\color{blue}{\ell \cdot \left(1 + \left(\ell \cdot \ell\right) \cdot 0.16666666666666666\right)}}} + U \]

   if 1.39999999999999999e143 < J

   1. Initial program 68.8%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 93.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) + J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{-1}{8} \cdot \left(J \cdot \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \frac{-1}{8} \cdot \left(\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) + \left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right), U\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot \left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      11. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left(\left(\frac{-1}{8} \cdot J\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right) \cdot {K}^{2}\right)\right)\right), U\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot J\right) \cdot \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \left(\left(\frac{-1}{8} \cdot J\right) \cdot \left({K}^{2} \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 42.7%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) + \left(\left(K \cdot K\right) \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \left(J \cdot -0.125\right)\right)} + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right) + 2 \cdot J\right)}\right), U\right) \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(2 \cdot J + \frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right), U\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(2 \cdot J\right), \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\left(J \cdot 2\right), \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \left(\frac{-1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)\right)\right), U\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \left(\left(J \cdot {K}^{2}\right) \cdot \frac{-1}{4}\right)\right)\right), U\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \mathsf{*.f64}\left(\left(J \cdot {K}^{2}\right), \frac{-1}{4}\right)\right)\right), U\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \left({K}^{2}\right)\right), \frac{-1}{4}\right)\right)\right), U\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \left(K \cdot K\right)\right), \frac{-1}{4}\right)\right)\right), U\right) \]

       9. *-lowering-*.f64 72.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, 2\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(K, K\right)\right), \frac{-1}{4}\right)\right)\right), U\right) \]

   11. Simplified 72.7%

       \[\leadsto \ell \cdot \color{blue}{\left(J \cdot 2 + \left(J \cdot \left(K \cdot K\right)\right) \cdot -0.25\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.4 \cdot 10^{+143}:\\ \;\;\;\;U + \frac{J}{\frac{0.5}{\ell \cdot \left(1 + \left(\ell \cdot \ell\right) \cdot 0.16666666666666666\right)}}\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J + \left(J \cdot \left(K \cdot K\right)\right) \cdot -0.25\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.4% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 5200:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l (* (* l l) 0.3333333333333333)))))
   (if (<= l -3.4e+52) t_0 (if (<= l 5200.0) (+ U (* l (* 2.0 J))) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * ((l * l) * 0.3333333333333333));
	double tmp;
	if (l <= -3.4e+52) {
		tmp = t_0;
	} else if (l <= 5200.0) {
		tmp = U + (l * (2.0 * J));
	} 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 = j * (l * ((l * l) * 0.3333333333333333d0))
    if (l <= (-3.4d+52)) then
        tmp = t_0
    else if (l <= 5200.0d0) then
        tmp = u + (l * (2.0d0 * j))
    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 = J * (l * ((l * l) * 0.3333333333333333));
	double tmp;
	if (l <= -3.4e+52) {
		tmp = t_0;
	} else if (l <= 5200.0) {
		tmp = U + (l * (2.0 * J));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l * ((l * l) * 0.3333333333333333))
	tmp = 0
	if l <= -3.4e+52:
		tmp = t_0
	elif l <= 5200.0:
		tmp = U + (l * (2.0 * J))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * Float64(Float64(l * l) * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -3.4e+52)
		tmp = t_0;
	elseif (l <= 5200.0)
		tmp = Float64(U + Float64(l * Float64(2.0 * J)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l * ((l * l) * 0.3333333333333333));
	tmp = 0.0;
	if (l <= -3.4e+52)
		tmp = t_0;
	elseif (l <= 5200.0)
		tmp = U + (l * (2.0 * J));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.4e+52], t$95$0, If[LessEqual[l, 5200.0], N[(U + N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.4e52 or 5200 < 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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 56.2%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{3} \cdot J\right) \cdot \color{blue}{{\ell}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \left(J \cdot \frac{1}{3}\right) \cdot {\color{blue}{\ell}}^{3} \]

       3. associate-*l* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       4. unpow3 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\ell}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 62.3%

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   11. Simplified 62.3%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -3.4e52 < l < 5200

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 88.8%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 76.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 76.0%

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

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(2 \cdot J\right)}\right), U\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot 2\right)\right), U\right) \]

       2. *-lowering-*.f64 76.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   11. Simplified 76.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq 5200:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 70.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\\ \mathbf{if}\;\ell \leq 950:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* l l) 0.3333333333333333)))
   (if (<= l 950.0) (+ U (* l (* J (+ 2.0 t_0)))) (* J (* l t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (l * l) * 0.3333333333333333;
	double tmp;
	if (l <= 950.0) {
		tmp = U + (l * (J * (2.0 + t_0)));
	} else {
		tmp = J * (l * 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 = (l * l) * 0.3333333333333333d0
    if (l <= 950.0d0) then
        tmp = u + (l * (j * (2.0d0 + t_0)))
    else
        tmp = j * (l * t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (l * l) * 0.3333333333333333;
	double tmp;
	if (l <= 950.0) {
		tmp = U + (l * (J * (2.0 + t_0)));
	} else {
		tmp = J * (l * t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (l * l) * 0.3333333333333333
	tmp = 0
	if l <= 950.0:
		tmp = U + (l * (J * (2.0 + t_0)))
	else:
		tmp = J * (l * t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(l * l) * 0.3333333333333333)
	tmp = 0.0
	if (l <= 950.0)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + t_0))));
	else
		tmp = Float64(J * Float64(l * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l * l) * 0.3333333333333333;
	tmp = 0.0;
	if (l <= 950.0)
		tmp = U + (l * (J * (2.0 + t_0)));
	else
		tmp = J * (l * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]}, If[LessEqual[l, 950.0], N[(U + N[(l * N[(J * N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 950

   1. Initial program 81.7%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 87.4%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 71.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 71.5%

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

   if 950 < 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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 68.1%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 54.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 54.0%

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

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{3} \cdot J\right) \cdot \color{blue}{{\ell}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \left(J \cdot \frac{1}{3}\right) \cdot {\color{blue}{\ell}}^{3} \]

       3. associate-*l* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       4. unpow3 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\ell}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right) \]

       6. associate-*l* N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 64.1%

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   11. Simplified 64.1%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 950:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 950:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 l))))
   (if (<= l -3.4e+52) t_0 (if (<= l 950.0) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double tmp;
	if (l <= -3.4e+52) {
		tmp = t_0;
	} else if (l <= 950.0) {
		tmp = U;
	} 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 = j * (2.0d0 * l)
    if (l <= (-3.4d+52)) then
        tmp = t_0
    else if (l <= 950.0d0) then
        tmp = u
    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 = J * (2.0 * l);
	double tmp;
	if (l <= -3.4e+52) {
		tmp = t_0;
	} else if (l <= 950.0) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 * l)
	tmp = 0
	if l <= -3.4e+52:
		tmp = t_0
	elif l <= 950.0:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(2.0 * l))
	tmp = 0.0
	if (l <= -3.4e+52)
		tmp = t_0;
	elseif (l <= 950.0)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (2.0 * l);
	tmp = 0.0;
	if (l <= -3.4e+52)
		tmp = t_0;
	elseif (l <= 950.0)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.4e+52], t$95$0, If[LessEqual[l, 950.0], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.4e52 or 950 < 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

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      7. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 56.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 56.2%

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

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(2 \cdot J\right)}\right), U\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot 2\right)\right), U\right) \]

       2. *-lowering-*.f64 23.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

   11. Simplified 23.5%

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

   12. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(2 \cdot \ell\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{2}\right)\right) \]

       6. *-lowering-*.f64 23.6%

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{2}\right)\right) \]

   14. Simplified 23.6%

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

   if -3.4e52 < l < 950

   1. Initial program 75.3%

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

   3. Taylor expanded in J around 0

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

   5. Simplified 63.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+52}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;\ell \leq 950:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 54.2% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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 * (2.0d0 * j))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.8%

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

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\left(\frac{1}{3} \cdot J\right) \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   7. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(\left(\frac{1}{3} \cdot J\right) \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

5. Simplified 83.1%

   \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

6. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

   5. *-lowering-*.f64 67.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

8. Simplified 67.6%

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

9. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(2 \cdot J\right)}\right), U\right) \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \left(J \cdot 2\right)\right), U\right) \]

    2. *-lowering-*.f64 53.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, 2\right)\right), U\right) \]

11. Simplified 53.6%

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

12. Final simplification 53.6%

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

Alternative 21: 36.9% 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 85.8%

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

3. Taylor expanded in J around 0

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

5. Simplified 37.3%

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

Reproduce

herbie shell --seed 2024164 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))