Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.8% → 99.9%

Time: 13.6s

Alternatives: 24

Speedup: 2.3×

• 49.4% 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: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. sinh-undef N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.998:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.998)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (+
          2.0
          (*
           (* l l)
           (+ 0.3333333333333333 (* l (* l 0.016666666666666666)))))))))
     (+ U (* 2.0 (* J (sinh l)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.998) {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))));
	} else {
		tmp = U + (2.0 * (J * sinh(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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.998d0) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * 0.016666666666666666d0))))))))
    else
        tmp = u + (2.0d0 * (j * sinh(l)))
    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 <= 0.998) {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))));
	} else {
		tmp = U + (2.0 * (J * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.998:
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))))
	else:
		tmp = U + (2.0 * (J * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.998)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * 0.016666666666666666)))))))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * sinh(l))));
	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 <= 0.998)
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))));
	else
		tmp = U + (2.0 * (J * sinh(l)));
	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, 0.998], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.998

   1. Initial program 85.2%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 94.4%

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

   5. Simplified 94.4%

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

   if 0.998 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.4%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.998:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.02)
     (+ U (* t_0 (* J (* l (+ 2.0 (* l (* l 0.3333333333333333)))))))
     (+ U (* 2.0 (* J (sinh l)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	} else {
		tmp = U + (2.0 * (J * sinh(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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.02d0)) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (l * (l * 0.3333333333333333d0))))))
    else
        tmp = u + (2.0d0 * (j * sinh(l)))
    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 <= -0.02) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	} else {
		tmp = U + (2.0 * (J * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.02:
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))))
	else:
		tmp = U + (2.0 * (J * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333)))))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * sinh(l))));
	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 <= -0.02)
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	else
		tmp = U + (2.0 * (J * sinh(l)));
	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, -0.02], 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[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 88.2%

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

   5. Simplified 88.2%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 97.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (+ U (* l (* (+ 2.0 (* l (* l 0.3333333333333333))) (* J (cos (* K 0.5))))))
   (+ U (* 2.0 (* J (sinh l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * cos((K * 0.5)))));
	} else {
		tmp = U + (2.0 * (J * sinh(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 (cos((k / 2.0d0)) <= (-0.02d0)) then
        tmp = u + (l * ((2.0d0 + (l * (l * 0.3333333333333333d0))) * (j * cos((k * 0.5d0)))))
    else
        tmp = u + (2.0d0 * (j * sinh(l)))
    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)) <= -0.02) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (2.0 * (J * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.02:
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * math.cos((K * 0.5)))))
	else:
		tmp = U + (2.0 * (J * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		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(2.0 * Float64(J * sinh(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.02)
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * cos((K * 0.5)))));
	else
		tmp = U + (2.0 * (J * sinh(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], 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[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 86.6%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 97.2%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (+
    U
    (*
     l
     (*
      (+ 2.0 (* l (* l 0.3333333333333333)))
      (+ J (* J (* -0.125 (* K K)))))))
   (+ U (* 2.0 (* J (sinh l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	} else {
		tmp = U + (2.0 * (J * sinh(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 (cos((k / 2.0d0)) <= (-0.02d0)) then
        tmp = u + (l * ((2.0d0 + (l * (l * 0.3333333333333333d0))) * (j + (j * ((-0.125d0) * (k * k))))))
    else
        tmp = u + (2.0d0 * (j * sinh(l)))
    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)) <= -0.02) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	} else {
		tmp = U + (2.0 * (J * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.02:
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))))
	else:
		tmp = U + (2.0 * (J * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		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(2.0 * Float64(J * sinh(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.02)
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	else
		tmp = U + (2.0 * (J * sinh(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], 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[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 86.6%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 65.2%

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

   8. Simplified 65.2%

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

   if -0.0200000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 97.2%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 4.1 \cdot 10^{-79}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 4.1e-79)
   (+ U (* 2.0 (* J (sinh l))))
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (+
        2.0
        (*
         (* l l)
         (+
          0.3333333333333333
          (*
           l
           (*
            l
            (+
             0.016666666666666666
             (* (* l l) 0.0003968253968253968)))))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 4.1e-79) {
		tmp = U + (2.0 * (J * sinh(l)));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 4.1d-79) then
        tmp = u + (2.0d0 * (j * sinh(l)))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 4.1e-79) {
		tmp = U + (2.0 * (J * Math.sinh(l)));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 4.1e-79:
		tmp = U + (2.0 * (J * math.sinh(l)))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 4.1e-79)
		tmp = Float64(U + Float64(2.0 * Float64(J * sinh(l))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 4.1e-79)
		tmp = U + (2.0 * (J * sinh(l)));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 4.1e-79], N[(U + N[(2.0 * N[(J * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 4.09999999999999994e-79

   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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 88.6%

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

   if 4.09999999999999994e-79 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 82.2%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 98.8%

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

   5. Simplified 98.8%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 4.1 \cdot 10^{-79}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+129}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 4e+129)
   (+
    U
    (*
     J
     (*
      l
      (+
       2.0
       (*
        (* l l)
        (+
         0.3333333333333333
         (*
          l
          (*
           l
           (+ 0.016666666666666666 (* (* l l) 0.0003968253968253968))))))))))
   (+
    U
    (*
     l
     (*
      (+ 2.0 (* l (* l 0.3333333333333333)))
      (+ J (* J (* -0.125 (* K K)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 4e+129) {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	} else {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 4d+129) then
        tmp = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))))
    else
        tmp = u + (l * ((2.0d0 + (l * (l * 0.3333333333333333d0))) * (j + (j * ((-0.125d0) * (k * k))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 4e+129) {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	} else {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 4e+129:
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))))
	else:
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 4e+129)
		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 = Float64(U + Float64(l * Float64(Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))) * Float64(J + Float64(J * Float64(-0.125 * Float64(K * K)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 4e+129)
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	else
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 4e+129], 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], 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]]

Derivation

1. Split input into 2 regimes

2. if l < 4e129

   1. Initial program 82.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

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

      15. *-lowering-*.f64 79.5%

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

   8. Simplified 79.5%

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

   if 4e129 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+129}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.1% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+129}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 5.4e+129)
   (*
    U
    (+
     1.0
     (*
      J
      (/
       (*
        l
        (+
         2.0
         (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))
       U))))
   (+
    U
    (*
     l
     (*
      (+ 2.0 (* l (* l 0.3333333333333333)))
      (+ J (* J (* -0.125 (* K K)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 5.4e+129) {
		tmp = U * (1.0 + (J * ((l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))) / U)));
	} else {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 5.4d+129) then
        tmp = u * (1.0d0 + (j * ((l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))) / u)))
    else
        tmp = u + (l * ((2.0d0 + (l * (l * 0.3333333333333333d0))) * (j + (j * ((-0.125d0) * (k * k))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 5.4e+129) {
		tmp = U * (1.0 + (J * ((l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))) / U)));
	} else {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 5.4e+129:
		tmp = U * (1.0 + (J * ((l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))) / U)))
	else:
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 5.4e+129)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))) / U))));
	else
		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)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 5.4e+129)
		tmp = U * (1.0 + (J * ((l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))) / U)));
	else
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 5.4e+129], N[(U * N[(1.0 + N[(J * N[(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] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]

Derivation

1. Split input into 2 regimes

2. if l < 5.4000000000000002e129

   1. Initial program 82.1%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 92.2%

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

   5. Simplified 92.2%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-lowering-*.f64 75.1%

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

   8. Simplified 75.1%

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. associate-+l+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 75.1%

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

   10. Applied egg-rr 75.1%

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

   11. Taylor expanded in U around inf

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

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

          \[\leadsto \mathsf{*.f64}\left(U, \color{blue}{\left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)}{U}\right)}\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{J \cdot \left(\ell \cdot \left(2 + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)}{U}\right)}\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(J \cdot \color{blue}{\frac{\ell \cdot \left(2 + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}{U}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \color{blue}{\left(\frac{\ell \cdot \left(2 + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}{U}\right)}\right)\right)\right) \]

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

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

   13. Simplified 76.8%

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

   if 5.4000000000000002e129 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.4 \cdot 10^{+129}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.8% accurate, 12.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+129}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 2.2e+129)
   (+
    U
    (*
     J
     (*
      l
      (+
       2.0
       (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
   (+
    U
    (*
     l
     (*
      (+ 2.0 (* l (* l 0.3333333333333333)))
      (+ J (* J (* -0.125 (* K K)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.2e+129) {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 2.2d+129) then
        tmp = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))
    else
        tmp = u + (l * ((2.0d0 + (l * (l * 0.3333333333333333d0))) * (j + (j * ((-0.125d0) * (k * k))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.2e+129) {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	} else {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 2.2e+129:
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))
	else:
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 2.2e+129)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))));
	else
		tmp = Float64(U + Float64(l * Float64(Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))) * Float64(J + Float64(J * Float64(-0.125 * Float64(K * K)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 2.2e+129)
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
	else
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J + (J * (-0.125 * (K * K))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 2.2e+129], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[(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]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1999999999999999e129

   1. Initial program 82.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 75.9%

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

   8. Simplified 75.9%

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

   if 2.1999999999999999e129 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 89.5%

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

   8. Simplified 89.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+129}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 7 \cdot 10^{+38}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 7e+38)
   (+ U (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333)))))
   (+
    U
    (*
     J
     (*
      l
      (+
       1.0
       (*
        l
        (+ 0.5 (* l (+ 0.16666666666666666 (* l 0.041666666666666664)))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 7e+38) {
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	} else {
		tmp = U + (J * (l * (1.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664))))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (k <= 7d+38) then
        tmp = u + (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0))))
    else
        tmp = u + (j * (l * (1.0d0 + (l * (0.5d0 + (l * (0.16666666666666666d0 + (l * 0.041666666666666664d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 7e+38) {
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	} else {
		tmp = U + (J * (l * (1.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 7e+38:
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))))
	else:
		tmp = U + (J * (l * (1.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 7e+38)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(1.0 + Float64(l * Float64(0.5 + Float64(l * Float64(0.16666666666666666 + Float64(l * 0.041666666666666664)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 7e+38)
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	else
		tmp = U + (J * (l * (1.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 7e+38], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(1.0 + N[(l * N[(0.5 + N[(l * N[(0.16666666666666666 + N[(l * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 7.00000000000000003e38

   1. Initial program 84.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 77.2%

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

   5. Simplified 77.2%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 79.8%

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

   8. Simplified 79.8%

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

   if 7.00000000000000003e38 < K

   1. Initial program 85.5%

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

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 54.8%

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

   5. Simplified 54.8%

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

   6. Taylor expanded in l around 0

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

   8. Simplified 42.6%

      \[\leadsto J \cdot \left(e^{\ell} + \color{blue}{-1}\right) + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \ell \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot \ell\right)\right)\right)\right)}\right), U\right) \]
   10. Step-by-step derivation

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(1 + \ell \cdot \left(\frac{1}{2} + \ell \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \left(\ell \cdot \left(\frac{1}{2} + \ell \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot \ell\right)\right)\right)\right)\right)\right), U\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \left(\frac{1}{2} + \ell \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot \ell\right)\right)\right)\right)\right)\right), U\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\ell \cdot \left(\frac{1}{6} + \frac{1}{24} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \left(\frac{1}{6} + \frac{1}{24} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{24} \cdot \ell\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{6}, \left(\ell \cdot \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

       8. *-lowering-*.f64 51.7%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\ell, \frac{1}{24}\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

   11. Simplified 51.7%

       \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 7 \cdot 10^{+38}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.3% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\\ \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+14}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J l) (* (* l l) (* (* l l) 0.016666666666666666)))))
   (if (<= l -7.2e+15) t_0 (if (<= l 1.25e+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) * ((l * l) * 0.016666666666666666));
	double tmp;
	if (l <= -7.2e+15) {
		tmp = t_0;
	} else if (l <= 1.25e+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) * ((l * l) * 0.016666666666666666d0))
    if (l <= (-7.2d+15)) then
        tmp = t_0
    else if (l <= 1.25d+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) * ((l * l) * 0.016666666666666666));
	double tmp;
	if (l <= -7.2e+15) {
		tmp = t_0;
	} else if (l <= 1.25e+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) * ((l * l) * 0.016666666666666666))
	tmp = 0
	if l <= -7.2e+15:
		tmp = t_0
	elif l <= 1.25e+14:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * l) * Float64(Float64(l * l) * Float64(Float64(l * l) * 0.016666666666666666)))
	tmp = 0.0
	if (l <= -7.2e+15)
		tmp = t_0;
	elseif (l <= 1.25e+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) * ((l * l) * 0.016666666666666666));
	tmp = 0.0;
	if (l <= -7.2e+15)
		tmp = t_0;
	elseif (l <= 1.25e+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[(N[(J * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.2e+15], t$95$0, If[LessEqual[l, 1.25e+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 < -7.2e15 or 1.25e14 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 89.5%

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

   5. Simplified 89.5%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-lowering-*.f64 64.9%

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

   8. Simplified 64.9%

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

   9. Taylor expanded in J around inf

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

       1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 64.9%

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

   11. Simplified 64.9%

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

   12. Taylor expanded in l around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{4}\right)}\right) \]
   13. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left(\frac{1}{60} \cdot {\ell}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       2. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left(\frac{1}{60} \cdot \left({\ell}^{2} \cdot \color{blue}{{\ell}^{2}}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left(\left(\frac{1}{60} \cdot {\ell}^{2}\right) \cdot \color{blue}{{\ell}^{2}}\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left({\ell}^{2} \cdot \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{\frac{1}{60}} \cdot {\ell}^{2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{\frac{1}{60}} \cdot {\ell}^{2}\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left({\ell}^{2} \cdot \color{blue}{\frac{1}{60}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\frac{1}{60}}\right)\right)\right) \]

       10. unpow2 N/A

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

       11. *-lowering-*.f64 64.9%

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

   14. Simplified 64.9%

       \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)} \]

   if -7.2e15 < l < 1.25e14

   1. Initial program 70.0%

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

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 68.1%

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

   5. Simplified 68.1%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 83.2%

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

   8. Simplified 83.2%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;\left(J \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+14}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.6% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.036:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 0.036)
   (+ U (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333)))))
   (*
    (+ 2.0 (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))
    (* J l))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 0.036) {
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	} else {
		tmp = (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (J * 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 <= 0.036d0) then
        tmp = u + (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0))))
    else
        tmp = (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))) * (j * l)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 0.036) {
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	} else {
		tmp = (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (J * l);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 0.036:
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))))
	else:
		tmp = (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (J * l)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 0.036)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333)))));
	else
		tmp = Float64(Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))) * Float64(J * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 0.036)
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	else
		tmp = (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (J * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 0.036], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.0359999999999999973

   1. Initial program 79.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 70.4%

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

   5. Simplified 70.4%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.7%

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

   8. Simplified 76.7%

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

   if 0.0359999999999999973 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-lowering-*.f64 65.6%

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

   8. Simplified 65.6%

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

   9. Taylor expanded in J around inf

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

       1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 65.5%

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

   11. Simplified 65.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.036:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.0% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.036:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 0.036)
   (+ U (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333)))))
   (*
    J
    (*
     l
     (+
      2.0
      (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 0.036) {
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	} else {
		tmp = J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 0.036d0) then
        tmp = u + (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0))))
    else
        tmp = j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 0.036) {
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	} else {
		tmp = J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 0.036:
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))))
	else:
		tmp = J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 0.036)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333)))));
	else
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 0.036)
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	else
		tmp = J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 0.036], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.0359999999999999973

   1. Initial program 79.1%

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

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

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

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

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 70.4%

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

   5. Simplified 70.4%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 76.7%

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

   8. Simplified 76.7%

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

   if 0.0359999999999999973 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 86.0%

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

   5. Simplified 86.0%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. unpow2 N/A

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

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

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

      13. *-lowering-*.f64 65.6%

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

   8. Simplified 65.6%

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

   9. Taylor expanded in J around inf

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

       1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 65.5%

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

   11. Simplified 65.5%

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

   12. Taylor expanded in J around 0

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

       1. associate-*r* N/A

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

       2. distribute-rgt-in N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

       5. associate-*l* N/A

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

       6. pow-sqr N/A

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

       7. metadata-eval N/A

          \[\leadsto 2 \cdot \left(J \cdot \ell\right) + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \left(J \cdot \ell\right) \]

       8. distribute-rgt-in N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{\left(2 + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)\right)} \]

       9. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)} \]

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

           \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)}\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{\left(2 + \left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right)\right) \]

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

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{4} + \frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right)\right) \]

   14. Simplified 65.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.036:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 68.6% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \left(J \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq 0.036:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* l (* l 0.3333333333333333)))))
   (if (<= l -1.85e-7)
     (* l (* J t_0))
     (if (<= l 0.036) (+ U (* J (* 2.0 l))) (* t_0 (* J l))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 + (l * (l * 0.3333333333333333));
	double tmp;
	if (l <= -1.85e-7) {
		tmp = l * (J * t_0);
	} else if (l <= 0.036) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0 * (J * 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) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 + (l * (l * 0.3333333333333333d0))
    if (l <= (-1.85d-7)) then
        tmp = l * (j * t_0)
    else if (l <= 0.036d0) then
        tmp = u + (j * (2.0d0 * l))
    else
        tmp = t_0 * (j * l)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 2.0 + (l * (l * 0.3333333333333333));
	double tmp;
	if (l <= -1.85e-7) {
		tmp = l * (J * t_0);
	} else if (l <= 0.036) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0 * (J * l);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 + (l * (l * 0.3333333333333333))
	tmp = 0
	if l <= -1.85e-7:
		tmp = l * (J * t_0)
	elif l <= 0.036:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = t_0 * (J * l)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -1.85e-7)
		tmp = Float64(l * Float64(J * t_0));
	elseif (l <= 0.036)
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	else
		tmp = Float64(t_0 * Float64(J * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 + (l * (l * 0.3333333333333333));
	tmp = 0.0;
	if (l <= -1.85e-7)
		tmp = l * (J * t_0);
	elseif (l <= 0.036)
		tmp = U + (J * (2.0 * l));
	else
		tmp = t_0 * (J * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.85e-7], N[(l * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.036], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(J * l), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.85000000000000002e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 88.6%

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

   5. Simplified 88.6%

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

   6. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      13. *-lowering-*.f64 62.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(J, \ell\right)\right), U\right) \]

   8. Simplified 62.2%

      \[\leadsto \color{blue}{\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \left(J \cdot \ell\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(J \cdot \ell\right), \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left(\color{blue}{2} + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 62.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right) \]

   11. Simplified 62.2%

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)} \]

   12. Taylor expanded in l around 0

       \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\left(J \cdot {\ell}^{2}\right) \cdot \frac{1}{3} + \color{blue}{2} \cdot J\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \left({\ell}^{2} \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot J\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right) + 2 \cdot J\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right) + J \cdot \color{blue}{2}\right)\right) \]

       6. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \color{blue}{\frac{1}{3} \cdot {\ell}^{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{3}\right)}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\ell}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot \ell\right)}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 54.8%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

   14. Simplified 54.8%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} \]

   if -1.85000000000000002e-7 < l < 0.0359999999999999973

   1. Initial program 68.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 68.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 68.3%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 85.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 85.9%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   if 0.0359999999999999973 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      11. *-lowering-*.f64 86.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 86.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \left(J \cdot \ell\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      13. *-lowering-*.f64 65.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(J, \ell\right)\right), U\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \left(J \cdot \ell\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(J \cdot \ell\right), \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left(\color{blue}{2} + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 65.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right) \]

   11. Simplified 65.5%

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)} \]

   12. Taylor expanded in l around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{3}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{3}\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot \ell\right)}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 54.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{3}}\right)\right)\right)\right) \]

   14. Simplified 54.4%

       \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 0.036:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.6% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.036:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J (+ 2.0 (* l (* l 0.3333333333333333)))))))
   (if (<= l -1.85e-7) t_0 (if (<= l 0.036) (+ U (* J (* 2.0 l))) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * (2.0 + (l * (l * 0.3333333333333333))));
	double tmp;
	if (l <= -1.85e-7) {
		tmp = t_0;
	} else if (l <= 0.036) {
		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 = l * (j * (2.0d0 + (l * (l * 0.3333333333333333d0))))
    if (l <= (-1.85d-7)) then
        tmp = t_0
    else if (l <= 0.036d0) 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 = l * (J * (2.0 + (l * (l * 0.3333333333333333))));
	double tmp;
	if (l <= -1.85e-7) {
		tmp = t_0;
	} else if (l <= 0.036) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * (2.0 + (l * (l * 0.3333333333333333))))
	tmp = 0
	if l <= -1.85e-7:
		tmp = t_0
	elif l <= 0.036:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333)))))
	tmp = 0.0
	if (l <= -1.85e-7)
		tmp = t_0;
	elseif (l <= 0.036)
		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 = l * (J * (2.0 + (l * (l * 0.3333333333333333))));
	tmp = 0.0;
	if (l <= -1.85e-7)
		tmp = t_0;
	elseif (l <= 0.036)
		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[(l * N[(J * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.85e-7], t$95$0, If[LessEqual[l, 0.036], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.85000000000000002e-7 or 0.0359999999999999973 < l

   1. Initial program 99.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

      11. *-lowering-*.f64 87.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]

   5. Simplified 87.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \left(J \cdot \ell\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \left(J \cdot \ell\right)\right), U\right) \]

      13. *-lowering-*.f64 63.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(J, \ell\right)\right), U\right) \]

   8. Simplified 63.9%

      \[\leadsto \color{blue}{\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \left(J \cdot \ell\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(J \cdot \ell\right), \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \left(\color{blue}{2} + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 63.9%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right) \]

   11. Simplified 63.9%

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)} \]

   12. Taylor expanded in l around 0

       \[\leadsto \color{blue}{\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(\left(J \cdot {\ell}^{2}\right) \cdot \frac{1}{3} + \color{blue}{2} \cdot J\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \left({\ell}^{2} \cdot \frac{1}{3}\right) + \color{blue}{2} \cdot J\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right) + 2 \cdot J\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right) + J \cdot \color{blue}{2}\right)\right) \]

       6. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2} + 2\right)}\right)\right) \]

       7. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \left(2 + \color{blue}{\frac{1}{3} \cdot {\ell}^{2}}\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       9. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right) \]

       12. associate-*l* N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{3}\right)}\right)\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\ell}\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot \ell\right)}\right)\right)\right)\right) \]

       15. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

       16. *-lowering-*.f64 54.6%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

   14. Simplified 54.6%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} \]

   if -1.85000000000000002e-7 < l < 0.0359999999999999973

   1. Initial program 68.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 68.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 68.3%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 85.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 85.9%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-7}:\\ \;\;\;\;\ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 0.036:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.2% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   J
   (*
    l
    (+
     2.0
     (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * (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 + (j * (l * (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 + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

   9. exp-lowering-exp.f64 71.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

5. Simplified 71.7%

   \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), U\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

   10. *-lowering-*.f64 75.2%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

8. Simplified 75.2%

   \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)} + U \]

9. Final simplification 75.2%

   \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \]
10. Add Preprocessing

Alternative 17: 42.2% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \ell\right)\\ \mathbf{if}\;J \leq -1.96 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{+39}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 l))))
   (if (<= J -1.96e+80) t_0 (if (<= J 3.6e+39) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double tmp;
	if (J <= -1.96e+80) {
		tmp = t_0;
	} else if (J <= 3.6e+39) {
		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 (j <= (-1.96d+80)) then
        tmp = t_0
    else if (j <= 3.6d+39) 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 (J <= -1.96e+80) {
		tmp = t_0;
	} else if (J <= 3.6e+39) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 * l)
	tmp = 0
	if J <= -1.96e+80:
		tmp = t_0
	elif J <= 3.6e+39:
		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 (J <= -1.96e+80)
		tmp = t_0;
	elseif (J <= 3.6e+39)
		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 (J <= -1.96e+80)
		tmp = t_0;
	elseif (J <= 3.6e+39)
		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[J, -1.96e+80], t$95$0, If[LessEqual[J, 3.6e+39], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if J < -1.9599999999999999e80 or 3.59999999999999984e39 < J

   1. Initial program 72.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 61.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 61.0%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 54.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 54.4%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{2} \]

       2. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

       3. *-commutative N/A

          \[\leadsto J \cdot \left(2 \cdot \color{blue}{\ell}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(2 \cdot \ell\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{2}\right)\right) \]

       6. *-lowering-*.f64 42.8%

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{2}\right)\right) \]

   11. Simplified 42.8%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

   if -1.9599999999999999e80 < J < 3.59999999999999984e39

   1. Initial program 93.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 47.6%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -1.96 \cdot 10^{+80}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;J \leq 3.6 \cdot 10^{+39}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 54.9% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+26}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.7e+26) (* J (+ (* 2.0 l) (/ U J))) (+ U (* J (* 2.0 l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.7e+26) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U + (J * (2.0 * 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 <= (-1.7d+26)) then
        tmp = j * ((2.0d0 * l) + (u / j))
    else
        tmp = u + (j * (2.0d0 * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.7e+26) {
		tmp = J * ((2.0 * l) + (U / J));
	} else {
		tmp = U + (J * (2.0 * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.7e+26:
		tmp = J * ((2.0 * l) + (U / J))
	else:
		tmp = U + (J * (2.0 * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.7e+26)
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.7e+26)
		tmp = J * ((2.0 * l) + (U / J));
	else
		tmp = U + (J * (2.0 * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.7e+26], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.7000000000000001e26

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 73.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 73.2%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 20.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 20.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(2 \cdot \ell + \frac{U}{J}\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\frac{U}{J} + \color{blue}{2 \cdot \ell}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(\frac{U}{J}\right), \color{blue}{\left(2 \cdot \ell\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\color{blue}{2} \cdot \ell\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\ell \cdot \color{blue}{2}\right)\right)\right) \]

       6. *-lowering-*.f64 28.9%

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{*.f64}\left(\ell, \color{blue}{2}\right)\right)\right) \]

   11. Simplified 28.9%

       \[\leadsto \color{blue}{J \cdot \left(\frac{U}{J} + \ell \cdot 2\right)} \]

   if -1.7000000000000001e26 < l

   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 K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 71.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 71.3%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 62.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 62.0%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+26}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 71.2% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \end{array} \]

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(Float64(l * 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[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

   9. exp-lowering-exp.f64 71.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

5. Simplified 71.7%

   \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

   5. *-lowering-*.f64 72.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

8. Simplified 72.1%

   \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} + U \]

9. Final simplification 72.1%

   \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \]
10. Add Preprocessing

Alternative 20: 54.9% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+44}:\\ \;\;\;\;J \cdot \left(\ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.8e+44) (* J (+ l (/ U J))) (+ U (* J (* 2.0 l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.8e+44) {
		tmp = J * (l + (U / J));
	} else {
		tmp = U + (J * (2.0 * 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 <= (-2.8d+44)) then
        tmp = j * (l + (u / j))
    else
        tmp = u + (j * (2.0d0 * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.8e+44) {
		tmp = J * (l + (U / J));
	} else {
		tmp = U + (J * (2.0 * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.8e+44:
		tmp = J * (l + (U / J))
	else:
		tmp = U + (J * (2.0 * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.8e+44)
		tmp = Float64(J * Float64(l + Float64(U / J)));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.8e+44)
		tmp = J * (l + (U / J));
	else
		tmp = U + (J * (2.0 * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.8e+44], N[(J * N[(l + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.8000000000000001e44

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 72.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \color{blue}{-1}\right)\right), U\right) \]
   7. Step-by-step derivation

   8. Simplified 3.1%

      \[\leadsto J \cdot \left(e^{\ell} + \color{blue}{-1}\right) + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} + \frac{U}{J}\right) - 1\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto J \cdot \left(\left(\frac{U}{J} + e^{\ell}\right) - 1\right) \]

       2. associate--l+ N/A

          \[\leadsto J \cdot \left(\frac{U}{J} + \color{blue}{\left(e^{\ell} - 1\right)}\right) \]

       3. +-commutative N/A

          \[\leadsto J \cdot \left(\left(e^{\ell} - 1\right) + \color{blue}{\frac{U}{J}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\left(e^{\ell} - 1\right) + \frac{U}{J}\right)}\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\frac{U}{J} + \color{blue}{\left(e^{\ell} - 1\right)}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(\frac{U}{J}\right), \color{blue}{\left(e^{\ell} - 1\right)}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\color{blue}{e^{\ell}} - 1\right)\right)\right) \]

       8. expm1-define N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\mathsf{expm1}\left(\ell\right)\right)\right)\right) \]

       9. expm1-lowering-expm1.f64 11.8%

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{expm1.f64}\left(\ell\right)\right)\right) \]

   11. Simplified 11.8%

       \[\leadsto \color{blue}{J \cdot \left(\frac{U}{J} + \mathsf{expm1}\left(\ell\right)\right)} \]

   12. Taylor expanded in l around 0

       \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \color{blue}{\ell}\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 29.8%

       \[\leadsto J \cdot \left(\frac{U}{J} + \color{blue}{\ell}\right) \]

   if -2.8000000000000001e44 < l

   1. Initial program 80.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 71.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 61.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 61.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+44}:\\ \;\;\;\;J \cdot \left(\ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 48.6% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+45}:\\ \;\;\;\;J \cdot \left(\ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \ell\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4e+45) (* J (+ l (/ U J))) (+ U (* J l))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4e+45) {
		tmp = J * (l + (U / J));
	} else {
		tmp = U + (J * 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 <= (-4d+45)) then
        tmp = j * (l + (u / j))
    else
        tmp = u + (j * l)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4e+45) {
		tmp = J * (l + (U / J));
	} else {
		tmp = U + (J * l);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4e+45:
		tmp = J * (l + (U / J))
	else:
		tmp = U + (J * l)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4e+45)
		tmp = Float64(J * Float64(l + Float64(U / J)));
	else
		tmp = Float64(U + Float64(J * l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -4e+45)
		tmp = J * (l + (U / J));
	else
		tmp = U + (J * l);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4e+45], N[(J * N[(l + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * l), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.9999999999999997e45

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 72.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 72.2%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \color{blue}{-1}\right)\right), U\right) \]
   7. Step-by-step derivation

   8. Simplified 3.1%

      \[\leadsto J \cdot \left(e^{\ell} + \color{blue}{-1}\right) + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\left(e^{\ell} + \frac{U}{J}\right) - 1\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto J \cdot \left(\left(\frac{U}{J} + e^{\ell}\right) - 1\right) \]

       2. associate--l+ N/A

          \[\leadsto J \cdot \left(\frac{U}{J} + \color{blue}{\left(e^{\ell} - 1\right)}\right) \]

       3. +-commutative N/A

          \[\leadsto J \cdot \left(\left(e^{\ell} - 1\right) + \color{blue}{\frac{U}{J}}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\left(e^{\ell} - 1\right) + \frac{U}{J}\right)}\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\frac{U}{J} + \color{blue}{\left(e^{\ell} - 1\right)}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(\frac{U}{J}\right), \color{blue}{\left(e^{\ell} - 1\right)}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\color{blue}{e^{\ell}} - 1\right)\right)\right) \]

       8. expm1-define N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\mathsf{expm1}\left(\ell\right)\right)\right)\right) \]

       9. expm1-lowering-expm1.f64 11.8%

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{expm1.f64}\left(\ell\right)\right)\right) \]

   11. Simplified 11.8%

       \[\leadsto \color{blue}{J \cdot \left(\frac{U}{J} + \mathsf{expm1}\left(\ell\right)\right)} \]

   12. Taylor expanded in l around 0

       \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \color{blue}{\ell}\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 29.8%

       \[\leadsto J \cdot \left(\frac{U}{J} + \color{blue}{\ell}\right) \]

   if -3.9999999999999997e45 < l

   1. Initial program 80.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 71.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 71.6%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \color{blue}{-1}\right)\right), U\right) \]
   7. Step-by-step derivation

   8. Simplified 67.3%

      \[\leadsto J \cdot \left(e^{\ell} + \color{blue}{-1}\right) + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \ell\right)}, U\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 52.2%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \ell\right), U\right) \]

   11. Simplified 52.2%

       \[\leadsto \color{blue}{J \cdot \ell} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+45}:\\ \;\;\;\;J \cdot \left(\ell + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \ell\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 56.8% accurate, 28.4× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + \frac{J \cdot \left(2 \cdot \ell\right)}{U}\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* U (+ 1.0 (/ (* J (* 2.0 l)) U))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + ((J * (2.0 * l)) / 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 * (1.0d0 + ((j * (2.0d0 * l)) / u))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + ((J * (2.0 * l)) / U));
}

Python

def code(J, l, K, U):
	return U * (1.0 + ((J * (2.0 * l)) / U))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(Float64(J * Float64(2.0 * l)) / U)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + ((J * (2.0 * l)) / U));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

   9. exp-lowering-exp.f64 71.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

5. Simplified 71.7%

   \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

   6. *-lowering-*.f64 52.9%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

8. Simplified 52.9%

   \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

9. Taylor expanded in U around inf

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \ell}{U}\right)} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(U, \color{blue}{\left(1 + 2 \cdot \frac{J \cdot \ell}{U}\right)}\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \color{blue}{\left(2 \cdot \frac{J \cdot \ell}{U}\right)}\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot \left(J \cdot \ell\right)}{\color{blue}{U}}\right)\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\left(2 \cdot J\right) \cdot \ell}{U}\right)\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(2 \cdot J\right) \cdot \ell\right), \color{blue}{U}\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(J \cdot 2\right) \cdot \ell\right), U\right)\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right)\right)\right) \]

    10. *-lowering-*.f64 57.3%

        \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right)\right)\right) \]

11. Simplified 57.3%

    \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot 2\right)}{U}\right)} \]

12. Final simplification 57.3%

    \[\leadsto U \cdot \left(1 + \frac{J \cdot \left(2 \cdot \ell\right)}{U}\right) \]
13. Add Preprocessing

Alternative 23: 47.2% accurate, 62.4× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \ell \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* J l)))

C

double code(double J, double l, double K, double U) {
	return U + (J * l);
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * l)
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * l);
}

Python

def code(J, l, K, U):
	return U + (J * l)

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * l))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * l);
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * l), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

   9. exp-lowering-exp.f64 71.7%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

5. Simplified 71.7%

   \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \color{blue}{-1}\right)\right), U\right) \]
7. Step-by-step derivation

8. Simplified 53.7%

   \[\leadsto J \cdot \left(e^{\ell} + \color{blue}{-1}\right) + U \]

9. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \ell\right)}, U\right) \]
10. Step-by-step derivation

    1. *-lowering-*.f64 45.6%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \ell\right), U\right) \]

11. Simplified 45.6%

    \[\leadsto \color{blue}{J \cdot \ell} + U \]

12. Final simplification 45.6%

    \[\leadsto U + J \cdot \ell \]
13. Add Preprocessing

Alternative 24: 36.5% 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 84.8%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 33.8%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024152 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))