Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.9%

Time: 14.6s

Alternatives: 24

Speedup: 2.6×

• 48.7% 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: 86.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

   1. associate-*l* N/A

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

   2. *-commutative N/A

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

   3. fma-define N/A

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

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

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

   5. *-commutative N/A

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

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

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

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

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

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

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

   9. sinh-undef N/A

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

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

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

   11. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 2: 92.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.04)
   (+ U (* l (* (+ 2.0 (* l (* l 0.3333333333333333))) (* J (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)) <= -0.04) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * 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)) <= (-0.04d0)) then
        tmp = u + (l * ((2.0d0 + (l * (l * 0.3333333333333333d0))) * (j * 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)) <= -0.04) {
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * 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)) <= -0.04:
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * 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)) <= -0.04)
		tmp = Float64(U + Float64(l * Float64(Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333))) * Float64(J * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(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)) <= -0.04)
		tmp = U + (l * ((2.0 + (l * (l * 0.3333333333333333))) * (J * 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], -0.04], N[(U + N[(l * N[(N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(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))) < -0.0400000000000000008

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 86.8%

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

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

   1. Initial program 87.9%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 87.6%

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

   5. Simplified 87.6%

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

      1. *-commutative N/A

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

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

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

      3. div-inv N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. rec-exp N/A

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

      7. sinh-undef N/A

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

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

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

      9. sinh-lowering-sinh.f64 96.2%

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

   7. Applied egg-rr 96.2%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + \ell \cdot \left(\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.04:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.04)
     (+ U (* t_0 (* J (* 2.0 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 <= -0.04) {
		tmp = U + (t_0 * (J * (2.0 * 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 <= (-0.04d0)) then
        tmp = u + (t_0 * (j * (2.0d0 * 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 <= -0.04) {
		tmp = U + (t_0 * (J * (2.0 * 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 <= -0.04:
		tmp = U + (t_0 * (J * (2.0 * 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 <= -0.04)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(2.0 * 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 <= -0.04)
		tmp = U + (t_0 * (J * (2.0 * 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, -0.04], N[(U + N[(t$95$0 * N[(J * N[(2.0 * l), $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))) < -0.0400000000000000008

   1. Initial program 89.4%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(2 \cdot \ell\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   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 68.2%

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

   5. Simplified 68.2%

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

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

   1. Initial program 87.9%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 87.6%

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

   5. Simplified 87.6%

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

      1. *-commutative N/A

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

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

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

      3. div-inv N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. rec-exp N/A

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

      7. sinh-undef N/A

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

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

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

      9. sinh-lowering-sinh.f64 96.2%

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

   7. Applied egg-rr 96.2%

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

4. Final simplification 88.3%

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

Alternative 4: 99.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. sinh-undef N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

   11. sinh-lowering-sinh.f64 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 5: 82.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+15}:\\ \;\;\;\;U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e+15)
   (+ U (* 2.0 (* (sinh l) (+ J (* J (* (* K K) -0.125))))))
   (+
    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) <= 1e+15) {
		tmp = U + (2.0 * (sinh(l) * (J + (J * ((K * K) * -0.125)))));
	} 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) <= 1d+15) then
        tmp = u + (2.0d0 * (sinh(l) * (j + (j * ((k * k) * (-0.125d0))))))
    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) <= 1e+15) {
		tmp = U + (2.0 * (Math.sinh(l) * (J + (J * ((K * K) * -0.125)))));
	} 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) <= 1e+15:
		tmp = U + (2.0 * (math.sinh(l) * (J + (J * ((K * K) * -0.125)))))
	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) <= 1e+15)
		tmp = Float64(U + Float64(2.0 * Float64(sinh(l) * Float64(J + Float64(J * Float64(Float64(K * K) * -0.125))))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968)))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 1e+15)
		tmp = U + (2.0 * (sinh(l) * (J + (J * ((K * K) * -0.125)))));
	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], 1e+15], N[(U + N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * N[(J + N[(J * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 1e15

   1. Initial program 88.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 79.8%

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

   7. Simplified 79.8%

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

   if 1e15 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 88.7%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. *-commutative N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+15}:\\ \;\;\;\;U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ t_1 := U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -330:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 225:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{+129}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (* J (* (* K K) -0.125))))))))
   (if (<= l -2.4e+94)
     t_0
     (if (<= l -330.0)
       t_1
       (if (<= l 225.0)
         (+ U (* (cos (/ K 2.0)) (* J (* 2.0 l))))
         (if (<= l 1.36e+129) 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 + (J * ((K * K) * -0.125)))));
	double tmp;
	if (l <= -2.4e+94) {
		tmp = t_0;
	} else if (l <= -330.0) {
		tmp = t_1;
	} else if (l <= 225.0) {
		tmp = U + (cos((K / 2.0)) * (J * (2.0 * l)));
	} else if (l <= 1.36e+129) {
		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 + (j * ((k * k) * (-0.125d0))))))
    if (l <= (-2.4d+94)) then
        tmp = t_0
    else if (l <= (-330.0d0)) then
        tmp = t_1
    else if (l <= 225.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (2.0d0 * l)))
    else if (l <= 1.36d+129) 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 + (J * ((K * K) * -0.125)))));
	double tmp;
	if (l <= -2.4e+94) {
		tmp = t_0;
	} else if (l <= -330.0) {
		tmp = t_1;
	} else if (l <= 225.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (2.0 * l)));
	} else if (l <= 1.36e+129) {
		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 + (J * ((K * K) * -0.125)))))
	tmp = 0
	if l <= -2.4e+94:
		tmp = t_0
	elif l <= -330.0:
		tmp = t_1
	elif l <= 225.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (2.0 * l)))
	elif l <= 1.36e+129:
		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(Float64(l * l) * 0.3333333333333333))))
	t_1 = Float64(U + Float64(2.0 * Float64(sinh(l) * Float64(J + Float64(J * Float64(Float64(K * K) * -0.125))))))
	tmp = 0.0
	if (l <= -2.4e+94)
		tmp = t_0;
	elseif (l <= -330.0)
		tmp = t_1;
	elseif (l <= 225.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(2.0 * l))));
	elseif (l <= 1.36e+129)
		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 + (J * ((K * K) * -0.125)))));
	tmp = 0.0;
	if (l <= -2.4e+94)
		tmp = t_0;
	elseif (l <= -330.0)
		tmp = t_1;
	elseif (l <= 225.0)
		tmp = U + (cos((K / 2.0)) * (J * (2.0 * l)));
	elseif (l <= 1.36e+129)
		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[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * N[(J + N[(J * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.4e+94], t$95$0, If[LessEqual[l, -330.0], t$95$1, If[LessEqual[l, 225.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.36e+129], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.39999999999999983e94 or 1.3599999999999999e129 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 94.1%

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

   6. Taylor expanded in 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. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 94.1%

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

   8. Simplified 94.1%

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

   if -2.39999999999999983e94 < l < -330 or 225 < l < 1.3599999999999999e129

   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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 84.4%

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

   7. Simplified 84.4%

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

   if -330 < l < 225

   1. Initial program 77.0%

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

   3. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(2 \cdot \ell\right)}\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(K, 2\right)\right)\right), U\right) \]
   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 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -330:\\ \;\;\;\;U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 225:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{elif}\;\ell \leq 1.36 \cdot 10^{+129}:\\ \;\;\;\;U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+15}:\\ \;\;\;\;U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e+15)
   (+ U (* 2.0 (* (sinh l) (+ J (* J (* (* K K) -0.125))))))
   (+
    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) <= 1e+15) {
		tmp = U + (2.0 * (sinh(l) * (J + (J * ((K * K) * -0.125)))));
	} 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) <= 1d+15) then
        tmp = u + (2.0d0 * (sinh(l) * (j + (j * ((k * k) * (-0.125d0))))))
    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) <= 1e+15) {
		tmp = U + (2.0 * (Math.sinh(l) * (J + (J * ((K * K) * -0.125)))));
	} 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) <= 1e+15:
		tmp = U + (2.0 * (math.sinh(l) * (J + (J * ((K * K) * -0.125)))))
	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) <= 1e+15)
		tmp = Float64(U + Float64(2.0 * Float64(sinh(l) * Float64(J + Float64(J * Float64(Float64(K * K) * -0.125))))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * 0.016666666666666666)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 1e+15)
		tmp = U + (2.0 * (sinh(l) * (J + (J * ((K * K) * -0.125)))));
	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], 1e+15], N[(U + N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * N[(J + N[(J * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * 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)) < 1e15

   1. Initial program 88.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 79.8%

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

   7. Simplified 79.8%

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

   if 1e15 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 88.7%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 93.7%

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

   5. Simplified 93.7%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+15}:\\ \;\;\;\;U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+15}:\\ \;\;\;\;U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 1e+15)
   (+ U (* 2.0 (* (sinh l) (+ J (* J (* (* K K) -0.125))))))
   (+
    U
    (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* l (* l 0.3333333333333333)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e+15) {
		tmp = U + (2.0 * (sinh(l) * (J + (J * ((K * K) * -0.125)))));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 1d+15) then
        tmp = u + (2.0d0 * (sinh(l) * (j + (j * ((k * k) * (-0.125d0))))))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (l * (l * 0.3333333333333333d0))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 1e+15) {
		tmp = U + (2.0 * (Math.sinh(l) * (J + (J * ((K * K) * -0.125)))));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 1e+15:
		tmp = U + (2.0 * (math.sinh(l) * (J + (J * ((K * K) * -0.125)))))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 1e+15)
		tmp = Float64(U + Float64(2.0 * Float64(sinh(l) * Float64(J + Float64(J * Float64(Float64(K * K) * -0.125))))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * 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 ((K / 2.0) <= 1e+15)
		tmp = U + (2.0 * (sinh(l) * (J + (J * ((K * K) * -0.125)))));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (l * (l * 0.3333333333333333))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 1e+15], N[(U + N[(2.0 * N[(N[Sinh[l], $MachinePrecision] * N[(J + N[(J * N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 1e15

   1. Initial program 88.2%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. sinh-undef N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      11. sinh-lowering-sinh.f64 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in K around 0

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

      1. associate-*r* N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

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

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

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

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

      8. unpow2 N/A

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

      9. *-lowering-*.f64 79.8%

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

   7. Simplified 79.8%

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

   if 1e15 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 88.7%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

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

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

      7. *-commutative N/A

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

      8. *-lowering-*.f64 89.2%

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

   5. Simplified 89.2%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 10^{+15}:\\ \;\;\;\;U + 2 \cdot \left(\sinh \ell \cdot \left(J + J \cdot \left(\left(K \cdot K\right) \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -70000:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{elif}\;\ell \leq 9.2:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -70000.0)
   (+ U (* (* 2.0 (sinh l)) J))
   (if (<= l 9.2)
     (+ U (* J (* (cos (* K 0.5)) (* 2.0 l))))
     (+ U (* J (+ (exp l) -1.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -70000.0) {
		tmp = U + ((2.0 * sinh(l)) * J);
	} else if (l <= 9.2) {
		tmp = U + (J * (cos((K * 0.5)) * (2.0 * l)));
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	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) :: tmp
    if (l <= (-70000.0d0)) then
        tmp = u + ((2.0d0 * sinh(l)) * j)
    else if (l <= 9.2d0) then
        tmp = u + (j * (cos((k * 0.5d0)) * (2.0d0 * l)))
    else
        tmp = u + (j * (exp(l) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -70000.0) {
		tmp = U + ((2.0 * Math.sinh(l)) * J);
	} else if (l <= 9.2) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (2.0 * l)));
	} else {
		tmp = U + (J * (Math.exp(l) + -1.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -70000.0:
		tmp = U + ((2.0 * math.sinh(l)) * J)
	elif l <= 9.2:
		tmp = U + (J * (math.cos((K * 0.5)) * (2.0 * l)))
	else:
		tmp = U + (J * (math.exp(l) + -1.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -70000.0)
		tmp = Float64(U + Float64(Float64(2.0 * sinh(l)) * J));
	elseif (l <= 9.2)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(2.0 * l))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -70000.0)
		tmp = U + ((2.0 * sinh(l)) * J);
	elseif (l <= 9.2)
		tmp = U + (J * (cos((K * 0.5)) * (2.0 * l)));
	else
		tmp = U + (J * (exp(l) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -70000.0], N[(U + N[(N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.2], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -7e4

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 74.6%

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

   5. Simplified 74.6%

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

      1. *-commutative N/A

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

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

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

      3. div-inv N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. rec-exp N/A

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

      7. sinh-undef N/A

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

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

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

      9. sinh-lowering-sinh.f64 74.6%

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

   7. Applied egg-rr 74.6%

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

   if -7e4 < l < 9.1999999999999993

   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

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. associate-*l* N/A

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

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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt-out N/A

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

      9. +-commutative N/A

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

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

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

   5. Simplified 99.2%

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

   6. Taylor expanded in l around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

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

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

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

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

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

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

      11. *-commutative N/A

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

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

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

      13. *-commutative N/A

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

      14. *-lowering-*.f64 99.0%

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

   8. Simplified 99.0%

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

   if 9.1999999999999993 < 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 K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in l around 0

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

   8. Simplified 75.8%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -70000:\\ \;\;\;\;U + \left(2 \cdot \sinh \ell\right) \cdot J\\ \mathbf{elif}\;\ell \leq 9.2:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1.5)
   (*
    U
    (+
     1.0
     (*
      J
      (*
       (+
        2.0
        (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))
       (/ l U)))))
   (+ U (* J (+ (exp l) -1.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.5) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	} else {
		tmp = U + (J * (exp(l) + -1.0));
	}
	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) :: tmp
    if (l <= 1.5d0) then
        tmp = u * (1.0d0 + (j * ((2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))) * (l / u))))
    else
        tmp = u + (j * (exp(l) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.5) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	} else {
		tmp = U + (J * (Math.exp(l) + -1.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1.5:
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))))
	else:
		tmp = U + (J * (math.exp(l) + -1.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1.5)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))) * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1.5)
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	else
		tmp = U + (J * (exp(l) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1.5], N[(U * N[(1.0 + N[(J * N[(N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5

   1. Initial program 84.3%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 80.2%

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

   8. Simplified 80.2%

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

   9. Taylor expanded in U around inf

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

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

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

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

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

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

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

   11. Simplified 81.1%

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

   if 1.5 < 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 K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in l around 0

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

   8. Simplified 75.8%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.8% 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 88.3%

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

3. Taylor expanded in K around 0

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

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

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

   2. sub-neg N/A

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

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

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

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

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

   5. exp-neg N/A

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

   6. distribute-neg-frac N/A

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

   7. metadata-eval N/A

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

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

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

   9. exp-lowering-exp.f64 75.6%

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

5. Simplified 75.6%

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

   1. *-commutative N/A

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

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

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

   3. div-inv N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. rec-exp N/A

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

   7. sinh-undef N/A

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

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

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

   9. sinh-lowering-sinh.f64 81.7%

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

7. Applied egg-rr 81.7%

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

8. Final simplification 81.7%

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

Alternative 12: 76.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\\ t_1 := \ell \cdot t\_0\\ t_2 := \ell \cdot t\_1\\ \mathbf{if}\;\ell \leq -220000000:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot t\_0\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;U + \frac{\left(\ell \cdot J\right) \cdot \left(8 + t\_2 \cdot \left(t\_1 \cdot \left(\left(\ell \cdot \ell\right) \cdot t\_1\right)\right)\right)}{4 + t\_2 \cdot \left(\ell \cdot \left(\ell \cdot 0.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \left(1 + \left(K \cdot K\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))
        (t_1 (* l t_0))
        (t_2 (* l t_1)))
   (if (<= l -220000000.0)
     (* U (+ 1.0 (* J (* (+ 2.0 (* (* l l) t_0)) (/ l U)))))
     (if (<= l 6.2e+31)
       (+
        U
        (/
         (* (* l J) (+ 8.0 (* t_2 (* t_1 (* (* l l) t_1)))))
         (+ 4.0 (* t_2 (- (* l (* l 0.3333333333333333)) 2.0)))))
       (+
        U
        (*
         (*
          J
          (*
           l
           (+
            2.0
            (*
             (* l l)
             (+ 0.3333333333333333 (* l (* l 0.016666666666666666)))))))
         (+ 1.0 (* (* K K) -0.125))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 + ((l * l) * 0.016666666666666666);
	double t_1 = l * t_0;
	double t_2 = l * t_1;
	double tmp;
	if (l <= -220000000.0) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * t_0)) * (l / U))));
	} else if (l <= 6.2e+31) {
		tmp = U + (((l * J) * (8.0 + (t_2 * (t_1 * ((l * l) * t_1))))) / (4.0 + (t_2 * ((l * (l * 0.3333333333333333)) - 2.0))));
	} else {
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))) * (1.0 + ((K * K) * -0.125)));
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)
    t_1 = l * t_0
    t_2 = l * t_1
    if (l <= (-220000000.0d0)) then
        tmp = u * (1.0d0 + (j * ((2.0d0 + ((l * l) * t_0)) * (l / u))))
    else if (l <= 6.2d+31) then
        tmp = u + (((l * j) * (8.0d0 + (t_2 * (t_1 * ((l * l) * t_1))))) / (4.0d0 + (t_2 * ((l * (l * 0.3333333333333333d0)) - 2.0d0))))
    else
        tmp = u + ((j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * 0.016666666666666666d0))))))) * (1.0d0 + ((k * k) * (-0.125d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 + ((l * l) * 0.016666666666666666);
	double t_1 = l * t_0;
	double t_2 = l * t_1;
	double tmp;
	if (l <= -220000000.0) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * t_0)) * (l / U))));
	} else if (l <= 6.2e+31) {
		tmp = U + (((l * J) * (8.0 + (t_2 * (t_1 * ((l * l) * t_1))))) / (4.0 + (t_2 * ((l * (l * 0.3333333333333333)) - 2.0))));
	} else {
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))) * (1.0 + ((K * K) * -0.125)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 + ((l * l) * 0.016666666666666666)
	t_1 = l * t_0
	t_2 = l * t_1
	tmp = 0
	if l <= -220000000.0:
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * t_0)) * (l / U))))
	elif l <= 6.2e+31:
		tmp = U + (((l * J) * (8.0 + (t_2 * (t_1 * ((l * l) * t_1))))) / (4.0 + (t_2 * ((l * (l * 0.3333333333333333)) - 2.0))))
	else:
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))) * (1.0 + ((K * K) * -0.125)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))
	t_1 = Float64(l * t_0)
	t_2 = Float64(l * t_1)
	tmp = 0.0
	if (l <= -220000000.0)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(2.0 + Float64(Float64(l * l) * t_0)) * Float64(l / U)))));
	elseif (l <= 6.2e+31)
		tmp = Float64(U + Float64(Float64(Float64(l * J) * Float64(8.0 + Float64(t_2 * Float64(t_1 * Float64(Float64(l * l) * t_1))))) / Float64(4.0 + Float64(t_2 * Float64(Float64(l * Float64(l * 0.3333333333333333)) - 2.0)))));
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * 0.016666666666666666))))))) * Float64(1.0 + Float64(Float64(K * K) * -0.125))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 + ((l * l) * 0.016666666666666666);
	t_1 = l * t_0;
	t_2 = l * t_1;
	tmp = 0.0;
	if (l <= -220000000.0)
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * t_0)) * (l / U))));
	elseif (l <= 6.2e+31)
		tmp = U + (((l * J) * (8.0 + (t_2 * (t_1 * ((l * l) * t_1))))) / (4.0 + (t_2 * ((l * (l * 0.3333333333333333)) - 2.0))));
	else
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))) * (1.0 + ((K * K) * -0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(l * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(l * t$95$1), $MachinePrecision]}, If[LessEqual[l, -220000000.0], N[(U * N[(1.0 + N[(J * N[(N[(2.0 + N[(N[(l * l), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e+31], N[(U + N[(N[(N[(l * J), $MachinePrecision] * N[(8.0 + N[(t$95$2 * N[(t$95$1 * N[(N[(l * l), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(4.0 + N[(t$95$2 * N[(N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.2e8

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 65.3%

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

   8. Simplified 65.3%

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

   9. Taylor expanded in U around inf

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

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

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

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

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

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

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

   11. Simplified 73.7%

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

   if -2.2e8 < l < 6.2000000000000004e31

   1. Initial program 78.5%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 76.6%

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

   5. Simplified 76.6%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 83.0%

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

   8. Simplified 83.0%

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

      1. associate-*r* N/A

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

      2. flip3-+ N/A

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

      3. associate-*r/ N/A

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

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

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

   10. Applied egg-rr 84.4%

       \[\leadsto \color{blue}{\frac{\left(J \cdot \ell\right) \cdot \left(8 + \left(\ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)}{4 + \left(\ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) - 2\right)}} + U \]

   11. Taylor expanded in l around 0

       \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot \ell\right)}\right), 2\right)\right)\right)\right), U\right) \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{3}\right)\right), 2\right)\right)\right)\right), U\right) \]

       2. *-lowering-*.f64 84.5%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(J, \ell\right), \mathsf{+.f64}\left(8, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), \mathsf{+.f64}\left(4, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{3}\right)\right), 2\right)\right)\right)\right), U\right) \]

   13. Simplified 84.5%

       \[\leadsto \frac{\left(J \cdot \ell\right) \cdot \left(8 + \left(\ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot \left(\left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)}{4 + \left(\ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot \left(\ell \cdot \color{blue}{\left(\ell \cdot 0.3333333333333333\right)} - 2\right)} + U \]

   if 6.2000000000000004e31 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 96.9%

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

   5. Simplified 96.9%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 80.0%

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

   8. Simplified 80.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -220000000:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+31}:\\ \;\;\;\;U + \frac{\left(\ell \cdot J\right) \cdot \left(8 + \left(\ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot \left(\left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\right)}{4 + \left(\ell \cdot \left(\ell \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot \left(\ell \cdot \left(\ell \cdot 0.3333333333333333\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \left(1 + \left(K \cdot K\right) \cdot -0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.3% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1550:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \left(1 + \left(K \cdot K\right) \cdot -0.125\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1550.0)
   (*
    U
    (+
     1.0
     (*
      J
      (*
       (+
        2.0
        (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))
       (/ l U)))))
   (+
    U
    (*
     (*
      J
      (*
       l
       (+
        2.0
        (* (* l l) (+ 0.3333333333333333 (* l (* l 0.016666666666666666)))))))
     (+ 1.0 (* (* K K) -0.125))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1550.0) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	} else {
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))) * (1.0 + ((K * K) * -0.125)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 1550.0d0) then
        tmp = u * (1.0d0 + (j * ((2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))) * (l / u))))
    else
        tmp = u + ((j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * 0.016666666666666666d0))))))) * (1.0d0 + ((k * k) * (-0.125d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1550.0) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	} else {
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))) * (1.0 + ((K * K) * -0.125)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1550.0:
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))))
	else:
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))) * (1.0 + ((K * K) * -0.125)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1550.0)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))) * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * 0.016666666666666666))))))) * Float64(1.0 + Float64(Float64(K * K) * -0.125))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1550.0)
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	else
		tmp = U + ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666))))))) * (1.0 + ((K * K) * -0.125)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1550.0], N[(U * N[(1.0 + N[(J * N[(N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[(K * K), $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1550

   1. Initial program 84.3%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 80.2%

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

   8. Simplified 80.2%

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

   9. Taylor expanded in U around inf

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

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

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

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

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

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

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

   11. Simplified 81.1%

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

   if 1550 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

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

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

      11. *-lowering-*.f64 88.3%

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

   5. Simplified 88.3%

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

   6. Taylor expanded in K around 0

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

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

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

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

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 74.4%

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

   8. Simplified 74.4%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \color{blue}{\left(1 + -0.125 \cdot \left(K \cdot K\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1550:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\right) \cdot \left(1 + \left(K \cdot K\right) \cdot -0.125\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.9% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -95000000:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -95000000.0)
   (*
    U
    (+
     1.0
     (*
      J
      (*
       (+
        2.0
        (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))
       (/ l U)))))
   (+
    U
    (*
     J
     (*
      l
      (+
       2.0
       (*
        (* l l)
        (+
         0.3333333333333333
         (*
          l
          (*
           l
           (+ 0.016666666666666666 (* (* l l) 0.0003968253968253968))))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -95000000.0) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	} else {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-95000000.0d0)) then
        tmp = u * (1.0d0 + (j * ((2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))) * (l / u))))
    else
        tmp = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -95000000.0) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	} else {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -95000000.0:
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))))
	else:
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -95000000.0)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))) * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -95000000.0)
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	else
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -95000000.0], N[(U * N[(1.0 + N[(J * N[(N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / U), $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 l < -9.5e7

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 73.7%

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

   5. Simplified 73.7%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. *-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

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

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

      12. *-commutative N/A

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

      13. *-lowering-*.f64 65.3%

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

   8. Simplified 65.3%

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

   9. Taylor expanded in U around inf

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

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

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

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

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

       3. associate-/l* N/A

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

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

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

       5. *-commutative N/A

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

       6. associate-/l* N/A

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

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

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

   11. Simplified 73.7%

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

   if -9.5e7 < l

   1. Initial program 85.0%

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

   3. Taylor expanded in K around 0

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

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

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

      2. sub-neg N/A

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

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

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

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

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

      5. exp-neg N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

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

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

      9. exp-lowering-exp.f64 76.1%

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

   5. Simplified 76.1%

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

   6. Taylor expanded in l around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

      7. unpow2 N/A

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

      8. associate-*l* N/A

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

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

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

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

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

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

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

      12. *-commutative N/A

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

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

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

      14. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

      15. *-lowering-*.f64 80.6%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{60}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{2520}\right)\right)\right)\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 80.6%

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -95000000:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 76.1% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -200000000:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -200000000.0)
   (*
    U
    (+
     1.0
     (*
      J
      (*
       (+
        2.0
        (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))
       (/ l U)))))
   (+
    U
    (*
     J
     (*
      l
      (+
       2.0
       (*
        (* l l)
        (+ 0.3333333333333333 (* l (* l 0.016666666666666666))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -200000000.0) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	} else {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666)))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-200000000.0d0)) then
        tmp = u * (1.0d0 + (j * ((2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))) * (l / u))))
    else
        tmp = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * 0.016666666666666666d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -200000000.0) {
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	} else {
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666)))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -200000000.0:
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))))
	else:
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666)))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -200000000.0)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)))) * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * 0.016666666666666666))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -200000000.0)
		tmp = U * (1.0 + (J * ((2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))) * (l / U))));
	else
		tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -200000000.0], N[(U * N[(1.0 + N[(J * N[(N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / U), $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 * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2e8

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 73.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 73.7%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      13. *-lowering-*.f64 65.3%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 65.3%

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)} + U \]

   9. Taylor expanded in U around inf

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}{U}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \color{blue}{\left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}{U}\right)}\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}{U}\right)}\right)\right) \]

       3. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(J \cdot \color{blue}{\frac{\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}{U}}\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \color{blue}{\left(\frac{\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}{U}\right)}\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \left(\frac{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \ell}{U}\right)\right)\right)\right) \]

       6. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \cdot \color{blue}{\frac{\ell}{U}}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right), \color{blue}{\left(\frac{\ell}{U}\right)}\right)\right)\right)\right) \]

   11. Simplified 73.7%

       \[\leadsto \color{blue}{U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)} \]

   if -2e8 < l

   1. Initial program 85.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 76.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 76.1%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      13. *-lowering-*.f64 80.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 80.1%

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -200000000:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right) \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 73.7% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{+22}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1.7e+22)
   (+ U (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333)))))
   (*
    J
    (*
     l
     (+
      2.0
      (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.7e+22) {
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	} else {
		tmp = J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 1.7d+22) then
        tmp = u + (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0))))
    else
        tmp = j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.7e+22) {
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	} else {
		tmp = J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1.7e+22:
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))))
	else:
		tmp = J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1.7e+22)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333)))));
	else
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1.7e+22)
		tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
	else
		tmp = J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1.7e+22], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.7e22

   1. Initial program 84.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 75.5%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 75.5%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 77.2%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 77.2%

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} + U \]

   if 1.7e22 < 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 K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 75.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 75.8%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      13. *-lowering-*.f64 71.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 71.1%

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\ell \cdot J\right) \cdot \left(\color{blue}{2} + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 66.7%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 66.7%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \ell \cdot \left(\left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right) \cdot \color{blue}{J}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right) \cdot \color{blue}{J} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right), \color{blue}{J}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(2 + \left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right), J\right) \]

       5. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \left(2 + \ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right), J\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\ell \cdot \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right), J\right) \]

       7. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right), J\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right), J\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right), J\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right), J\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right), J\right) \]

       12. *-lowering-*.f64 71.2%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right), J\right) \]

   13. Applied egg-rr 71.2%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right) \cdot J} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.7 \cdot 10^{+22}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.6% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{if}\;\ell \leq -2800000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+20}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J (+ 2.0 (* l (* l 0.3333333333333333)))))))
   (if (<= l -2800000.0) t_0 (if (<= l 2.3e+20) (+ U (* J (* 2.0 l))) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * (2.0 + (l * (l * 0.3333333333333333))));
	double tmp;
	if (l <= -2800000.0) {
		tmp = t_0;
	} else if (l <= 2.3e+20) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = l * (j * (2.0d0 + (l * (l * 0.3333333333333333d0))))
    if (l <= (-2800000.0d0)) then
        tmp = t_0
    else if (l <= 2.3d+20) then
        tmp = u + (j * (2.0d0 * l))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = l * (J * (2.0 + (l * (l * 0.3333333333333333))));
	double tmp;
	if (l <= -2800000.0) {
		tmp = t_0;
	} else if (l <= 2.3e+20) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * (2.0 + (l * (l * 0.3333333333333333))))
	tmp = 0
	if l <= -2800000.0:
		tmp = t_0
	elif l <= 2.3e+20:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * Float64(2.0 + Float64(l * Float64(l * 0.3333333333333333)))))
	tmp = 0.0
	if (l <= -2800000.0)
		tmp = t_0;
	elseif (l <= 2.3e+20)
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (J * (2.0 + (l * (l * 0.3333333333333333))));
	tmp = 0.0;
	if (l <= -2800000.0)
		tmp = t_0;
	elseif (l <= 2.3e+20)
		tmp = U + (J * (2.0 * l));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * N[(2.0 + N[(l * N[(l * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2800000.0], t$95$0, If[LessEqual[l, 2.3e+20], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -2.8e6 or 2.3e20 < 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 K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 75.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 75.0%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), U\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

      9. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

      13. *-lowering-*.f64 67.8%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

   8. Simplified 67.8%

      \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \left(\ell \cdot J\right) \cdot \left(\color{blue}{2} + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \ell \cdot \color{blue}{\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \color{blue}{\left(J \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \color{blue}{\left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \color{blue}{\left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)}\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\color{blue}{\frac{1}{3}} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right) \]

       10. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \color{blue}{\left(\frac{1}{60} \cdot {\ell}^{2}\right)}\right)\right)\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \color{blue}{\frac{1}{60}}\right)\right)\right)\right)\right)\right) \]

       13. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right) \]

       14. *-lowering-*.f64 65.5%

           \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{60}\right)\right)\right)\right)\right)\right) \]

   11. Simplified 65.5%

       \[\leadsto \color{blue}{\ell \cdot \left(J \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)} \]

   12. Taylor expanded in l around 0

       \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \color{blue}{\left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \color{blue}{\left(\frac{1}{3} \cdot {\ell}^{2}\right)}\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{3}\right)\right)\right)\right) \]

       4. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \color{blue}{\left(\ell \cdot \frac{1}{3}\right)}\right)\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \left(\ell \cdot \left(\frac{1}{3} \cdot \color{blue}{\ell}\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{3} \cdot \ell\right)}\right)\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f64 54.4%

          \[\leadsto \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{3}}\right)\right)\right)\right)\right) \]

   14. Simplified 54.4%

       \[\leadsto \ell \cdot \left(J \cdot \color{blue}{\left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)}\right) \]

   if -2.8e6 < l < 2.3e20

   1. Initial program 78.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 76.1%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 76.1%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 84.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 84.8%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2800000:\\ \;\;\;\;\ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+20}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot \left(2 + \ell \cdot \left(\ell \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 75.6% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   J
   (*
    l
    (+
     2.0
     (* (* l l) (+ 0.3333333333333333 (* l (* l 0.016666666666666666)))))))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666)))))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + (l * (l * 0.016666666666666666d0)))))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666)))))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666)))))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(l * Float64(l * 0.016666666666666666))))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + (l * (l * 0.016666666666666666)))))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(l * N[(l * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

   9. exp-lowering-exp.f64 75.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

5. Simplified 75.6%

   \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)}\right), U\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + {\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left({\ell}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left({\ell}^{2}\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\frac{1}{3} + \frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\frac{1}{60} \cdot {\ell}^{2}\right)\right)\right)\right)\right)\right), U\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left({\ell}^{2} \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\left(\ell \cdot \ell\right) \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right), U\right) \]

   9. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \left(\ell \cdot \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\frac{1}{60} \cdot \ell\right)\right)\right)\right)\right)\right)\right), U\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \left(\ell \cdot \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

   13. *-lowering-*.f64 76.8%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{+.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \mathsf{*.f64}\left(\ell, \frac{1}{60}\right)\right)\right)\right)\right)\right)\right), U\right) \]

8. Simplified 76.8%

   \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right)} + U \]

9. Final simplification 76.8%

   \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \ell \cdot \left(\ell \cdot 0.016666666666666666\right)\right)\right)\right) \]
10. Add Preprocessing

Alternative 19: 46.6% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -3950000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-24}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 l))))
   (if (<= l -3950000.0) t_0 (if (<= l 1.7e-24) U t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double tmp;
	if (l <= -3950000.0) {
		tmp = t_0;
	} else if (l <= 1.7e-24) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (2.0d0 * l)
    if (l <= (-3950000.0d0)) then
        tmp = t_0
    else if (l <= 1.7d-24) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double tmp;
	if (l <= -3950000.0) {
		tmp = t_0;
	} else if (l <= 1.7e-24) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 * l)
	tmp = 0
	if l <= -3950000.0:
		tmp = t_0
	elif l <= 1.7e-24:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(2.0 * l))
	tmp = 0.0
	if (l <= -3950000.0)
		tmp = t_0;
	elseif (l <= 1.7e-24)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (2.0 * l);
	tmp = 0.0;
	if (l <= -3950000.0)
		tmp = t_0;
	elseif (l <= 1.7e-24)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3950000.0], t$95$0, If[LessEqual[l, 1.7e-24], U, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if l < -3.95e6 or 1.69999999999999996e-24 < l

   1. Initial program 97.8%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 73.3%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 73.3%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 21.6%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 21.6%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \ell\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(J \cdot \ell\right) \cdot \color{blue}{2} \]

       2. associate-*r* N/A

          \[\leadsto J \cdot \color{blue}{\left(\ell \cdot 2\right)} \]

       3. *-commutative N/A

          \[\leadsto J \cdot \left(2 \cdot \color{blue}{\ell}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(2 \cdot \ell\right)}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\ell \cdot \color{blue}{2}\right)\right) \]

       6. *-lowering-*.f64 21.1%

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \color{blue}{2}\right)\right) \]

   11. Simplified 21.1%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

   if -3.95e6 < l < 1.69999999999999996e-24

   1. Initial program 79.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in J around 0

      \[\leadsto \color{blue}{U} \]
   4. Step-by-step derivation

   5. Simplified 76.9%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3950000:\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;\ell \leq 1.7 \cdot 10^{-24}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 55.8% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-25}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 5e-25) (+ U (* J (* 2.0 l))) (* J (+ (* 2.0 l) (/ U J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 5e-25) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = J * ((2.0 * l) + (U / 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 (l <= 5d-25) then
        tmp = u + (j * (2.0d0 * l))
    else
        tmp = j * ((2.0d0 * l) + (u / j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 5e-25) {
		tmp = U + (J * (2.0 * l));
	} else {
		tmp = J * ((2.0 * l) + (U / J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 5e-25:
		tmp = U + (J * (2.0 * l))
	else:
		tmp = J * ((2.0 * l) + (U / J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 5e-25)
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	else
		tmp = Float64(J * Float64(Float64(2.0 * l) + Float64(U / J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 5e-25)
		tmp = U + (J * (2.0 * l));
	else
		tmp = J * ((2.0 * l) + (U / J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 5e-25], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(2.0 * l), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.99999999999999962e-25

   1. Initial program 85.5%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 76.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 76.7%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 67.8%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 67.8%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   if 4.99999999999999962e-25 < l

   1. Initial program 95.9%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in K around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      4. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

      5. exp-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

      6. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

      9. exp-lowering-exp.f64 72.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

   5. Simplified 72.7%

      \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

   6. Taylor expanded in l around 0

      \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

      6. *-lowering-*.f64 20.7%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

   8. Simplified 20.7%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

   9. Taylor expanded in J around inf

      \[\leadsto \color{blue}{J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \color{blue}{\left(2 \cdot \ell + \frac{U}{J}\right)}\right) \]

       2. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \left(\frac{U}{J} + \color{blue}{2 \cdot \ell}\right)\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(\frac{U}{J}\right), \color{blue}{\left(2 \cdot \ell\right)}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\color{blue}{2} \cdot \ell\right)\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \left(\ell \cdot \color{blue}{2}\right)\right)\right) \]

       6. *-lowering-*.f64 27.5%

          \[\leadsto \mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{/.f64}\left(U, J\right), \mathsf{*.f64}\left(\ell, \color{blue}{2}\right)\right)\right) \]

   11. Simplified 27.5%

       \[\leadsto \color{blue}{J \cdot \left(\frac{U}{J} + \ell \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{-25}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(2 \cdot \ell + \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 71.8% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333))))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * (2.0 + ((l * l) * 0.3333333333333333))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

   9. exp-lowering-exp.f64 75.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

5. Simplified 75.6%

   \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \color{blue}{\left(\ell \cdot \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)}\right), U\right) \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \left(2 + \frac{1}{3} \cdot {\ell}^{2}\right)\right)\right), U\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \left(\frac{1}{3} \cdot {\ell}^{2}\right)\right)\right)\right), U\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left({\ell}^{2}\right)\right)\right)\right)\right), U\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \left(\ell \cdot \ell\right)\right)\right)\right)\right), U\right) \]

   5. *-lowering-*.f64 73.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(2, \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right)\right), U\right) \]

8. Simplified 73.1%

   \[\leadsto J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)} + U \]

9. Final simplification 73.1%

   \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right) \]
10. Add Preprocessing

Alternative 22: 58.0% accurate, 28.4× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + \frac{J \cdot \left(2 \cdot \ell\right)}{U}\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* U (+ 1.0 (/ (* J (* 2.0 l)) U))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + ((J * (2.0 * l)) / U));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u * (1.0d0 + ((j * (2.0d0 * l)) / u))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + ((J * (2.0 * l)) / U));
}

Python

def code(J, l, K, U):
	return U * (1.0 + ((J * (2.0 * l)) / U))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(Float64(J * Float64(2.0 * l)) / U)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + ((J * (2.0 * l)) / U));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

   9. exp-lowering-exp.f64 75.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

5. Simplified 75.6%

   \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

   6. *-lowering-*.f64 55.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

8. Simplified 55.1%

   \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

9. Taylor expanded in U around inf

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \ell}{U}\right)} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(U, \color{blue}{\left(1 + 2 \cdot \frac{J \cdot \ell}{U}\right)}\right) \]

    2. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \color{blue}{\left(2 \cdot \frac{J \cdot \ell}{U}\right)}\right)\right) \]

    3. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{2 \cdot \left(J \cdot \ell\right)}{\color{blue}{U}}\right)\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \left(\frac{\left(2 \cdot J\right) \cdot \ell}{U}\right)\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(2 \cdot J\right) \cdot \ell\right), \color{blue}{U}\right)\right)\right) \]

    6. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left(J \cdot 2\right) \cdot \ell\right), U\right)\right)\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right)\right)\right) \]

    10. *-lowering-*.f64 57.9%

        \[\leadsto \mathsf{*.f64}\left(U, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right)\right)\right) \]

11. Simplified 57.9%

    \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot 2\right)}{U}\right)} \]

12. Final simplification 57.9%

    \[\leadsto U \cdot \left(1 + \frac{J \cdot \left(2 \cdot \ell\right)}{U}\right) \]
13. Add Preprocessing

Alternative 23: 54.9% 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 88.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in K around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(J \cdot \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right)}, U\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} - e^{\mathsf{neg}\left(\ell\right)}\right)\right), U\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(e^{\ell} + \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\left(e^{\ell}\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   4. exp-lowering-exp.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(e^{\mathsf{neg}\left(\ell\right)}\right)\right)\right)\right), U\right) \]

   5. exp-neg N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\mathsf{neg}\left(\frac{1}{e^{\ell}}\right)\right)\right)\right), U\right) \]

   6. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{\mathsf{neg}\left(1\right)}{e^{\ell}}\right)\right)\right), U\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \left(\frac{-1}{e^{\ell}}\right)\right)\right), U\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \left(e^{\ell}\right)\right)\right)\right), U\right) \]

   9. exp-lowering-exp.f64 75.6%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{+.f64}\left(\mathsf{exp.f64}\left(\ell\right), \mathsf{/.f64}\left(-1, \mathsf{exp.f64}\left(\ell\right)\right)\right)\right), U\right) \]

5. Simplified 75.6%

   \[\leadsto \color{blue}{J \cdot \left(e^{\ell} + \frac{-1}{e^{\ell}}\right)} + U \]

6. Taylor expanded in l around 0

   \[\leadsto \mathsf{+.f64}\left(\color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)}, U\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\left(J \cdot \ell\right) \cdot 2\right), U\right) \]

   2. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(\ell \cdot 2\right)\right), U\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\left(J \cdot \left(2 \cdot \ell\right)\right), U\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(2 \cdot \ell\right)\right), U\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \left(\ell \cdot 2\right)\right), U\right) \]

   6. *-lowering-*.f64 55.1%

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(J, \mathsf{*.f64}\left(\ell, 2\right)\right), U\right) \]

8. Simplified 55.1%

   \[\leadsto \color{blue}{J \cdot \left(\ell \cdot 2\right)} + U \]

9. Final simplification 55.1%

   \[\leadsto U + J \cdot \left(2 \cdot \ell\right) \]
10. Add Preprocessing

Alternative 24: 38.0% 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 88.3%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in J around 0

   \[\leadsto \color{blue}{U} \]
4. Step-by-step derivation

5. Simplified 39.8%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))