Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.1% → 100.0%

Time: 14.6s

Alternatives: 18

Speedup: 2.4×

• 49.5% 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: 86.1% 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: 100.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

   5. sinh-undef N/A

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

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

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

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

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

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

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

   9. /-lowering-/.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 91.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.95:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{t\_0 \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.95)
     (*
      U
      (+ 1.0 (* J (/ (* t_0 (* l (+ 2.0 (* l (* l 0.3333333333333333))))) U))))
     (+
      U
      (*
       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 t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.95) {
		tmp = U * (1.0 + (J * ((t_0 * (l * (2.0 + (l * (l * 0.3333333333333333))))) / U)));
	} else {
		tmp = U + (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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.95d0) then
        tmp = u * (1.0d0 + (j * ((t_0 * (l * (2.0d0 + (l * (l * 0.3333333333333333d0))))) / u)))
    else
        tmp = u + (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 t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.95) {
		tmp = U * (1.0 + (J * ((t_0 * (l * (2.0 + (l * (l * 0.3333333333333333))))) / U)));
	} else {
		tmp = U + (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):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.95:
		tmp = U * (1.0 + (J * ((t_0 * (l * (2.0 + (l * (l * 0.3333333333333333))))) / U)))
	else:
		tmp = U + (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)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.95)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(t_0 * Float64(l * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))))) / U))));
	else
		tmp = Float64(U + 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)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.95)
		tmp = U * (1.0 + (J * ((t_0 * (l * (2.0 + (l * (l * 0.3333333333333333))))) / U)));
	else
		tmp = U + (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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.95], N[(U * N[(1.0 + N[(J * N[(N[(t$95$0 * N[(l * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + 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]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.94999999999999996

   1. Initial program 86.1%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 83.2%

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

   6. Taylor expanded in U around inf

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

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

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

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

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

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

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

   8. Simplified 90.5%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{U}\right)} \]
   9. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

   10. Applied egg-rr 91.5%

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

   if 0.94999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 96.7

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

   5. Simplified 96.7%

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

   6. Taylor expanded in K around 0

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

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

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      5. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      8. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

      9. associate-*l* N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) + U \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

      15. unpow2 N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right) + U \]

      16. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.95:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{\cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.95:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.95)
     (+ U (* t_0 (* J (* l (+ 2.0 (* l (* l 0.3333333333333333)))))))
     (+
      U
      (*
       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 t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.95) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	} else {
		tmp = U + (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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.95d0) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (l * (l * 0.3333333333333333d0))))))
    else
        tmp = u + (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 t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.95) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	} else {
		tmp = U + (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):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.95:
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))))
	else:
		tmp = U + (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)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.95)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333)))))));
	else
		tmp = Float64(U + 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)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.95)
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	else
		tmp = U + (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_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.95], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + 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]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.94999999999999996

   1. Initial program 86.1%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 87.9

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

   5. Simplified 87.9%

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

   if 0.94999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 96.7

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

   5. Simplified 96.7%

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

   6. Taylor expanded in K around 0

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

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

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      5. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      8. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

      9. associate-*l* N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) + U \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

      15. unpow2 N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right) + U \]

      16. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.95:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.95:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.95)
   (+ U (* l (* (+ 2.0 (* l (* l 0.3333333333333333))) (* J (cos (* K 0.5))))))
   (+
    U
    (*
     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 (cos((K / 2.0)) <= 0.95) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * cos((K * 0.5)))));
	} else {
		tmp = U + (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 (cos((k / 2.0d0)) <= 0.95d0) then
        tmp = u + (l * ((2.0d0 + (l * (l * 0.3333333333333333d0))) * (j * cos((k * 0.5d0)))))
    else
        tmp = u + (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 (Math.cos((K / 2.0)) <= 0.95) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (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 math.cos((K / 2.0)) <= 0.95:
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * math.cos((K * 0.5)))))
	else:
		tmp = U + (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 (cos(Float64(K / 2.0)) <= 0.95)
		tmp = Float64(U + Float64(l * Float64(Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))) * Float64(J * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + 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 (cos((K / 2.0)) <= 0.95)
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * cos((K * 0.5)))));
	else
		tmp = U + (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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.95], N[(U + N[(l * N[(N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + 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]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.94999999999999996

   1. Initial program 86.1%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 83.2%

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

   if 0.94999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.2%

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

   3. Taylor expanded in l around 0

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 96.7

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

   5. Simplified 96.7%

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

   6. Taylor expanded in K around 0

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

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

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      5. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      8. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

      9. associate-*l* N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) + U \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

      15. unpow2 N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right) + U \]

      16. *-lowering-*.f64 96.4

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

   8. Simplified 96.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.95:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \left(\ell \cdot 0.3333333333333333\right) \cdot \left(t\_0 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+203}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1200:\\ \;\;\;\;J \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 550000:\\ \;\;\;\;U + \ell \cdot \left(t\_0 \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+128}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (* (* l 0.3333333333333333) (* t_0 (* J (* l l))))))
   (if (<= l -4.2e+203)
     t_1
     (if (<= l -1200.0)
       (*
        J
        (*
         (+ 1.0 (* K (* K -0.125)))
         (*
          l
          (+
           2.0
           (*
            (* l l)
            (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
       (if (<= l 550000.0)
         (+ U (* l (* t_0 (* 2.0 J))))
         (if (<= l 1.05e+128)
           (+
            U
            (*
             J
             (*
              l
              (+
               2.0
               (*
                (* l l)
                (+
                 0.3333333333333333
                 (*
                  l
                  (*
                   l
                   (+
                    0.016666666666666666
                    (* (* l l) 0.0003968253968253968))))))))))
           t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = (l * 0.3333333333333333) * (t_0 * (J * (l * l)));
	double tmp;
	if (l <= -4.2e+203) {
		tmp = t_1;
	} else if (l <= -1200.0) {
		tmp = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else if (l <= 550000.0) {
		tmp = U + (l * (t_0 * (2.0 * J)));
	} else if (l <= 1.05e+128) {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = (l * 0.3333333333333333d0) * (t_0 * (j * (l * l)))
    if (l <= (-4.2d+203)) then
        tmp = t_1
    else if (l <= (-1200.0d0)) then
        tmp = j * ((1.0d0 + (k * (k * (-0.125d0)))) * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))
    else if (l <= 550000.0d0) then
        tmp = u + (l * (t_0 * (2.0d0 * j)))
    else if (l <= 1.05d+128) then
        tmp = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = (l * 0.3333333333333333) * (t_0 * (J * (l * l)));
	double tmp;
	if (l <= -4.2e+203) {
		tmp = t_1;
	} else if (l <= -1200.0) {
		tmp = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else if (l <= 550000.0) {
		tmp = U + (l * (t_0 * (2.0 * J)));
	} else if (l <= 1.05e+128) {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = (l * 0.3333333333333333) * (t_0 * (J * (l * l)))
	tmp = 0
	if l <= -4.2e+203:
		tmp = t_1
	elif l <= -1200.0:
		tmp = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))
	elif l <= 550000.0:
		tmp = U + (l * (t_0 * (2.0 * J)))
	elif l <= 1.05e+128:
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(Float64(l * 0.3333333333333333) * Float64(t_0 * Float64(J * Float64(l * l))))
	tmp = 0.0
	if (l <= -4.2e+203)
		tmp = t_1;
	elseif (l <= -1200.0)
		tmp = Float64(J * Float64(Float64(1.0 + Float64(K * Float64(K * -0.125))) * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))));
	elseif (l <= 550000.0)
		tmp = Float64(U + Float64(l * Float64(t_0 * Float64(2.0 * J))));
	elseif (l <= 1.05e+128)
		tmp = Float64(U + 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))))))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = (l * 0.3333333333333333) * (t_0 * (J * (l * l)));
	tmp = 0.0;
	if (l <= -4.2e+203)
		tmp = t_1;
	elseif (l <= -1200.0)
		tmp = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	elseif (l <= 550000.0)
		tmp = U + (l * (t_0 * (2.0 * J)));
	elseif (l <= 1.05e+128)
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(l * 0.3333333333333333), $MachinePrecision] * N[(t$95$0 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.2e+203], t$95$1, If[LessEqual[l, -1200.0], N[(J * N[(N[(1.0 + N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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], If[LessEqual[l, 550000.0], N[(U + N[(l * N[(t$95$0 * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.05e+128], N[(U + 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], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.19999999999999967e203 or 1.05e128 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 98.2%

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

   6. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

   8. Simplified 98.2%

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

   if -4.19999999999999967e203 < l < -1200

   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

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \left(J \cdot \color{blue}{\left(\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)}\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \color{blue}{\left(\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot {K}^{2}\right)}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot {K}^{2}}\right) + U \]

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

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot {K}^{2}\right)} + U \]

   8. Simplified 20.6%

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

   9. Taylor expanded in J around inf

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

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

          \[\leadsto \color{blue}{J \cdot \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) + \ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + \ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \]

       3. distribute-lft1-in N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot {K}^{2} + 1\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto J \cdot \color{blue}{\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} \]

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

          \[\leadsto J \cdot \left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{{K}^{2} \cdot \frac{-1}{8}}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{\left(K \cdot K\right)} \cdot \frac{-1}{8}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{K \cdot \left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + \color{blue}{K \cdot \left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + K \cdot \color{blue}{\left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + K \cdot \left(K \cdot \frac{-1}{8}\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \]

   11. Simplified 73.2%

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

   if -1200 < l < 5.5e5

   1. Initial program 79.8%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 98.8%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 98.7

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

   8. Simplified 98.7%

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

   if 5.5e5 < l < 1.05e128

   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

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 84.8

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

   5. Simplified 84.8%

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

   6. Taylor expanded in K around 0

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

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

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      5. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      8. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

      9. associate-*l* N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) + U \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

      15. unpow2 N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right) + U \]

      16. *-lowering-*.f64 64.6

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

   8. Simplified 64.6%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+203}:\\ \;\;\;\;\left(\ell \cdot 0.3333333333333333\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1200:\\ \;\;\;\;J \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 550000:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+128}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot 0.3333333333333333\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+204}:\\ \;\;\;\;\left(\ell \cdot 0.3333333333333333\right) \cdot \left(t\_0 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1150:\\ \;\;\;\;J \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{+14}:\\ \;\;\;\;U + \ell \cdot \left(t\_0 \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(\left(t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \frac{0.3333333333333333}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (if (<= l -1e+204)
     (* (* l 0.3333333333333333) (* t_0 (* J (* l l))))
     (if (<= l -1150.0)
       (*
        J
        (*
         (+ 1.0 (* K (* K -0.125)))
         (*
          l
          (+
           2.0
           (*
            (* l l)
            (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
       (if (<= l 1.26e+14)
         (+ U (* l (* t_0 (* 2.0 J))))
         (* U (* (* t_0 (* J (* l (* l l)))) (/ 0.3333333333333333 U))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double tmp;
	if (l <= -1e+204) {
		tmp = (l * 0.3333333333333333) * (t_0 * (J * (l * l)));
	} else if (l <= -1150.0) {
		tmp = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else if (l <= 1.26e+14) {
		tmp = U + (l * (t_0 * (2.0 * J)));
	} else {
		tmp = U * ((t_0 * (J * (l * (l * l)))) * (0.3333333333333333 / U));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    if (l <= (-1d+204)) then
        tmp = (l * 0.3333333333333333d0) * (t_0 * (j * (l * l)))
    else if (l <= (-1150.0d0)) then
        tmp = j * ((1.0d0 + (k * (k * (-0.125d0)))) * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))
    else if (l <= 1.26d+14) then
        tmp = u + (l * (t_0 * (2.0d0 * j)))
    else
        tmp = u * ((t_0 * (j * (l * (l * l)))) * (0.3333333333333333d0 / u))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -1e+204) {
		tmp = (l * 0.3333333333333333) * (t_0 * (J * (l * l)));
	} else if (l <= -1150.0) {
		tmp = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else if (l <= 1.26e+14) {
		tmp = U + (l * (t_0 * (2.0 * J)));
	} else {
		tmp = U * ((t_0 * (J * (l * (l * l)))) * (0.3333333333333333 / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	tmp = 0
	if l <= -1e+204:
		tmp = (l * 0.3333333333333333) * (t_0 * (J * (l * l)))
	elif l <= -1150.0:
		tmp = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))
	elif l <= 1.26e+14:
		tmp = U + (l * (t_0 * (2.0 * J)))
	else:
		tmp = U * ((t_0 * (J * (l * (l * l)))) * (0.3333333333333333 / U))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -1e+204)
		tmp = Float64(Float64(l * 0.3333333333333333) * Float64(t_0 * Float64(J * Float64(l * l))));
	elseif (l <= -1150.0)
		tmp = Float64(J * Float64(Float64(1.0 + Float64(K * Float64(K * -0.125))) * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))));
	elseif (l <= 1.26e+14)
		tmp = Float64(U + Float64(l * Float64(t_0 * Float64(2.0 * J))));
	else
		tmp = Float64(U * Float64(Float64(t_0 * Float64(J * Float64(l * Float64(l * l)))) * Float64(0.3333333333333333 / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -1e+204)
		tmp = (l * 0.3333333333333333) * (t_0 * (J * (l * l)));
	elseif (l <= -1150.0)
		tmp = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	elseif (l <= 1.26e+14)
		tmp = U + (l * (t_0 * (2.0 * J)));
	else
		tmp = U * ((t_0 * (J * (l * (l * l)))) * (0.3333333333333333 / U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1e+204], N[(N[(l * 0.3333333333333333), $MachinePrecision] * N[(t$95$0 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1150.0], N[(J * N[(N[(1.0 + N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 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], If[LessEqual[l, 1.26e+14], N[(U + N[(l * N[(t$95$0 * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(N[(t$95$0 * N[(J * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.99999999999999989e203

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

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

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

   8. Simplified 100.0%

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

   if -9.99999999999999989e203 < l < -1150

   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

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 66.7

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

   5. Simplified 66.7%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \left(J \cdot \color{blue}{\left(\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)}\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \color{blue}{\left(\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot {K}^{2}\right)}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot {K}^{2}}\right) + U \]

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

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot {K}^{2}\right)} + U \]

   8. Simplified 20.6%

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

   9. Taylor expanded in J around inf

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

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

          \[\leadsto \color{blue}{J \cdot \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) + \ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + \ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \]

       3. distribute-lft1-in N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot {K}^{2} + 1\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto J \cdot \color{blue}{\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} \]

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

          \[\leadsto J \cdot \left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{{K}^{2} \cdot \frac{-1}{8}}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{\left(K \cdot K\right)} \cdot \frac{-1}{8}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{K \cdot \left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + \color{blue}{K \cdot \left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + K \cdot \color{blue}{\left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + K \cdot \left(K \cdot \frac{-1}{8}\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \]

   11. Simplified 73.2%

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

   if -1150 < l < 1.26e14

   1. Initial program 80.2%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 96.8%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-lowering-*.f64 96.7

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

   8. Simplified 96.7%

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

   if 1.26e14 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 79.6%

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

   6. Taylor expanded in U around inf

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

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

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

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

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

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

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

   8. Simplified 93.1%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{U}\right)} \]

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

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

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

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

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

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

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

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

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. /-lowering-/.f64 93.2

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

   11. Simplified 93.2%

       \[\leadsto U \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \frac{0.3333333333333333}{U}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+204}:\\ \;\;\;\;\left(\ell \cdot 0.3333333333333333\right) \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -1150:\\ \;\;\;\;J \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.26 \cdot 10^{+14}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(\left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \frac{0.3333333333333333}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+
    U
    (*
     l
     (*
      (+ 2.0 (* l (* l 0.3333333333333333)))
      (+ J (* J (* -0.125 (* K K)))))))
   (+
    U
    (*
     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 (cos((K / 2.0)) <= -0.005) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	} else {
		tmp = U + (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 (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = u + (l * ((2.0d0 + (l * (l * 0.3333333333333333d0))) * (j + (j * ((-0.125d0) * (k * k))))))
    else
        tmp = u + (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 (Math.cos((K / 2.0)) <= -0.005) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	} else {
		tmp = U + (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 math.cos((K / 2.0)) <= -0.005:
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))))
	else:
		tmp = U + (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 (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(U + Float64(l * Float64(Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))) * Float64(J + Float64(J * Float64(-0.125 * Float64(K * K)))))));
	else
		tmp = Float64(U + 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 (cos((K / 2.0)) <= -0.005)
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	else
		tmp = U + (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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(U + N[(l * N[(N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J + N[(J * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + 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]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 76.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 63.0

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

   8. Simplified 63.0%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      12. *-commutative N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      14. unpow2 N/A

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      15. *-lowering-*.f64 96.0

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

   5. Simplified 96.0%

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

   6. Taylor expanded in K around 0

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

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

         \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      5. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

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

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

      8. unpow2 N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right) + U \]

      9. associate-*l* N/A

         \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) + U \]

      13. *-commutative N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

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

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right) + U \]

      15. unpow2 N/A

          \[\leadsto J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right) + U \]

      16. *-lowering-*.f64 92.6

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

   8. Simplified 92.6%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  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) {
	return U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
}

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 + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0))))))))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))))

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
end

Wolfram

code[J_, l_, K_, U_] := 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. Initial program 89.2%

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

3. Taylor expanded in l around 0

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

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

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + {\ell}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. associate-*l* N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)\right)}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {\ell}^{2}\right)}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   12. *-commutative N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{{\ell}^{2} \cdot \frac{1}{2520}}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   14. unpow2 N/A

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \ell \cdot \left(\ell \cdot \left(\frac{1}{60} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{2520}\right)\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   15. *-lowering-*.f64 94.6

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

5. Simplified 94.6%

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

6. Final simplification 94.6%

   \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right) \]
7. Add Preprocessing

Alternative 9: 76.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\\ t_1 := J \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot t\_0\right)\right)\right)\\ \mathbf{if}\;\ell \leq -920:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+67}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))
        (t_1
         (* J (* (+ 1.0 (* K (* K -0.125))) (* l (+ 2.0 (* (* l l) t_0)))))))
   (if (<= l -920.0)
     t_1
     (if (<= l 2.4e+67) (+ U (* l (* J (+ 2.0 (* l (* l t_0)))))) t_1))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 + ((l * l) * 0.016666666666666666);
	double t_1 = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * t_0))));
	double tmp;
	if (l <= -920.0) {
		tmp = t_1;
	} else if (l <= 2.4e+67) {
		tmp = U + (l * (J * (2.0 + (l * (l * t_0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)
    t_1 = j * ((1.0d0 + (k * (k * (-0.125d0)))) * (l * (2.0d0 + ((l * l) * t_0))))
    if (l <= (-920.0d0)) then
        tmp = t_1
    else if (l <= 2.4d+67) then
        tmp = u + (l * (j * (2.0d0 + (l * (l * t_0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 + ((l * l) * 0.016666666666666666);
	double t_1 = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * t_0))));
	double tmp;
	if (l <= -920.0) {
		tmp = t_1;
	} else if (l <= 2.4e+67) {
		tmp = U + (l * (J * (2.0 + (l * (l * t_0)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 + ((l * l) * 0.016666666666666666)
	t_1 = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * t_0))))
	tmp = 0
	if l <= -920.0:
		tmp = t_1
	elif l <= 2.4e+67:
		tmp = U + (l * (J * (2.0 + (l * (l * t_0)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))
	t_1 = Float64(J * Float64(Float64(1.0 + Float64(K * Float64(K * -0.125))) * Float64(l * Float64(2.0 + Float64(Float64(l * l) * t_0)))))
	tmp = 0.0
	if (l <= -920.0)
		tmp = t_1;
	elseif (l <= 2.4e+67)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + Float64(l * Float64(l * t_0))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 + ((l * l) * 0.016666666666666666);
	t_1 = J * ((1.0 + (K * (K * -0.125))) * (l * (2.0 + ((l * l) * t_0))));
	tmp = 0.0;
	if (l <= -920.0)
		tmp = t_1;
	elseif (l <= 2.4e+67)
		tmp = U + (l * (J * (2.0 + (l * (l * t_0)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[(N[(1.0 + N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -920.0], t$95$1, If[LessEqual[l, 2.4e+67], N[(U + N[(l * N[(J * N[(2.0 + N[(l * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if l < -920 or 2.40000000000000002e67 < 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

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 87.9

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

   5. Simplified 87.9%

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

   6. Taylor expanded in K around 0

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \left(J \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \left(J \cdot \color{blue}{\left(\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \cdot {K}^{2}\right)}\right)\right) + U \]

      3. associate-*r* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \frac{-1}{8} \cdot \color{blue}{\left(\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot {K}^{2}\right)}\right) + U \]

      4. associate-*l* N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \color{blue}{\left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot {K}^{2}}\right) + U \]

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

         \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + \left(\frac{-1}{8} \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \cdot {K}^{2}\right)} + U \]

   8. Simplified 7.5%

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

   9. Taylor expanded in J around inf

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

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

          \[\leadsto \color{blue}{J \cdot \left(\frac{-1}{8} \cdot \left({K}^{2} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) + \ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]

       2. associate-*r* N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(\frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + \ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \]

       3. distribute-lft1-in N/A

          \[\leadsto J \cdot \color{blue}{\left(\left(\frac{-1}{8} \cdot {K}^{2} + 1\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} \]

       4. +-commutative N/A

          \[\leadsto J \cdot \left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

          \[\leadsto J \cdot \color{blue}{\left(\left(1 + \frac{-1}{8} \cdot {K}^{2}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)} \]

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

          \[\leadsto J \cdot \left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)} \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{{K}^{2} \cdot \frac{-1}{8}}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{\left(K \cdot K\right)} \cdot \frac{-1}{8}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto J \cdot \left(\left(1 + \color{blue}{K \cdot \left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + \color{blue}{K \cdot \left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + K \cdot \color{blue}{\left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

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

           \[\leadsto J \cdot \left(\left(1 + K \cdot \left(K \cdot \frac{-1}{8}\right)\right) \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \]

   11. Simplified 75.7%

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

   if -920 < l < 2.40000000000000002e67

   1. Initial program 81.0%

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

   3. Taylor expanded in l around 0

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 95.0

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

   5. Simplified 95.0%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\ell \cdot J\right)} \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) + U \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

         \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

         \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

         \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

         \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) + U \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) + U \]

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

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) + U \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) + U \]

      15. *-lowering-*.f64 85.5

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

   8. Simplified 85.5%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -920:\\ \;\;\;\;J \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+67}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.2% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+118}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(\frac{0.3333333333333333}{U} \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1e+118)
   (+
    U
    (*
     l
     (*
      J
      (+
       2.0
       (* l (* l (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))
   (*
    U
    (*
     (/ 0.3333333333333333 U)
     (* (+ 1.0 (* K (* K -0.125))) (* J (* l (* l l))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1e+118) {
		tmp = U + (l * (J * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	} else {
		tmp = U * ((0.3333333333333333 / U) * ((1.0 + (K * (K * -0.125))) * (J * (l * (l * l)))));
	}
	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 <= 1d+118) then
        tmp = u + (l * (j * (2.0d0 + (l * (l * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))))
    else
        tmp = u * ((0.3333333333333333d0 / u) * ((1.0d0 + (k * (k * (-0.125d0)))) * (j * (l * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1e+118) {
		tmp = U + (l * (J * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	} else {
		tmp = U * ((0.3333333333333333 / U) * ((1.0 + (K * (K * -0.125))) * (J * (l * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1e+118:
		tmp = U + (l * (J * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))))
	else:
		tmp = U * ((0.3333333333333333 / U) * ((1.0 + (K * (K * -0.125))) * (J * (l * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1e+118)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + Float64(l * Float64(l * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))));
	else
		tmp = Float64(U * Float64(Float64(0.3333333333333333 / U) * Float64(Float64(1.0 + Float64(K * Float64(K * -0.125))) * Float64(J * Float64(l * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1e+118)
		tmp = U + (l * (J * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	else
		tmp = U * ((0.3333333333333333 / U) * ((1.0 + (K * (K * -0.125))) * (J * (l * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1e+118], N[(U + N[(l * N[(J * N[(2.0 + N[(l * N[(l * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(N[(0.3333333333333333 / U), $MachinePrecision] * N[(N[(1.0 + N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9.99999999999999967e117

   1. Initial program 87.3%

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

   3. Taylor expanded in l around 0

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

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

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      4. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      7. *-commutative N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      9. unpow2 N/A

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      10. *-lowering-*.f64 90.6

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

   5. Simplified 90.6%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\ell \cdot J\right)} \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) + U \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

         \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

         \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

         \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right) + U \]

      7. unpow2 N/A

         \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + U \]

      8. associate-*l* N/A

         \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

         \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) + U \]

      12. *-commutative N/A

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) + U \]

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

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) + U \]

      14. unpow2 N/A

          \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) + U \]

      15. *-lowering-*.f64 77.7

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

   8. Simplified 77.7%

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

   if 9.99999999999999967e117 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 90.1%

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

   6. Taylor expanded in U around inf

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

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

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

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

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

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

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{U}\right)} \]

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

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

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

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

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

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

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

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

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. /-lowering-/.f64 100.0

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

   11. Simplified 100.0%

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

   12. Taylor expanded in K around 0

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

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

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

       2. *-commutative N/A

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

       3. unpow2 N/A

          \[\leadsto U \cdot \left(\left(\left(1 + \color{blue}{\left(K \cdot K\right)} \cdot \frac{-1}{8}\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       4. associate-*l* N/A

          \[\leadsto U \cdot \left(\left(\left(1 + \color{blue}{K \cdot \left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

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

          \[\leadsto U \cdot \left(\left(\left(1 + \color{blue}{K \cdot \left(K \cdot \frac{-1}{8}\right)}\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       6. *-lowering-*.f64 84.2

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

   14. Simplified 84.2%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 10^{+118}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(\frac{0.3333333333333333}{U} \cdot \left(\left(1 + K \cdot \left(K \cdot -0.125\right)\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.2% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1300:\\ \;\;\;\;U \cdot \left(\left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \frac{0.3333333333333333}{U}\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+15}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1300.0)
   (* U (* (* J (* l (* l l))) (/ 0.3333333333333333 U)))
   (if (<= l 4e+15)
     (+ U (* J (* 2.0 l)))
     (* J (* l (* (* l l) 0.3333333333333333))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1300.0) {
		tmp = U * ((J * (l * (l * l))) * (0.3333333333333333 / U));
	} else if (l <= 4e+15) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = J * (l * ((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 (l <= (-1300.0d0)) then
        tmp = u * ((j * (l * (l * l))) * (0.3333333333333333d0 / u))
    else if (l <= 4d+15) then
        tmp = u + (j * (2.0d0 * l))
    else
        tmp = j * (l * ((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 (l <= -1300.0) {
		tmp = U * ((J * (l * (l * l))) * (0.3333333333333333 / U));
	} else if (l <= 4e+15) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = J * (l * ((l * l) * 0.3333333333333333));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1300.0:
		tmp = U * ((J * (l * (l * l))) * (0.3333333333333333 / U))
	elif l <= 4e+15:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = J * (l * ((l * l) * 0.3333333333333333))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1300.0)
		tmp = Float64(U * Float64(Float64(J * Float64(l * Float64(l * l))) * Float64(0.3333333333333333 / U)));
	elseif (l <= 4e+15)
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	else
		tmp = Float64(J * Float64(l * Float64(Float64(l * l) * 0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1300.0)
		tmp = U * ((J * (l * (l * l))) * (0.3333333333333333 / U));
	elseif (l <= 4e+15)
		tmp = U + (J * (2.0 * l));
	else
		tmp = J * (l * ((l * l) * 0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1300.0], N[(U * N[(N[(J * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+15], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1300

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 61.3%

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

   6. Taylor expanded in U around inf

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

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

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

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

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

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

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

   8. Simplified 74.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{U}\right)} \]

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. *-commutative N/A

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

       7. associate-*l* N/A

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

       8. *-commutative N/A

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

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

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

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

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

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

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

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

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

       13. cube-mult N/A

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

       14. unpow2 N/A

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

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

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

       16. unpow2 N/A

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

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

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

       18. /-lowering-/.f64 74.1

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

   11. Simplified 74.1%

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

   12. Taylor expanded in K around 0

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

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

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

       2. cube-mult N/A

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 56.9

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

   14. Simplified 56.9%

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

   if -1300 < l < 4e15

   1. Initial program 80.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 96.0

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

   7. Simplified 96.0%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 86.2

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

   10. Simplified 86.2%

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

   if 4e15 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 79.6%

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

   6. Taylor expanded in K around 0

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

   8. Simplified 56.1%

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

   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

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

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

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. cube-mult N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. associate-*r* N/A

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

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

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 62.8

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

   11. Simplified 62.8%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1300:\\ \;\;\;\;U \cdot \left(\left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \frac{0.3333333333333333}{U}\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+15}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.4% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+14}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l (* (* l l) 0.3333333333333333)))))
   (if (<= l -2.3e+40) t_0 (if (<= l 1.45e+14) (+ U (* J (* 2.0 l))) 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 <= -2.3e+40) {
		tmp = t_0;
	} else if (l <= 1.45e+14) {
		tmp = U + (J * (2.0 * l));
	} 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 <= (-2.3d+40)) then
        tmp = t_0
    else if (l <= 1.45d+14) then
        tmp = u + (j * (2.0d0 * l))
    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 <= -2.3e+40) {
		tmp = t_0;
	} else if (l <= 1.45e+14) {
		tmp = U + (J * (2.0 * l));
	} 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 <= -2.3e+40:
		tmp = t_0
	elif l <= 1.45e+14:
		tmp = U + (J * (2.0 * l))
	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 <= -2.3e+40)
		tmp = t_0;
	elseif (l <= 1.45e+14)
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	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 <= -2.3e+40)
		tmp = t_0;
	elseif (l <= 1.45e+14)
		tmp = U + (J * (2.0 * l));
	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, -2.3e+40], t$95$0, If[LessEqual[l, 1.45e+14], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.29999999999999994e40 or 1.45e14 < 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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 75.6%

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

   6. Taylor expanded in K around 0

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

   8. Simplified 56.2%

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

   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

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

       2. *-commutative N/A

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

       3. associate-*l* N/A

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

       4. unpow3 N/A

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

       5. unpow2 N/A

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

       6. associate-*r* N/A

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

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

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

       8. associate-*r* N/A

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

       9. unpow2 N/A

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

       10. unpow3 N/A

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

       11. cube-mult N/A

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

       12. unpow2 N/A

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

       13. associate-*r* N/A

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

       14. *-commutative N/A

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

       15. associate-*r* N/A

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

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

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

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

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

       18. unpow2 N/A

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

       19. *-lowering-*.f64 61.6

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

   11. Simplified 61.6%

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

   if -2.29999999999999994e40 < l < 1.45e14

   1. Initial program 81.6%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. sinh-undef N/A

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

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

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

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

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

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

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

      9. /-lowering-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in l around 0

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

      1. *-lowering-*.f64 90.6

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

   7. Simplified 90.6%

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

   8. Taylor expanded in K around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 81.4

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

   10. Simplified 81.4%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+40}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+14}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.3% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.2%

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

3. Taylor expanded in l around 0

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

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

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. *-commutative N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   9. unpow2 N/A

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   10. *-lowering-*.f64 92.0

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

5. Simplified 92.0%

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

6. Taylor expanded in K around 0

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

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} + U \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\ell \cdot J\right)} \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) + U \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

      \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} + U \]

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

      \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right) + U \]

   7. unpow2 N/A

      \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) + U \]

   8. associate-*l* N/A

      \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

      \[\leadsto \ell \cdot \left(J \cdot \left(2 + \color{blue}{\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

       \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \color{blue}{\left(\ell \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) + U \]

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

       \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) + U \]

   12. *-commutative N/A

       \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) + U \]

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

       \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{{\ell}^{2} \cdot \frac{1}{60}}\right)\right)\right)\right) + U \]

   14. unpow2 N/A

       \[\leadsto \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \frac{1}{60}\right)\right)\right)\right) + U \]

   15. *-lowering-*.f64 76.0

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

8. Simplified 76.0%

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

9. Final simplification 76.0%

   \[\leadsto U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 14: 72.3% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1850:\\ \;\;\;\;U \cdot \left(\left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \frac{0.3333333333333333}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1850.0)
   (* U (* (* J (* l (* l l))) (/ 0.3333333333333333 U)))
   (+ U (* l (* J (+ 2.0 (* l (* l 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1850.0) {
		tmp = U * ((J * (l * (l * l))) * (0.3333333333333333 / U));
	} else {
		tmp = U + (l * (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 (l <= (-1850.0d0)) then
        tmp = u * ((j * (l * (l * l))) * (0.3333333333333333d0 / u))
    else
        tmp = u + (l * (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 (l <= -1850.0) {
		tmp = U * ((J * (l * (l * l))) * (0.3333333333333333 / U));
	} else {
		tmp = U + (l * (J * (2.0 + (l * (l * 0.3333333333333333)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1850.0:
		tmp = U * ((J * (l * (l * l))) * (0.3333333333333333 / U))
	else:
		tmp = U + (l * (J * (2.0 + (l * (l * 0.3333333333333333)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1850.0)
		tmp = Float64(U * Float64(Float64(J * Float64(l * Float64(l * l))) * Float64(0.3333333333333333 / U)));
	else
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1850.0)
		tmp = U * ((J * (l * (l * l))) * (0.3333333333333333 / U));
	else
		tmp = U + (l * (J * (2.0 + (l * (l * 0.3333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1850.0], N[(U * N[(N[(J * N[(l * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1850

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 61.3%

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

   6. Taylor expanded in U around inf

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

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

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

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

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

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

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

   8. Simplified 74.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)}{U}\right)} \]

   9. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. associate-/l* N/A

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

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

          \[\leadsto U \cdot \color{blue}{\left(\left(J \cdot \left({\ell}^{3} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \frac{\frac{1}{3}}{U}\right)} \]

       5. *-commutative N/A

          \[\leadsto U \cdot \left(\color{blue}{\left(\left({\ell}^{3} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot J\right)} \cdot \frac{\frac{1}{3}}{U}\right) \]

       6. *-commutative N/A

          \[\leadsto U \cdot \left(\left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{3}\right)} \cdot J\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       7. associate-*l* N/A

          \[\leadsto U \cdot \left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \frac{\frac{1}{3}}{U}\right) \]

       8. *-commutative N/A

          \[\leadsto U \cdot \left(\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot {\ell}^{3}\right)}\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto U \cdot \left(\color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \frac{\frac{1}{3}}{U}\right) \]

       10. cos-lowering-cos.f64 N/A

           \[\leadsto U \cdot \left(\left(\color{blue}{\cos \left(\frac{1}{2} \cdot K\right)} \cdot \left(J \cdot {\ell}^{3}\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto U \cdot \left(\left(\cos \color{blue}{\left(\frac{1}{2} \cdot K\right)} \cdot \left(J \cdot {\ell}^{3}\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto U \cdot \left(\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\left(J \cdot {\ell}^{3}\right)}\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       13. cube-mult N/A

           \[\leadsto U \cdot \left(\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)}\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       14. unpow2 N/A

           \[\leadsto U \cdot \left(\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right)\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto U \cdot \left(\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(\ell \cdot {\ell}^{2}\right)}\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       16. unpow2 N/A

           \[\leadsto U \cdot \left(\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       17. *-lowering-*.f64 N/A

           \[\leadsto U \cdot \left(\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       18. /-lowering-/.f64 74.1

           \[\leadsto U \cdot \left(\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \color{blue}{\frac{0.3333333333333333}{U}}\right) \]

   11. Simplified 74.1%

       \[\leadsto U \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \frac{0.3333333333333333}{U}\right)} \]

   12. Taylor expanded in K around 0

       \[\leadsto U \cdot \left(\color{blue}{\left(J \cdot {\ell}^{3}\right)} \cdot \frac{\frac{1}{3}}{U}\right) \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto U \cdot \left(\color{blue}{\left(J \cdot {\ell}^{3}\right)} \cdot \frac{\frac{1}{3}}{U}\right) \]

       2. cube-mult N/A

          \[\leadsto U \cdot \left(\left(J \cdot \color{blue}{\left(\ell \cdot \left(\ell \cdot \ell\right)\right)}\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       3. unpow2 N/A

          \[\leadsto U \cdot \left(\left(J \cdot \left(\ell \cdot \color{blue}{{\ell}^{2}}\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto U \cdot \left(\left(J \cdot \color{blue}{\left(\ell \cdot {\ell}^{2}\right)}\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       5. unpow2 N/A

          \[\leadsto U \cdot \left(\left(J \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \cdot \frac{\frac{1}{3}}{U}\right) \]

       6. *-lowering-*.f64 56.9

          \[\leadsto U \cdot \left(\left(J \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \cdot \frac{0.3333333333333333}{U}\right) \]

   14. Simplified 56.9%

       \[\leadsto U \cdot \left(\color{blue}{\left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right)} \cdot \frac{0.3333333333333333}{U}\right) \]

   if -1850 < l

   1. Initial program 85.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      2. associate-*r* N/A

         \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      3. associate-*l* N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

      5. *-commutative N/A

         \[\leadsto \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      6. associate-*r* N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{3}\right) \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      7. *-commutative N/A

         \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

      8. distribute-rgt-out N/A

         \[\leadsto \ell \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)\right)} + U \]

      9. +-commutative N/A

         \[\leadsto \ell \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right) + U \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} + U \]

   5. Simplified 91.4%

      \[\leadsto \color{blue}{\ell \cdot \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \ell \cdot \left(\color{blue}{J} \cdot \left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right) + U \]
   7. Step-by-step derivation

   8. Simplified 77.7%

      \[\leadsto \ell \cdot \left(\color{blue}{J} \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1850:\\ \;\;\;\;U \cdot \left(\left(J \cdot \left(\ell \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot \frac{0.3333333333333333}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 46.3% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -3.6 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+14}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 l))))
   (if (<= l -3.6e-23) t_0 (if (<= l 1.4e+14) 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 <= -3.6e-23) {
		tmp = t_0;
	} else if (l <= 1.4e+14) {
		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 <= (-3.6d-23)) then
        tmp = t_0
    else if (l <= 1.4d+14) 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 <= -3.6e-23) {
		tmp = t_0;
	} else if (l <= 1.4e+14) {
		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 <= -3.6e-23:
		tmp = t_0
	elif l <= 1.4e+14:
		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 <= -3.6e-23)
		tmp = t_0;
	elseif (l <= 1.4e+14)
		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 <= -3.6e-23)
		tmp = t_0;
	elseif (l <= 1.4e+14)
		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, -3.6e-23], t$95$0, If[LessEqual[l, 1.4e+14], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.5999999999999998e-23 or 1.4e14 < l

   1. Initial program 98.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot J + U \]

      5. sinh-undef N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

      7. sinh-lowering-sinh.f64 N/A

         \[\leadsto \left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot J + U \]

      9. /-lowering-/.f64 100.0

         \[\leadsto \left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot J + U \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

   5. Taylor expanded in l around 0

      \[\leadsto \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 29.7

         \[\leadsto \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

   7. Simplified 29.7%

      \[\leadsto \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

   8. Taylor expanded in K around 0

      \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot J + U \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

      2. *-lowering-*.f64 24.8

         \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

   10. Simplified 24.8%

       \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

   11. Taylor expanded in l around inf

       \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 2} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

       3. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(2 \cdot \ell\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell\right)} \]

       5. *-commutative N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

       6. *-lowering-*.f64 22.3

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

   13. Simplified 22.3%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

   if -3.5999999999999998e-23 < l < 1.4e14

   1. Initial program 80.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 77.5%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.6 \cdot 10^{-23}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+14}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 72.6% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* J (* l (+ 2.0 (* l (* l 0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
}

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 * (l * (2.0d0 + (l * (l * 0.3333333333333333d0)))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)}\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   2. associate-*r* N/A

      \[\leadsto \ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   3. associate-*l* N/A

      \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + U \]

   5. *-commutative N/A

      \[\leadsto \ell \cdot \left(\color{blue}{{\ell}^{2} \cdot \left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   6. associate-*r* N/A

      \[\leadsto \ell \cdot \left(\color{blue}{\left({\ell}^{2} \cdot \frac{1}{3}\right) \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   7. *-commutative N/A

      \[\leadsto \ell \cdot \left(\color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + U \]

   8. distribute-rgt-out N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)\right)} + U \]

   9. +-commutative N/A

      \[\leadsto \ell \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right) + U \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \ell \cdot \color{blue}{\left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} + U \]

5. Simplified 84.5%

   \[\leadsto \color{blue}{\ell \cdot \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

6. Taylor expanded in K around 0

   \[\leadsto \ell \cdot \left(\color{blue}{J} \cdot \left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right) + U \]
7. Step-by-step derivation

8. Simplified 71.1%

   \[\leadsto \ell \cdot \left(\color{blue}{J} \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) + U \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \ell \cdot \color{blue}{\left(\left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right) \cdot J\right)} + U \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right) \cdot J} + U \]

   3. associate-*r* N/A

      \[\leadsto \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{3}}\right)\right) \cdot J + U \]

   4. *-commutative N/A

      \[\leadsto \left(\ell \cdot \left(2 + \color{blue}{\frac{1}{3} \cdot \left(\ell \cdot \ell\right)}\right)\right) \cdot J + U \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot J} + U \]

   6. *-commutative N/A

      \[\leadsto \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{1}{3}}\right)\right) \cdot J + U \]

   7. associate-*r* N/A

      \[\leadsto \left(\ell \cdot \left(2 + \color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)}\right)\right) \cdot J + U \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)} \cdot J + U \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \left(\ell \cdot \color{blue}{\left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)}\right) \cdot J + U \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \left(\ell \cdot \left(2 + \color{blue}{\ell \cdot \left(\ell \cdot \frac{1}{3}\right)}\right)\right) \cdot J + U \]

   11. *-lowering-*.f64 73.3

       \[\leadsto \left(\ell \cdot \left(2 + \ell \cdot \color{blue}{\left(\ell \cdot 0.3333333333333333\right)}\right)\right) \cdot J + U \]

10. Applied egg-rr 73.3%

    \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot J} + U \]

11. Final simplification 73.3%

    \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \]
12. Add Preprocessing

Alternative 17: 54.8% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(2 \cdot \ell\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* J (* 2.0 l))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (2.0 * 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 * (2.0d0 * l))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (2.0 * l));
}

Python

def code(J, l, K, U):
	return U + (J * (2.0 * l))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(2.0 * l)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (2.0 * l));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 89.2%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} + U \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot J + U \]

   5. sinh-undef N/A

      \[\leadsto \left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(2 \cdot \sinh \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

   7. sinh-lowering-sinh.f64 N/A

      \[\leadsto \left(\left(2 \cdot \color{blue}{\sinh \ell}\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

   8. cos-lowering-cos.f64 N/A

      \[\leadsto \left(\left(2 \cdot \sinh \ell\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}\right) \cdot J + U \]

   9. /-lowering-/.f64 100.0

      \[\leadsto \left(\left(2 \cdot \sinh \ell\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot J + U \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \sinh \ell\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J} + U \]

5. Taylor expanded in l around 0

   \[\leadsto \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]
6. Step-by-step derivation

   1. *-lowering-*.f64 64.7

      \[\leadsto \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

7. Simplified 64.7%

   \[\leadsto \left(\color{blue}{\left(2 \cdot \ell\right)} \cdot \cos \left(\frac{K}{2}\right)\right) \cdot J + U \]

8. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\left(2 \cdot \ell\right)} \cdot J + U \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

   2. *-lowering-*.f64 57.0

      \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

10. Simplified 57.0%

    \[\leadsto \color{blue}{\left(\ell \cdot 2\right)} \cdot J + U \]

11. Final simplification 57.0%

    \[\leadsto U + J \cdot \left(2 \cdot \ell\right) \]
12. Add Preprocessing

Alternative 18: 36.7% 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 89.2%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 42.4%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))