Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.9%

Time: 14.4s

Alternatives: 22

Speedup: 2.4×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. sinh-undef N/A

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

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

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

   10. sinh-lowering-sinh.f64 99.9%

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

4. Applied egg-rr 99.9%

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

Alternative 2: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.75:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;U + t\_0 \cdot \frac{J}{\frac{0.5}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.75)
     (+
      U
      (*
       (* J (* l (+ 2.0 (* l (* l (* (* l l) 0.016666666666666666))))))
       (+ 1.0 (* -0.125 (* K K)))))
     (if (<= t_0 0.01)
       (+ U (* t_0 (/ J (/ 0.5 l))))
       (+ U (* (* 2.0 (sinh l)) J))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.75) {
		tmp = U + ((J * (l * (2.0 + (l * (l * ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	} else if (t_0 <= 0.01) {
		tmp = U + (t_0 * (J / (0.5 / l)));
	} else {
		tmp = U + ((2.0 * sinh(l)) * J);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.75d0)) then
        tmp = u + ((j * (l * (2.0d0 + (l * (l * ((l * l) * 0.016666666666666666d0)))))) * (1.0d0 + ((-0.125d0) * (k * k))))
    else if (t_0 <= 0.01d0) then
        tmp = u + (t_0 * (j / (0.5d0 / l)))
    else
        tmp = u + ((2.0d0 * sinh(l)) * j)
    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 <= -0.75) {
		tmp = U + ((J * (l * (2.0 + (l * (l * ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	} else if (t_0 <= 0.01) {
		tmp = U + (t_0 * (J / (0.5 / l)));
	} else {
		tmp = U + ((2.0 * Math.sinh(l)) * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.75:
		tmp = U + ((J * (l * (2.0 + (l * (l * ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))))
	elif t_0 <= 0.01:
		tmp = U + (t_0 * (J / (0.5 / l)))
	else:
		tmp = U + ((2.0 * math.sinh(l)) * J)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.75)
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * Float64(Float64(l * l) * 0.016666666666666666)))))) * Float64(1.0 + Float64(-0.125 * Float64(K * K)))));
	elseif (t_0 <= 0.01)
		tmp = Float64(U + Float64(t_0 * Float64(J / Float64(0.5 / l))));
	else
		tmp = Float64(U + Float64(Float64(2.0 * sinh(l)) * J));
	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 <= -0.75)
		tmp = U + ((J * (l * (2.0 + (l * (l * ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	elseif (t_0 <= 0.01)
		tmp = U + (t_0 * (J / (0.5 / l)));
	else
		tmp = U + ((2.0 * sinh(l)) * J);
	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, -0.75], N[(U + N[(N[(J * N[(l * N[(2.0 + N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(U + N[(t$95$0 * N[(J / N[(0.5 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.75

   1. Initial program 84.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 77.7%

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

   5. Simplified 77.7%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.6%

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

   8. Simplified 73.6%

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

   9. Taylor expanded in l around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 73.6%

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

   11. Simplified 73.6%

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

   if -0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 75.3%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. clear-num N/A

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

      6. flip-- N/A

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

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

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

      8. sinh-undef N/A

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

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

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

      10. sinh-lowering-sinh.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in l around 0

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

      1. /-lowering-/.f64 75.3%

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

   7. Simplified 75.3%

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

   if 0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. sinh-undef N/A

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

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

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

      10. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 96.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.75:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.01:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \frac{J}{\frac{0.5}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.75:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;t\_0 \leq 0.01:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.75)
     (+
      U
      (*
       (* J (* l (+ 2.0 (* l (* l (* (* l l) 0.016666666666666666))))))
       (+ 1.0 (* -0.125 (* K K)))))
     (if (<= t_0 0.01)
       (+ U (* l (* (cos (* K 0.5)) (* 2.0 J))))
       (+ U (* (* 2.0 (sinh l)) J))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.75) {
		tmp = U + ((J * (l * (2.0 + (l * (l * ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	} else if (t_0 <= 0.01) {
		tmp = U + (l * (cos((K * 0.5)) * (2.0 * J)));
	} else {
		tmp = U + ((2.0 * sinh(l)) * J);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.75d0)) then
        tmp = u + ((j * (l * (2.0d0 + (l * (l * ((l * l) * 0.016666666666666666d0)))))) * (1.0d0 + ((-0.125d0) * (k * k))))
    else if (t_0 <= 0.01d0) then
        tmp = u + (l * (cos((k * 0.5d0)) * (2.0d0 * j)))
    else
        tmp = u + ((2.0d0 * sinh(l)) * j)
    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 <= -0.75) {
		tmp = U + ((J * (l * (2.0 + (l * (l * ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	} else if (t_0 <= 0.01) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (2.0 * J)));
	} else {
		tmp = U + ((2.0 * Math.sinh(l)) * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.75:
		tmp = U + ((J * (l * (2.0 + (l * (l * ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))))
	elif t_0 <= 0.01:
		tmp = U + (l * (math.cos((K * 0.5)) * (2.0 * J)))
	else:
		tmp = U + ((2.0 * math.sinh(l)) * J)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.75)
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * Float64(Float64(l * l) * 0.016666666666666666)))))) * Float64(1.0 + Float64(-0.125 * Float64(K * K)))));
	elseif (t_0 <= 0.01)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(2.0 * J))));
	else
		tmp = Float64(U + Float64(Float64(2.0 * sinh(l)) * J));
	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 <= -0.75)
		tmp = U + ((J * (l * (2.0 + (l * (l * ((l * l) * 0.016666666666666666)))))) * (1.0 + (-0.125 * (K * K))));
	elseif (t_0 <= 0.01)
		tmp = U + (l * (cos((K * 0.5)) * (2.0 * J)));
	else
		tmp = U + ((2.0 * sinh(l)) * J);
	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, -0.75], N[(U + N[(N[(J * N[(l * N[(2.0 + N[(l * N[(l * N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.75

   1. Initial program 84.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 77.7%

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

   5. Simplified 77.7%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 73.6%

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

   8. Simplified 73.6%

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

   9. Taylor expanded in l around inf

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

       1. unpow3 N/A

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

       2. unpow2 N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

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

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

       6. *-commutative N/A

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

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

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

       8. unpow2 N/A

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

       9. *-lowering-*.f64 73.6%

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

   11. Simplified 73.6%

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

   if -0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0

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

      1. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 96.2%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

      7. *-commutative N/A

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

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

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

      9. *-commutative N/A

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

      10. *-lowering-*.f64 75.2%

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

   8. Simplified 75.2%

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

   if 0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. sinh-undef N/A

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

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

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

      10. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 96.1%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.75:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.01:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.995:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.995)
     (+ U (* t_0 (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333))))))
     (+ U (* (* 2.0 (sinh l)) J)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.995) {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else {
		tmp = U + ((2.0 * sinh(l)) * J);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.995d0) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0)))))
    else
        tmp = u + ((2.0d0 * sinh(l)) * j)
    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 <= 0.995) {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else {
		tmp = U + ((2.0 * Math.sinh(l)) * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.995:
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))))
	else:
		tmp = U + ((2.0 * math.sinh(l)) * J)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.995)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))))));
	else
		tmp = Float64(U + Float64(Float64(2.0 * sinh(l)) * J));
	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 <= 0.995)
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	else
		tmp = U + ((2.0 * sinh(l)) * J);
	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, 0.995], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.994999999999999996

   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

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 86.5%

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

   5. Simplified 86.5%

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

   if 0.994999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 91.0%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. sinh-undef N/A

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

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

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

      10. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 100.0%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.995:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.01:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.01)
   (+ U (* l (* (* J (+ 2.0 (* (* l l) 0.3333333333333333))) (cos (* K 0.5)))))
   (+ U (* (* 2.0 (sinh l)) J))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.01) {
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * cos((K * 0.5))));
	} else {
		tmp = U + ((2.0 * sinh(l)) * J);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.01d0) then
        tmp = u + (l * ((j * (2.0d0 + ((l * l) * 0.3333333333333333d0))) * cos((k * 0.5d0))))
    else
        tmp = u + ((2.0d0 * sinh(l)) * j)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.01) {
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * Math.cos((K * 0.5))));
	} else {
		tmp = U + ((2.0 * Math.sinh(l)) * J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.01:
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * math.cos((K * 0.5))))
	else:
		tmp = U + ((2.0 * math.sinh(l)) * J)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.01)
		tmp = Float64(U + Float64(l * Float64(Float64(J * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))) * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U + Float64(Float64(2.0 * sinh(l)) * J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.01)
		tmp = U + (l * ((J * (2.0 + ((l * l) * 0.3333333333333333))) * cos((K * 0.5))));
	else
		tmp = U + ((2.0 * sinh(l)) * J);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.01], N[(U + N[(l * N[(N[(J * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0100000000000000002

   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. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 86.4%

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

   if 0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. sinh-undef N/A

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

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

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

      10. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 96.1%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.01:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-59}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-59)
   (+ U (* (* 2.0 (sinh l)) J))
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (+
        2.0
        (*
         (* l l)
         (+
          0.3333333333333333
          (*
           (* l l)
           (+ 0.016666666666666666 (* l (* l 0.0003968253968253968)))))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-59) {
		tmp = U + ((2.0 * sinh(l)) * J);
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 5d-59) then
        tmp = u + ((2.0d0 * sinh(l)) * j)
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * (0.016666666666666666d0 + (l * (l * 0.0003968253968253968d0))))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-59) {
		tmp = U + ((2.0 * Math.sinh(l)) * J);
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-59:
		tmp = U + ((2.0 * math.sinh(l)) * J)
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-59)
		tmp = Float64(U + Float64(Float64(2.0 * sinh(l)) * J));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(l * Float64(l * 0.0003968253968253968)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-59)
		tmp = U + ((2.0 * sinh(l)) * J);
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + (l * (l * 0.0003968253968253968))))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-59], N[(U + N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(l * N[(l * 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)) < 5.0000000000000001e-59

   1. Initial program 90.5%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. sinh-undef N/A

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

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

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

      10. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 88.2%

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

   if 5.0000000000000001e-59 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 78.1%

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

   3. Taylor expanded in l around 0

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 95.9%

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

   5. Simplified 95.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-59}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \ell \cdot \left(\ell \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ t_1 := U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -0.0024:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 240:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+138}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (*
          (* l (cos (* K 0.5)))
          (* J (+ 2.0 (* l (* l 0.3333333333333333))))))
        (t_1 (+ U (* (* 2.0 (sinh l)) J))))
   (if (<= l -4e+155)
     t_0
     (if (<= l -0.0024)
       t_1
       (if (<= l 240.0)
         (+ U (* (cos (/ K 2.0)) (* J (* 2.0 l))))
         (if (<= l 1e+138) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (l * cos((K * 0.5))) * (J * (2.0 + (l * (l * 0.3333333333333333))));
	double t_1 = U + ((2.0 * sinh(l)) * J);
	double tmp;
	if (l <= -4e+155) {
		tmp = t_0;
	} else if (l <= -0.0024) {
		tmp = t_1;
	} else if (l <= 240.0) {
		tmp = U + (cos((K / 2.0)) * (J * (2.0 * l)));
	} else if (l <= 1e+138) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (l * cos((k * 0.5d0))) * (j * (2.0d0 + (l * (l * 0.3333333333333333d0))))
    t_1 = u + ((2.0d0 * sinh(l)) * j)
    if (l <= (-4d+155)) then
        tmp = t_0
    else if (l <= (-0.0024d0)) then
        tmp = t_1
    else if (l <= 240.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (2.0d0 * l)))
    else if (l <= 1d+138) then
        tmp = t_1
    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 * Math.cos((K * 0.5))) * (J * (2.0 + (l * (l * 0.3333333333333333))));
	double t_1 = U + ((2.0 * Math.sinh(l)) * J);
	double tmp;
	if (l <= -4e+155) {
		tmp = t_0;
	} else if (l <= -0.0024) {
		tmp = t_1;
	} else if (l <= 240.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (2.0 * l)));
	} else if (l <= 1e+138) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (l * math.cos((K * 0.5))) * (J * (2.0 + (l * (l * 0.3333333333333333))))
	t_1 = U + ((2.0 * math.sinh(l)) * J)
	tmp = 0
	if l <= -4e+155:
		tmp = t_0
	elif l <= -0.0024:
		tmp = t_1
	elif l <= 240.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (2.0 * l)))
	elif l <= 1e+138:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(l * cos(Float64(K * 0.5))) * Float64(J * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333)))))
	t_1 = Float64(U + Float64(Float64(2.0 * sinh(l)) * J))
	tmp = 0.0
	if (l <= -4e+155)
		tmp = t_0;
	elseif (l <= -0.0024)
		tmp = t_1;
	elseif (l <= 240.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(2.0 * l))));
	elseif (l <= 1e+138)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l * cos((K * 0.5))) * (J * (2.0 + (l * (l * 0.3333333333333333))));
	t_1 = U + ((2.0 * sinh(l)) * J);
	tmp = 0.0;
	if (l <= -4e+155)
		tmp = t_0;
	elseif (l <= -0.0024)
		tmp = t_1;
	elseif (l <= 240.0)
		tmp = U + (cos((K / 2.0)) * (J * (2.0 * l)));
	elseif (l <= 1e+138)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(J * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e+155], t$95$0, If[LessEqual[l, -0.0024], t$95$1, If[LessEqual[l, 240.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+138], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.00000000000000003e155 or 1e138 < 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. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 100.0%

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

   6. Taylor expanded in J around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. *-commutative N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. associate-*l* N/A

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

      15. *-commutative N/A

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

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

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

      17. *-commutative N/A

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

      18. *-lowering-*.f64 100.0%

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

   8. Simplified 100.0%

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

   if -4.00000000000000003e155 < l < -0.00239999999999999979 or 240 < l < 1e138

   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. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. sinh-undef N/A

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

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

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

      10. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 84.9%

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

   if -0.00239999999999999979 < l < 240

   1. Initial program 71.9%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+155}:\\ \;\;\;\;\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0024:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{elif}\;\ell \leq 240:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+138}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 2e-26)
   (+ U (* (* 2.0 (sinh l)) J))
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (+
        2.0
        (*
         l
         (* l (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 2e-26) {
		tmp = U + ((2.0 * sinh(l)) * J);
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 2d-26) then
        tmp = u + ((2.0d0 * sinh(l)) * j)
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (l * (l * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 2e-26) {
		tmp = U + ((2.0 * Math.sinh(l)) * J);
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 2e-26:
		tmp = U + ((2.0 * math.sinh(l)) * J)
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 2e-26)
		tmp = Float64(U + Float64(Float64(2.0 * sinh(l)) * J));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 2e-26)
		tmp = U + ((2.0 * sinh(l)) * J);
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 2e-26], N[(U + N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(l * N[(l * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 2.0000000000000001e-26

   1. Initial program 90.3%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

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

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

      4. *-commutative N/A

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

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

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

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

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

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

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

      8. sinh-undef N/A

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

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

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

      10. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

   7. Simplified 88.6%

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

   if 2.0000000000000001e-26 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 77.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 92.4%

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

   5. Simplified 92.4%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 2 \cdot 10^{-26}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

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

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

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

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

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

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

   8. sinh-undef N/A

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

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

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

   10. sinh-lowering-sinh.f64 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in K around 0

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

7. Simplified 81.5%

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

8. Final simplification 81.5%

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

Alternative 10: 75.7% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+156}:\\ \;\;\;\;U + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J -8e+156)
   (+ U (* (* J (* 2.0 l)) (+ 1.0 (* K (* K -0.125)))))
   (+
    U
    (*
     J
     (*
      l
      (+
       2.0
       (*
        (* l l)
        (+
         0.3333333333333333
         (*
          l
          (*
           l
           (+ 0.016666666666666666 (* (* l l) 0.0003968253968253968))))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -8e+156) {
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	} else {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-8d+156)) then
        tmp = u + ((j * (2.0d0 * l)) * (1.0d0 + (k * (k * (-0.125d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -8e+156) {
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	} else {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= -8e+156:
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))))
	else:
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= -8e+156)
		tmp = Float64(U + Float64(Float64(J * Float64(2.0 * l)) * Float64(1.0 + Float64(K * Float64(K * -0.125)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= -8e+156)
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	else
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, -8e+156], N[(U + N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -7.9999999999999999e156

   1. Initial program 62.2%

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

   3. Taylor expanded in l around 0

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

      1. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 96.2%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 77.4%

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

   8. Simplified 77.4%

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

   9. Taylor expanded in l around 0

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

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

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

       15. *-lowering-*.f64 77.4%

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

   11. Simplified 77.4%

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

   if -7.9999999999999999e156 < J

   1. Initial program 89.9%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

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

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

      11. *-commutative N/A

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

      12. unpow2 N/A

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

      13. associate-*l* N/A

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

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

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

      15. *-lowering-*.f64 93.7%

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

   5. Simplified 93.7%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

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

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

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

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

      13. *-commutative N/A

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

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

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

      15. unpow2 N/A

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

      16. *-lowering-*.f64 79.4%

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

   8. Simplified 79.4%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+156}:\\ \;\;\;\;U + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.4% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+156}:\\ \;\;\;\;U + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J -8e+156)
   (+ U (* (* J (* 2.0 l)) (+ 1.0 (* K (* K -0.125)))))
   (+
    U
    (*
     J
     (*
      l
      (+
       2.0
       (*
        l
        (* l (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -8e+156) {
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	} else {
		tmp = U + (J * (l * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-8d+156)) then
        tmp = u + ((j * (2.0d0 * l)) * (1.0d0 + (k * (k * (-0.125d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + (l * (l * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -8e+156) {
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	} else {
		tmp = U + (J * (l * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= -8e+156:
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))))
	else:
		tmp = U + (J * (l * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= -8e+156)
		tmp = Float64(U + Float64(Float64(J * Float64(2.0 * l)) * Float64(1.0 + Float64(K * Float64(K * -0.125)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(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 (J <= -8e+156)
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	else
		tmp = U + (J * (l * (2.0 + (l * (l * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, -8e+156], N[(U + N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(l * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -7.9999999999999999e156

   1. Initial program 62.2%

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

   3. Taylor expanded in l around 0

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

      1. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 96.2%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 77.4%

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

   8. Simplified 77.4%

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

   9. Taylor expanded in l around 0

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

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

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

       15. *-lowering-*.f64 77.4%

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

   11. Simplified 77.4%

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

   if -7.9999999999999999e156 < J

   1. Initial program 89.9%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 91.3%

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

   5. Simplified 91.3%

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

   6. Taylor expanded in K around 0

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

   8. Simplified 77.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+156}:\\ \;\;\;\;U + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.6% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+156}:\\ \;\;\;\;U + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J -8e+156)
   (+ U (* (* J (* 2.0 l)) (+ 1.0 (* K (* K -0.125)))))
   (+
    U
    (*
     (* l J)
     (+
      2.0
      (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -8e+156) {
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	} else {
		tmp = U + ((l * J) * (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 (j <= (-8d+156)) then
        tmp = u + ((j * (2.0d0 * l)) * (1.0d0 + (k * (k * (-0.125d0)))))
    else
        tmp = u + ((l * j) * (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 (J <= -8e+156) {
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	} else {
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= -8e+156:
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))))
	else:
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= -8e+156)
		tmp = Float64(U + Float64(Float64(J * Float64(2.0 * l)) * Float64(1.0 + Float64(K * Float64(K * -0.125)))));
	else
		tmp = Float64(U + Float64(Float64(l * J) * 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 (J <= -8e+156)
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	else
		tmp = U + ((l * J) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, -8e+156], N[(U + N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -7.9999999999999999e156

   1. Initial program 62.2%

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

   3. Taylor expanded in l around 0

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

      1. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 96.2%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 77.4%

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

   8. Simplified 77.4%

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

   9. Taylor expanded in l around 0

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

       1. associate-*r* N/A

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

       2. associate-*r* N/A

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

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

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

       4. *-commutative N/A

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

       5. associate-*l* N/A

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

       6. *-commutative N/A

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

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

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

       8. *-commutative N/A

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

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

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

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

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

       11. *-commutative N/A

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

       12. unpow2 N/A

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

       13. associate-*l* N/A

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

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

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

       15. *-lowering-*.f64 77.4%

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

   11. Simplified 77.4%

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

   if -7.9999999999999999e156 < J

   1. Initial program 89.9%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 91.3%

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

   5. Simplified 91.3%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 67.8%

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

   8. Simplified 67.8%

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

   9. Taylor expanded in K around 0

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

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

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

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

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

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

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

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

       6. unpow2 N/A

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

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

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

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

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

       9. *-commutative N/A

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

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

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

       11. unpow2 N/A

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

       12. *-lowering-*.f64 76.6%

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

   11. Simplified 76.6%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+156}:\\ \;\;\;\;U + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.8% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(J \cdot \frac{\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)}{U}\right)\\ \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3.05 \cdot 10^{+22}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* U (* J (/ (* l (* (* l l) 0.3333333333333333)) U)))))
   (if (<= l -1.05e+61)
     t_0
     (if (<= l 3.05e+22)
       (+ U (* l (* J (+ 2.0 (* l (* l 0.3333333333333333))))))
       t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (J * ((l * ((l * l) * 0.3333333333333333)) / U));
	double tmp;
	if (l <= -1.05e+61) {
		tmp = t_0;
	} else if (l <= 3.05e+22) {
		tmp = U + (l * (J * (2.0 + (l * (l * 0.3333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u * (j * ((l * ((l * l) * 0.3333333333333333d0)) / u))
    if (l <= (-1.05d+61)) then
        tmp = t_0
    else if (l <= 3.05d+22) then
        tmp = u + (l * (j * (2.0d0 + (l * (l * 0.3333333333333333d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U * (J * ((l * ((l * l) * 0.3333333333333333)) / U));
	double tmp;
	if (l <= -1.05e+61) {
		tmp = t_0;
	} else if (l <= 3.05e+22) {
		tmp = U + (l * (J * (2.0 + (l * (l * 0.3333333333333333)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (J * ((l * ((l * l) * 0.3333333333333333)) / U))
	tmp = 0
	if l <= -1.05e+61:
		tmp = t_0
	elif l <= 3.05e+22:
		tmp = U + (l * (J * (2.0 + (l * (l * 0.3333333333333333)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(J * Float64(Float64(l * Float64(Float64(l * l) * 0.3333333333333333)) / U)))
	tmp = 0.0
	if (l <= -1.05e+61)
		tmp = t_0;
	elseif (l <= 3.05e+22)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U * (J * ((l * ((l * l) * 0.3333333333333333)) / U));
	tmp = 0.0;
	if (l <= -1.05e+61)
		tmp = t_0;
	elseif (l <= 3.05e+22)
		tmp = U + (l * (J * (2.0 + (l * (l * 0.3333333333333333)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U * N[(J * N[(N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.05e+61], t$95$0, If[LessEqual[l, 3.05e+22], N[(U + N[(l * N[(J * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1.0500000000000001e61 or 3.0499999999999999e22 < 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. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 82.9%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 65.9%

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

   8. Simplified 65.9%

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

   9. Taylor expanded in U around inf

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

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

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

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

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

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

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

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

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

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

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

       10. unpow2 N/A

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

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

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

       12. /-lowering-/.f64 72.4%

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

   11. Simplified 72.4%

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

   12. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-/l* N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

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

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

       10. *-commutative N/A

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

       11. associate-*r/ N/A

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

       12. associate-*l/ N/A

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

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

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

       14. *-commutative N/A

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

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

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

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

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

       17. unpow2 N/A

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

       18. *-lowering-*.f64 76.5%

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

   14. Simplified 76.5%

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

   if -1.0500000000000001e61 < l < 3.0499999999999999e22

   1. Initial program 76.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 87.4%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 73.5%

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

   8. Simplified 73.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+61}:\\ \;\;\;\;U \cdot \left(J \cdot \frac{\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)}{U}\right)\\ \mathbf{elif}\;\ell \leq 3.05 \cdot 10^{+22}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(J \cdot \frac{\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.1% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(J \cdot \frac{\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)}{U}\right)\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* U (* J (/ (* l (* (* l l) 0.3333333333333333)) U)))))
   (if (<= l -5.4e+19) t_0 (if (<= l 2.9e+22) (+ U (* J (* 2.0 l))) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (J * ((l * ((l * l) * 0.3333333333333333)) / U));
	double tmp;
	if (l <= -5.4e+19) {
		tmp = t_0;
	} else if (l <= 2.9e+22) {
		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 = u * (j * ((l * ((l * l) * 0.3333333333333333d0)) / u))
    if (l <= (-5.4d+19)) then
        tmp = t_0
    else if (l <= 2.9d+22) 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 = U * (J * ((l * ((l * l) * 0.3333333333333333)) / U));
	double tmp;
	if (l <= -5.4e+19) {
		tmp = t_0;
	} else if (l <= 2.9e+22) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (J * ((l * ((l * l) * 0.3333333333333333)) / U))
	tmp = 0
	if l <= -5.4e+19:
		tmp = t_0
	elif l <= 2.9e+22:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(J * Float64(Float64(l * Float64(Float64(l * l) * 0.3333333333333333)) / U)))
	tmp = 0.0
	if (l <= -5.4e+19)
		tmp = t_0;
	elseif (l <= 2.9e+22)
		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 = U * (J * ((l * ((l * l) * 0.3333333333333333)) / U));
	tmp = 0.0;
	if (l <= -5.4e+19)
		tmp = t_0;
	elseif (l <= 2.9e+22)
		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[(U * N[(J * N[(N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.4e+19], t$95$0, If[LessEqual[l, 2.9e+22], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.4e19 or 2.9e22 < 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. distribute-rgt-in N/A

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

      2. fma-define N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. associate-*l* N/A

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

      6. fma-undefine N/A

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

      7. distribute-rgt-in N/A

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

      8. *-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. Simplified 77.7%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. unpow2 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

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

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in U around inf

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

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

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

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

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

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

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

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

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

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

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

       10. unpow2 N/A

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

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

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

       12. /-lowering-/.f64 68.0%

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

   11. Simplified 68.0%

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

   12. Taylor expanded in l around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

       3. associate-*l* N/A

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

       4. cube-mult N/A

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

       5. unpow2 N/A

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

       6. associate-/l* N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \frac{{\ell}^{2}}{U}\right)\right)}\right)\right) \]

       10. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \left(\left(\frac{1}{3} \cdot \frac{{\ell}^{2}}{U}\right) \cdot \color{blue}{\ell}\right)\right)\right) \]

       11. associate-*r/ N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \left(\frac{\frac{1}{3} \cdot {\ell}^{2}}{U} \cdot \ell\right)\right)\right) \]

       12. associate-*l/ N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \left(\frac{\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell}{\color{blue}{U}}\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right), \color{blue}{U}\right)\right)\right) \]

       14. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\left(\ell \cdot \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right), U\right)\right)\right) \]

       15. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right), U\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right), U\right)\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right), U\right)\right)\right) \]

       18. *-lowering-*.f64 71.8%

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{*.f64}\left(J, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), U\right)\right)\right) \]

   14. Simplified 71.8%

       \[\leadsto U \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)}{U}\right)} \]

   if -5.4e19 < l < 2.9e22

   1. Initial program 73.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 93.0%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 77.9%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 77.9%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   10. 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 77.9%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   11. Simplified 77.9%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+19}:\\ \;\;\;\;U \cdot \left(J \cdot \frac{\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)}{U}\right)\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+22}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(J \cdot \frac{\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.7% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -2.7 \cdot 10^{+155}:\\ \;\;\;\;U + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J -2.7e+155)
   (+ U (* (* J (* 2.0 l)) (+ 1.0 (* K (* K -0.125)))))
   (+ U (* J (* l (+ 2.0 (* l (* l 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -2.7e+155) {
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	} else {
		tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-2.7d+155)) then
        tmp = u + ((j * (2.0d0 * l)) * (1.0d0 + (k * (k * (-0.125d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + (l * (l * 0.3333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -2.7e+155) {
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	} else {
		tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= -2.7e+155:
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))))
	else:
		tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= -2.7e+155)
		tmp = Float64(U + Float64(Float64(J * Float64(2.0 * l)) * Float64(1.0 + Float64(K * Float64(K * -0.125)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= -2.7e+155)
		tmp = U + ((J * (2.0 * l)) * (1.0 + (K * (K * -0.125))));
	else
		tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, -2.7e+155], N[(U + N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -2.69999999999999994e155

   1. Initial program 62.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 96.2%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 77.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 77.4%

      \[\leadsto \ell \cdot \left(\color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \left(\ell \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right)\right)}, U\right) \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(2 \cdot \left(\left(J \cdot \ell\right) \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right)\right), U\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\left(2 \cdot \left(J \cdot \ell\right)\right) \cdot \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right), U\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot \left(J \cdot \ell\right)\right), \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right), U\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right), U\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right), U\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right), U\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right), U\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right), U\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), \left(1 + \frac{-1}{8} \cdot {K}^{2}\right)\right), U\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), \mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right)\right), U\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), \mathsf{+.f64}\left(1, \left({K}^{2} \cdot \frac{-1}{8}\right)\right)\right), U\right) \]

       12. unpow2 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), \mathsf{+.f64}\left(1, \left(\left(K \cdot K\right) \cdot \frac{-1}{8}\right)\right)\right), U\right) \]

       13. associate-*l* N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), \mathsf{+.f64}\left(1, \left(K \cdot \left(K \cdot \frac{-1}{8}\right)\right)\right)\right), U\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(K, \left(K \cdot \frac{-1}{8}\right)\right)\right)\right), U\right) \]

       15. *-lowering-*.f64 77.4%

           \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(K, \mathsf{*.f64}\left(K, \frac{-1}{8}\right)\right)\right)\right), U\right) \]

   11. Simplified 77.4%

       \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)} + U \]

   if -2.69999999999999994e155 < J

   1. Initial program 89.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 84.1%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 71.1%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right) \cdot J\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right) \cdot J\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot J\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot J\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right), J\right), U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right), J\right), U\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right)\right), J\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right), J\right), U\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

      11. *-lowering-*.f64 75.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

   10. Applied egg-rr 75.3%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot J} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.7 \cdot 10^{+155}:\\ \;\;\;\;U + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(1 + K \cdot \left(K \cdot -0.125\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.7% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+156}:\\ \;\;\;\;U + \ell \cdot \left(\left(1 + -0.125 \cdot \left(K \cdot K\right)\right) \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J -8e+156)
   (+ U (* l (* (+ 1.0 (* -0.125 (* K K))) (* 2.0 J))))
   (+ U (* J (* l (+ 2.0 (* l (* l 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -8e+156) {
		tmp = U + (l * ((1.0 + (-0.125 * (K * K))) * (2.0 * J)));
	} else {
		tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-8d+156)) then
        tmp = u + (l * ((1.0d0 + ((-0.125d0) * (k * k))) * (2.0d0 * j)))
    else
        tmp = u + (j * (l * (2.0d0 + (l * (l * 0.3333333333333333d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -8e+156) {
		tmp = U + (l * ((1.0 + (-0.125 * (K * K))) * (2.0 * J)));
	} else {
		tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= -8e+156:
		tmp = U + (l * ((1.0 + (-0.125 * (K * K))) * (2.0 * J)))
	else:
		tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= -8e+156)
		tmp = Float64(U + Float64(l * Float64(Float64(1.0 + Float64(-0.125 * Float64(K * K))) * Float64(2.0 * J))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= -8e+156)
		tmp = U + (l * ((1.0 + (-0.125 * (K * K))) * (2.0 * J)));
	else
		tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, -8e+156], N[(U + N[(l * N[(N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -7.9999999999999999e156

   1. Initial program 62.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 96.2%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\color{blue}{\left(1 + \frac{-1}{8} \cdot {K}^{2}\right)}, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left(\frac{-1}{8} \cdot {K}^{2}\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left({K}^{2}\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \left(K \cdot K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

      4. *-lowering-*.f64 77.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 77.4%

      \[\leadsto \ell \cdot \left(\color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \color{blue}{\left(2 \cdot J\right)}\right)\right), U\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \left(J \cdot 2\right)\right)\right), U\right) \]

       2. *-lowering-*.f64 77.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{-1}{8}, \mathsf{*.f64}\left(K, K\right)\right)\right), \mathsf{*.f64}\left(J, 2\right)\right)\right), U\right) \]

   11. Simplified 77.4%

       \[\leadsto \ell \cdot \left(\left(1 + -0.125 \cdot \left(K \cdot K\right)\right) \cdot \color{blue}{\left(J \cdot 2\right)}\right) + U \]

   if -7.9999999999999999e156 < J

   1. Initial program 89.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   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. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 84.1%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 71.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 71.1%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right) \cdot J\right)\right), U\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right) \cdot J\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot J\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot J\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right), J\right), U\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right), J\right), U\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right)\right), J\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right), J\right), U\right) \]

      9. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

      11. *-lowering-*.f64 75.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

   10. Applied egg-rr 75.3%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot J} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -8 \cdot 10^{+156}:\\ \;\;\;\;U + \ell \cdot \left(\left(1 + -0.125 \cdot \left(K \cdot K\right)\right) \cdot \left(2 \cdot J\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+62}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* l (* (* l l) 0.3333333333333333)))))
   (if (<= l -5.4e+19) t_0 (if (<= l 1.4e+62) (+ U (* J (* 2.0 l))) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (l * ((l * l) * 0.3333333333333333));
	double tmp;
	if (l <= -5.4e+19) {
		tmp = t_0;
	} else if (l <= 1.4e+62) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (l * ((l * l) * 0.3333333333333333d0))
    if (l <= (-5.4d+19)) then
        tmp = t_0
    else if (l <= 1.4d+62) then
        tmp = u + (j * (2.0d0 * l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (l * ((l * l) * 0.3333333333333333));
	double tmp;
	if (l <= -5.4e+19) {
		tmp = t_0;
	} else if (l <= 1.4e+62) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (l * ((l * l) * 0.3333333333333333))
	tmp = 0
	if l <= -5.4e+19:
		tmp = t_0
	elif l <= 1.4e+62:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(l * Float64(Float64(l * l) * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -5.4e+19)
		tmp = t_0;
	elseif (l <= 1.4e+62)
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l * ((l * l) * 0.3333333333333333));
	tmp = 0.0;
	if (l <= -5.4e+19)
		tmp = t_0;
	elseif (l <= 1.4e+62)
		tmp = U + (J * (2.0 * l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(l * N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.4e+19], t$95$0, If[LessEqual[l, 1.4e+62], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.4e19 or 1.40000000000000007e62 < 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. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 81.9%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 65.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 65.5%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

   9. Taylor expanded in U around inf

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}{U}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \color{blue}{\left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}{U}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{J \cdot \left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}{U}\right)}\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(J \cdot \color{blue}{\frac{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}{U}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \color{blue}{\left(\frac{\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}{U}\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \left(\frac{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell}{U}\right)\right)\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\frac{\ell}{U}}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right), \color{blue}{\left(\frac{\ell}{U}\right)}\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right), \left(\frac{\color{blue}{\ell}}{U}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right), \left(\frac{\ell}{U}\right)\right)\right)\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right), \left(\frac{\ell}{U}\right)\right)\right)\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \left(\frac{\ell}{U}\right)\right)\right)\right)\right) \]

       12. /-lowering-/.f64 70.9%

           \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right), \mathsf{/.f64}\left(\ell, \color{blue}{U}\right)\right)\right)\right)\right) \]

   11. Simplified 70.9%

       \[\leadsto \color{blue}{U \cdot \left(1 + J \cdot \left(\left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right) \cdot \frac{\ell}{U}\right)\right)} \]

   12. Taylor expanded in l around inf

       \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(\frac{1}{3} \cdot J\right) \cdot \color{blue}{{\ell}^{3}} \]

       2. *-commutative N/A

          \[\leadsto \left(J \cdot \frac{1}{3}\right) \cdot {\color{blue}{\ell}}^{3} \]

       3. associate-*l* N/A

          \[\leadsto J \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{3}\right)} \]

       4. unpow3 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\ell}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto J \cdot \left(\frac{1}{3} \cdot \left({\ell}^{2} \cdot \ell\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto J \cdot \left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \color{blue}{\ell}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(\left(\frac{1}{3} \cdot {\ell}^{2}\right) \cdot \ell\right)}\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       12. *-lowering-*.f64 73.3%

           \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   14. Simplified 73.3%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -5.4e19 < l < 1.40000000000000007e62

   1. Initial program 75.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 88.4%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 74.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 74.1%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   10. 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 74.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   11. Simplified 74.1%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+19}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+62}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 69.2% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{if}\;\ell \leq -9 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+62}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (* l (* J (* l l))))))
   (if (<= l -9e+19) t_0 (if (<= l 1.4e+62) (+ U (* J (* 2.0 l))) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * (l * (J * (l * l)));
	double tmp;
	if (l <= -9e+19) {
		tmp = t_0;
	} else if (l <= 1.4e+62) {
		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 = 0.3333333333333333d0 * (l * (j * (l * l)))
    if (l <= (-9d+19)) then
        tmp = t_0
    else if (l <= 1.4d+62) 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 = 0.3333333333333333 * (l * (J * (l * l)));
	double tmp;
	if (l <= -9e+19) {
		tmp = t_0;
	} else if (l <= 1.4e+62) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * (l * (J * (l * l)))
	tmp = 0
	if l <= -9e+19:
		tmp = t_0
	elif l <= 1.4e+62:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * Float64(l * Float64(J * Float64(l * l))))
	tmp = 0.0
	if (l <= -9e+19)
		tmp = t_0;
	elseif (l <= 1.4e+62)
		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 = 0.3333333333333333 * (l * (J * (l * l)));
	tmp = 0.0;
	if (l <= -9e+19)
		tmp = t_0;
	elseif (l <= 1.4e+62)
		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[(0.3333333333333333 * N[(l * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -9e+19], t$95$0, If[LessEqual[l, 1.4e+62], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -9e19 or 1.40000000000000007e62 < 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. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 81.9%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 65.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 65.5%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

   9. Taylor expanded in l around inf

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \left(J \cdot {\ell}^{3}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(J \cdot {\ell}^{3}\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{3} \cdot \color{blue}{J}\right)\right) \]

       3. cube-mult N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \left(\ell \cdot \ell\right)\right) \cdot J\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot {\ell}^{2}\right) \cdot J\right)\right) \]

       5. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \color{blue}{\left({\ell}^{2} \cdot J\right)}\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(J \cdot \color{blue}{{\ell}^{2}}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot {\ell}^{2}\right)}\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \color{blue}{\left({\ell}^{2}\right)}\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{\ell}\right)\right)\right)\right) \]

       10. *-lowering-*.f64 65.5%

           \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right)\right) \]

   11. Simplified 65.5%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)} \]

   if -9e19 < l < 1.40000000000000007e62

   1. Initial program 75.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 88.4%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 74.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 74.1%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   10. 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 74.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   11. Simplified 74.1%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9 \cdot 10^{+19}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+62}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\ell \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 46.7% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(2 \cdot J\right)\\ \mathbf{if}\;\ell \leq -1020:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.00013:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* 2.0 J))))
   (if (<= l -1020.0) t_0 (if (<= l 0.00013) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (2.0 * J);
	double tmp;
	if (l <= -1020.0) {
		tmp = t_0;
	} else if (l <= 0.00013) {
		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 = l * (2.0d0 * j)
    if (l <= (-1020.0d0)) then
        tmp = t_0
    else if (l <= 0.00013d0) 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 = l * (2.0 * J);
	double tmp;
	if (l <= -1020.0) {
		tmp = t_0;
	} else if (l <= 0.00013) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (2.0 * J)
	tmp = 0
	if l <= -1020.0:
		tmp = t_0
	elif l <= 0.00013:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(2.0 * J))
	tmp = 0.0
	if (l <= -1020.0)
		tmp = t_0;
	elseif (l <= 0.00013)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (2.0 * J);
	tmp = 0.0;
	if (l <= -1020.0)
		tmp = t_0;
	elseif (l <= 0.00013)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1020.0], t$95$0, If[LessEqual[l, 0.00013], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -1020 or 1.29999999999999989e-4 < l

   1. Initial program 99.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

      6. fma-undefine N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

      7. distribute-rgt-in N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

      8. *-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. Simplified 73.2%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

      9. *-lowering-*.f64 58.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 58.3%

      \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

   9. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   10. 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 18.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   11. Simplified 18.3%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   12. Taylor expanded in J around inf

       \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(2 \cdot J\right) \cdot \color{blue}{\ell} \]

       2. *-commutative N/A

          \[\leadsto \ell \cdot \color{blue}{\left(2 \cdot J\right)} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(2 \cdot J\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \left(J \cdot \color{blue}{2}\right)\right) \]

       5. *-lowering-*.f64 17.6%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \color{blue}{2}\right)\right) \]

   14. Simplified 17.6%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot 2\right)} \]

   if -1020 < l < 1.29999999999999989e-4

   1. Initial program 71.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 71.6%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1020:\\ \;\;\;\;\ell \cdot \left(2 \cdot J\right)\\ \mathbf{elif}\;\ell \leq 0.00013:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(2 \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 72.0% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* J (* l (+ 2.0 (* l (* l 0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (l * (2.0d0 + (l * (l * 0.3333333333333333d0)))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * (2.0 + (l * (l * 0.3333333333333333)))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

   2. fma-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

   6. fma-undefine N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   8. *-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. Simplified 85.3%

   \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

6. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   9. *-lowering-*.f64 70.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

8. Simplified 70.0%

   \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right) \cdot J\right)\right), U\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right) \cdot J\right)\right), U\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right) \cdot J\right)\right), U\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right) \cdot J\right), U\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right), J\right), U\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(2 + \frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right), J\right), U\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right)\right), J\right), U\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right), J\right), U\right) \]

   9. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

   11. *-lowering-*.f64 73.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right), J\right), U\right) \]

10. Applied egg-rr 73.8%

    \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \cdot J} + U \]

11. Final simplification 73.8%

    \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right) \]
12. Add Preprocessing

Alternative 21: 54.5% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(2 \cdot \ell\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* J (* 2.0 l))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (2.0 * l));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (2.0d0 * l))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (2.0 * l));
}

Python

def code(J, l, K, U):
	return U + (J * (2.0 * l))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(2.0 * l)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (2.0 * l));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(\ell \cdot \left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

   2. fma-define N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(J \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot {\ell}^{2}\right)\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\frac{1}{3} \cdot \left(\left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right) \cdot {\ell}^{2}\right), \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\mathsf{fma}\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}, \ell, \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right)\right), U\right) \]

   6. fma-undefine N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2}\right) \cdot \ell + \left(2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot \ell\right), U\right) \]

   7. distribute-rgt-in N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\ell \cdot \left(\left(\frac{1}{3} \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right) \cdot {\ell}^{2} + 2 \cdot \left(J \cdot \cos \left(\frac{1}{2} \cdot K\right)\right)\right)\right), U\right) \]

   8. *-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. Simplified 85.3%

   \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)} + U \]

6. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right), U\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\frac{1}{3} \cdot \ell\right)\right)\right)\right)\right), U\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

   9. *-lowering-*.f64 70.0%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right)\right)\right)\right), U\right) \]

8. Simplified 70.0%

   \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)} + U \]

9. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
10. 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 48.1%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

11. Simplified 48.1%

    \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

12. Final simplification 48.1%

    \[\leadsto U + J \cdot \left(2 \cdot \ell\right) \]
13. Add Preprocessing

Alternative 22: 37.4% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 87.1%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 33.7%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))