Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.8%

Time: 10.6s

Alternatives: 16

Speedup: 2.7×

• 50.2% 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: 86.4% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -0.1 \lor \neg \left(t\_1 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -0.1) (not (<= t_1 2e-12)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* l (* J 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -0.1) || !(t_1 <= 2e-12)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * (J * 2.0)));
	}
	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 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-0.1d0)) .or. (.not. (t_1 <= 2d-12))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    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 t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -0.1) || !(t_1 <= 2e-12)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -0.1) or not (t_1 <= 2e-12):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -0.1) || !(t_1 <= 2e-12))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -0.1) || ~((t_1 <= 2e-12)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (l * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.1], N[Not[LessEqual[t$95$1, 2e-12]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.10000000000000001 or 1.99999999999999996e-12 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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

   if -0.10000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999996e-12

   1. Initial program 81.6%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.1 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.67:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.055:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.67)
     (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U)))))
     (if (<= t_0 -0.055)
       (+ U (* (* l (pow K 2.0)) (* J -0.25)))
       (if (<= t_0 0.5)
         (+ U (* t_0 (* l (* J 2.0))))
         (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.67) {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	} else if (t_0 <= -0.055) {
		tmp = U + ((l * pow(K, 2.0)) * (J * -0.25));
	} else if (t_0 <= 0.5) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.67d0)) then
        tmp = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))))
    else if (t_0 <= (-0.055d0)) then
        tmp = u + ((l * (k ** 2.0d0)) * (j * (-0.25d0)))
    else if (t_0 <= 0.5d0) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.67) {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
	} else if (t_0 <= -0.055) {
		tmp = U + ((l * Math.pow(K, 2.0)) * (J * -0.25));
	} else if (t_0 <= 0.5) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.67:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))
	elif t_0 <= -0.055:
		tmp = U + ((l * math.pow(K, 2.0)) * (J * -0.25))
	elif t_0 <= 0.5:
		tmp = U + (t_0 * (l * (J * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.67)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))));
	elseif (t_0 <= -0.055)
		tmp = Float64(U + Float64(Float64(l * (K ^ 2.0)) * Float64(J * -0.25)));
	elseif (t_0 <= 0.5)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.67)
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	elseif (t_0 <= -0.055)
		tmp = U + ((l * (K ^ 2.0)) * (J * -0.25));
	elseif (t_0 <= 0.5)
		tmp = U + (t_0 * (l * (J * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.67], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.055], N[(U + N[(N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision] * N[(J * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.67000000000000004

   1. Initial program 90.5%

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

   3. Taylor expanded in l around 0 63.9%

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

      1. associate-*r* 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in U around inf 85.1%

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

      1. associate-/l* 88.2%

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

   8. Simplified 88.2%

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

   if -0.67000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0550000000000000003

   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 48.3%

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

      1. associate-*r* 48.3%

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

   5. Simplified 48.3%

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

   6. Taylor expanded in K around 0 74.0%

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

   7. Taylor expanded in K around inf 77.6%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   8. Step-by-step derivation

      1. associate-*r* 77.6%

         \[\leadsto \color{blue}{\left(-0.25 \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)} + U \]

      2. *-commutative 77.6%

         \[\leadsto \color{blue}{\left({K}^{2} \cdot \ell\right) \cdot \left(-0.25 \cdot J\right)} + U \]

      3. *-commutative 77.6%

         \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot \left(-0.25 \cdot J\right) + U \]

      4. *-commutative 77.6%

         \[\leadsto \left(\ell \cdot {K}^{2}\right) \cdot \color{blue}{\left(J \cdot -0.25\right)} + U \]

   9. Simplified 77.6%

      \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)} + U \]

   if -0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.5

   1. Initial program 69.2%

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

   3. Taylor expanded in l around 0 95.7%

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

      1. associate-*r* 95.7%

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

   5. Simplified 95.7%

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

   if 0.5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 92.8%

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

   3. Taylor expanded in l around 0 90.3%

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

      1. unpow2 90.3%

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

   5. Applied egg-rr 90.3%

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

   6. Taylor expanded in K around 0 89.2%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \color{blue}{1} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.67:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.055:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t\_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{if}\;t\_0 \leq -0.935:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.055:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (+ U (* t_0 (* l (* J 2.0))))))
   (if (<= t_0 -0.935)
     t_1
     (if (<= t_0 -0.055)
       (+ U (* (* l (pow K 2.0)) (* J -0.25)))
       (if (<= t_0 0.5)
         t_1
         (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (l * (J * 2.0)));
	double tmp;
	if (t_0 <= -0.935) {
		tmp = t_1;
	} else if (t_0 <= -0.055) {
		tmp = U + ((l * pow(K, 2.0)) * (J * -0.25));
	} else if (t_0 <= 0.5) {
		tmp = t_1;
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * (l * (j * 2.0d0)))
    if (t_0 <= (-0.935d0)) then
        tmp = t_1
    else if (t_0 <= (-0.055d0)) then
        tmp = u + ((l * (k ** 2.0d0)) * (j * (-0.25d0)))
    else if (t_0 <= 0.5d0) then
        tmp = t_1
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (t_0 * (l * (J * 2.0)));
	double tmp;
	if (t_0 <= -0.935) {
		tmp = t_1;
	} else if (t_0 <= -0.055) {
		tmp = U + ((l * Math.pow(K, 2.0)) * (J * -0.25));
	} else if (t_0 <= 0.5) {
		tmp = t_1;
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (l * (J * 2.0)))
	tmp = 0
	if t_0 <= -0.935:
		tmp = t_1
	elif t_0 <= -0.055:
		tmp = U + ((l * math.pow(K, 2.0)) * (J * -0.25))
	elif t_0 <= 0.5:
		tmp = t_1
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))))
	tmp = 0.0
	if (t_0 <= -0.935)
		tmp = t_1;
	elseif (t_0 <= -0.055)
		tmp = Float64(U + Float64(Float64(l * (K ^ 2.0)) * Float64(J * -0.25)));
	elseif (t_0 <= 0.5)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * (l * (J * 2.0)));
	tmp = 0.0;
	if (t_0 <= -0.935)
		tmp = t_1;
	elseif (t_0 <= -0.055)
		tmp = U + ((l * (K ^ 2.0)) * (J * -0.25));
	elseif (t_0 <= 0.5)
		tmp = t_1;
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.935], t$95$1, If[LessEqual[t$95$0, -0.055], N[(U + N[(N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision] * N[(J * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], t$95$1, N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.93500000000000005 or -0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.5

   1. Initial program 73.4%

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

   3. Taylor expanded in l around 0 90.8%

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

      1. associate-*r* 90.8%

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

   5. Simplified 90.8%

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

   if -0.93500000000000005 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0550000000000000003

   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 47.4%

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

      1. associate-*r* 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in K around 0 63.3%

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

   7. Taylor expanded in K around inf 67.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   8. Step-by-step derivation

      1. associate-*r* 67.9%

         \[\leadsto \color{blue}{\left(-0.25 \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)} + U \]

      2. *-commutative 67.9%

         \[\leadsto \color{blue}{\left({K}^{2} \cdot \ell\right) \cdot \left(-0.25 \cdot J\right)} + U \]

      3. *-commutative 67.9%

         \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot \left(-0.25 \cdot J\right) + U \]

      4. *-commutative 67.9%

         \[\leadsto \left(\ell \cdot {K}^{2}\right) \cdot \color{blue}{\left(J \cdot -0.25\right)} + U \]

   9. Simplified 67.9%

      \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)} + U \]

   if 0.5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 92.8%

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

   3. Taylor expanded in l around 0 90.3%

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

      1. unpow2 90.3%

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

   5. Applied egg-rr 90.3%

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

   6. Taylor expanded in K around 0 89.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.935:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.055:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.935:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.055:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.935)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (if (<= t_0 -0.055)
       (+ U (* (* l (pow K 2.0)) (* J -0.25)))
       (if (<= t_0 0.5)
         (+ U (* l (* (* J 2.0) (cos (* K -0.5)))))
         (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.935) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (t_0 <= -0.055) {
		tmp = U + ((l * pow(K, 2.0)) * (J * -0.25));
	} else if (t_0 <= 0.5) {
		tmp = U + (l * ((J * 2.0) * cos((K * -0.5))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.935d0)) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (t_0 <= (-0.055d0)) then
        tmp = u + ((l * (k ** 2.0d0)) * (j * (-0.25d0)))
    else if (t_0 <= 0.5d0) then
        tmp = u + (l * ((j * 2.0d0) * cos((k * (-0.5d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.935) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (t_0 <= -0.055) {
		tmp = U + ((l * Math.pow(K, 2.0)) * (J * -0.25));
	} else if (t_0 <= 0.5) {
		tmp = U + (l * ((J * 2.0) * Math.cos((K * -0.5))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.935:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif t_0 <= -0.055:
		tmp = U + ((l * math.pow(K, 2.0)) * (J * -0.25))
	elif t_0 <= 0.5:
		tmp = U + (l * ((J * 2.0) * math.cos((K * -0.5))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.935)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (t_0 <= -0.055)
		tmp = Float64(U + Float64(Float64(l * (K ^ 2.0)) * Float64(J * -0.25)));
	elseif (t_0 <= 0.5)
		tmp = Float64(U + Float64(l * Float64(Float64(J * 2.0) * cos(Float64(K * -0.5)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.935)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (t_0 <= -0.055)
		tmp = U + ((l * (K ^ 2.0)) * (J * -0.25));
	elseif (t_0 <= 0.5)
		tmp = U + (l * ((J * 2.0) * cos((K * -0.5))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.935], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.055], N[(U + N[(N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision] * N[(J * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], N[(U + N[(l * N[(N[(J * 2.0), $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.93500000000000005

   1. Initial program 80.3%

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

   3. Taylor expanded in l around 0 82.9%

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

   if -0.93500000000000005 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0550000000000000003

   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 47.4%

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

      1. associate-*r* 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in K around 0 63.3%

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

   7. Taylor expanded in K around inf 67.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   8. Step-by-step derivation

      1. associate-*r* 67.9%

         \[\leadsto \color{blue}{\left(-0.25 \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)} + U \]

      2. *-commutative 67.9%

         \[\leadsto \color{blue}{\left({K}^{2} \cdot \ell\right) \cdot \left(-0.25 \cdot J\right)} + U \]

      3. *-commutative 67.9%

         \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot \left(-0.25 \cdot J\right) + U \]

      4. *-commutative 67.9%

         \[\leadsto \left(\ell \cdot {K}^{2}\right) \cdot \color{blue}{\left(J \cdot -0.25\right)} + U \]

   9. Simplified 67.9%

      \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)} + U \]

   if -0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.5

   1. Initial program 69.2%

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

   3. Taylor expanded in l around 0 95.7%

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

      1. associate-*r* 95.7%

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

   5. Simplified 95.7%

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

      1. pow1 95.7%

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

      2. *-commutative 95.7%

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

      3. associate-*l* 95.7%

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

      4. *-commutative 95.7%

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

      5. div-inv 95.7%

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

      6. metadata-eval 95.7%

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

   7. Applied egg-rr 95.7%

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

      1. unpow1 95.7%

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

      2. *-commutative 95.7%

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

      3. metadata-eval 95.7%

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

      4. distribute-rgt-neg-in 95.7%

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

      5. cos-neg 95.7%

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

      6. *-commutative 95.7%

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

   9. Simplified 95.7%

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

   if 0.5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 92.8%

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

   3. Taylor expanded in l around 0 90.3%

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

      1. unpow2 90.3%

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

   5. Applied egg-rr 90.3%

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

   6. Taylor expanded in K around 0 89.2%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right) \cdot \color{blue}{1} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.935:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.055:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;t\_0 \leq -0.935:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq -0.055:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{elif}\;t\_0 \leq 0.5:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))
   (if (<= t_0 -0.935)
     t_1
     (if (<= t_0 -0.055)
       (+ U (* (* l (pow K 2.0)) (* J -0.25)))
       (if (<= t_0 0.5)
         t_1
         (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (2.0 * (J * (l * cos((K * 0.5)))));
	double tmp;
	if (t_0 <= -0.935) {
		tmp = t_1;
	} else if (t_0 <= -0.055) {
		tmp = U + ((l * pow(K, 2.0)) * (J * -0.25));
	} else if (t_0 <= 0.5) {
		tmp = t_1;
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    if (t_0 <= (-0.935d0)) then
        tmp = t_1
    else if (t_0 <= (-0.055d0)) then
        tmp = u + ((l * (k ** 2.0d0)) * (j * (-0.25d0)))
    else if (t_0 <= 0.5d0) then
        tmp = t_1
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	double tmp;
	if (t_0 <= -0.935) {
		tmp = t_1;
	} else if (t_0 <= -0.055) {
		tmp = U + ((l * Math.pow(K, 2.0)) * (J * -0.25));
	} else if (t_0 <= 0.5) {
		tmp = t_1;
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	tmp = 0
	if t_0 <= -0.935:
		tmp = t_1
	elif t_0 <= -0.055:
		tmp = U + ((l * math.pow(K, 2.0)) * (J * -0.25))
	elif t_0 <= 0.5:
		tmp = t_1
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))))
	tmp = 0.0
	if (t_0 <= -0.935)
		tmp = t_1;
	elseif (t_0 <= -0.055)
		tmp = Float64(U + Float64(Float64(l * (K ^ 2.0)) * Float64(J * -0.25)));
	elseif (t_0 <= 0.5)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (2.0 * (J * (l * cos((K * 0.5)))));
	tmp = 0.0;
	if (t_0 <= -0.935)
		tmp = t_1;
	elseif (t_0 <= -0.055)
		tmp = U + ((l * (K ^ 2.0)) * (J * -0.25));
	elseif (t_0 <= 0.5)
		tmp = t_1;
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.935], t$95$1, If[LessEqual[t$95$0, -0.055], N[(U + N[(N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision] * N[(J * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.5], t$95$1, N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.93500000000000005 or -0.0550000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.5

   1. Initial program 73.4%

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

   3. Taylor expanded in l around 0 90.7%

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

   if -0.93500000000000005 < (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0550000000000000003

   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 47.4%

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

      1. associate-*r* 47.4%

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

   5. Simplified 47.4%

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

   6. Taylor expanded in K around 0 63.3%

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

   7. Taylor expanded in K around inf 67.9%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   8. Step-by-step derivation

      1. associate-*r* 67.9%

         \[\leadsto \color{blue}{\left(-0.25 \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)} + U \]

      2. *-commutative 67.9%

         \[\leadsto \color{blue}{\left({K}^{2} \cdot \ell\right) \cdot \left(-0.25 \cdot J\right)} + U \]

      3. *-commutative 67.9%

         \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot \left(-0.25 \cdot J\right) + U \]

      4. *-commutative 67.9%

         \[\leadsto \left(\ell \cdot {K}^{2}\right) \cdot \color{blue}{\left(J \cdot -0.25\right)} + U \]

   9. Simplified 67.9%

      \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)} + U \]

   if 0.5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 92.8%

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

   3. Taylor expanded in l around 0 90.3%

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

      1. unpow2 90.3%

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

   5. Applied egg-rr 90.3%

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

   6. Taylor expanded in K around 0 89.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.935:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.055:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-10}:\\ \;\;\;\;U + t\_1 \cdot \left(\ell \cdot \left(J \cdot 2 + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+68}:\\ \;\;\;\;\left(e^{\ell} - t\_0\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -4.0)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (<= l 2e-10)
       (+
        U
        (* t_1 (* l (+ (* J 2.0) (* 0.3333333333333333 (* J (pow l 2.0)))))))
       (if (<= l 1e+68)
         (+ (* (- (exp l) t_0) J) U)
         (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -4.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 2e-10) {
		tmp = U + (t_1 * (l * ((J * 2.0) + (0.3333333333333333 * (J * pow(l, 2.0))))));
	} else if (l <= 1e+68) {
		tmp = ((exp(l) - t_0) * J) + U;
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-4.0d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if (l <= 2d-10) then
        tmp = u + (t_1 * (l * ((j * 2.0d0) + (0.3333333333333333d0 * (j * (l ** 2.0d0))))))
    else if (l <= 1d+68) then
        tmp = ((exp(l) - t_0) * j) + u
    else
        tmp = u + (t_1 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -4.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 2e-10) {
		tmp = U + (t_1 * (l * ((J * 2.0) + (0.3333333333333333 * (J * Math.pow(l, 2.0))))));
	} else if (l <= 1e+68) {
		tmp = ((Math.exp(l) - t_0) * J) + U;
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -4.0:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif l <= 2e-10:
		tmp = U + (t_1 * (l * ((J * 2.0) + (0.3333333333333333 * (J * math.pow(l, 2.0))))))
	elif l <= 1e+68:
		tmp = ((math.exp(l) - t_0) * J) + U
	else:
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -4.0)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif (l <= 2e-10)
		tmp = Float64(U + Float64(t_1 * Float64(l * Float64(Float64(J * 2.0) + Float64(0.3333333333333333 * Float64(J * (l ^ 2.0)))))));
	elseif (l <= 1e+68)
		tmp = Float64(Float64(Float64(exp(l) - t_0) * J) + U);
	else
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -4.0)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif (l <= 2e-10)
		tmp = U + (t_1 * (l * ((J * 2.0) + (0.3333333333333333 * (J * (l ^ 2.0))))));
	elseif (l <= 1e+68)
		tmp = ((exp(l) - t_0) * J) + U;
	else
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.0], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e-10], N[(U + N[(t$95$1 * N[(l * N[(N[(J * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+68], N[(N[(N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4

   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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -4 < l < 2.00000000000000007e-10

   1. Initial program 81.8%

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

   3. Taylor expanded in l around 0 99.6%

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

   if 2.00000000000000007e-10 < l < 9.99999999999999953e67

   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 84.6%

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

   if 9.99999999999999953e67 < 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 94.9%

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

      1. unpow2 94.9%

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

   5. Applied egg-rr 94.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-10}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2 + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+68}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-10} \lor \neg \left(\ell \leq 10^{+68}\right):\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - t\_0\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -4.0)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (or (<= l 2e-10) (not (<= l 1e+68)))
       (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))
       (+ (* (- (exp l) t_0) J) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -4.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if ((l <= 2e-10) || !(l <= 1e+68)) {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = ((exp(l) - t_0) * J) + U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-4.0d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if ((l <= 2d-10) .or. (.not. (l <= 1d+68))) then
        tmp = u + (t_1 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    else
        tmp = ((exp(l) - t_0) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -4.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if ((l <= 2e-10) || !(l <= 1e+68)) {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = ((Math.exp(l) - t_0) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -4.0:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif (l <= 2e-10) or not (l <= 1e+68):
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	else:
		tmp = ((math.exp(l) - t_0) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -4.0)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif ((l <= 2e-10) || !(l <= 1e+68))
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	else
		tmp = Float64(Float64(Float64(exp(l) - t_0) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -4.0)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif ((l <= 2e-10) || ~((l <= 1e+68)))
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	else
		tmp = ((exp(l) - t_0) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.0], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 2e-10], N[Not[LessEqual[l, 1e+68]], $MachinePrecision]], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4

   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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -4 < l < 2.00000000000000007e-10 or 9.99999999999999953e67 < l

   1. Initial program 87.4%

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

   3. Taylor expanded in l around 0 98.1%

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

      1. unpow2 98.1%

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

   5. Applied egg-rr 98.1%

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

   if 2.00000000000000007e-10 < l < 9.99999999999999953e67

   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 84.6%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-10} \lor \neg \left(\ell \leq 10^{+68}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (+ U (* (* l (pow K 2.0)) (* J -0.25)))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = U + ((l * pow(K, 2.0)) * (J * -0.25));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.02d0)) then
        tmp = u + ((l * (k ** 2.0d0)) * (j * (-0.25d0)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.02) {
		tmp = U + ((l * Math.pow(K, 2.0)) * (J * -0.25));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.02:
		tmp = U + ((l * math.pow(K, 2.0)) * (J * -0.25))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = Float64(U + Float64(Float64(l * (K ^ 2.0)) * Float64(J * -0.25)));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.02)
		tmp = U + ((l * (K ^ 2.0)) * (J * -0.25));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(U + N[(N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision] * N[(J * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.7%

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

   3. Taylor expanded in l around 0 58.2%

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

      1. associate-*r* 58.2%

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

   5. Simplified 58.2%

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

   6. Taylor expanded in K around 0 59.0%

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

   7. Taylor expanded in K around inf 60.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   8. Step-by-step derivation

      1. associate-*r* 60.7%

         \[\leadsto \color{blue}{\left(-0.25 \cdot J\right) \cdot \left({K}^{2} \cdot \ell\right)} + U \]

      2. *-commutative 60.7%

         \[\leadsto \color{blue}{\left({K}^{2} \cdot \ell\right) \cdot \left(-0.25 \cdot J\right)} + U \]

      3. *-commutative 60.7%

         \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot \left(-0.25 \cdot J\right) + U \]

      4. *-commutative 60.7%

         \[\leadsto \left(\ell \cdot {K}^{2}\right) \cdot \color{blue}{\left(J \cdot -0.25\right)} + U \]

   9. Simplified 60.7%

      \[\leadsto \color{blue}{\left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)} + U \]

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

   1. Initial program 90.8%

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

   3. Taylor expanded in l around 0 91.3%

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

      1. unpow2 91.3%

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

   5. Applied egg-rr 91.3%

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

   6. Taylor expanded in K around 0 87.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(\ell \cdot {K}^{2}\right) \cdot \left(J \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-16)
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+
    U
    (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-16) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 5d-16) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-16) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-16:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-16)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-16)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-16], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 5.0000000000000004e-16

   1. Initial program 92.6%

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

   3. Taylor expanded in K around 0 86.2%

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

   if 5.0000000000000004e-16 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 87.1%

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

   3. Taylor expanded in l around 0 89.7%

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

      1. unpow2 89.7%

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

   5. Applied egg-rr 89.7%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(\ell \cdot \left(J \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.02)
   (+ U (* (* l (* J 2.0)) (+ 1.0 (* -0.125 (* K K)))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.02) {
		tmp = U + ((l * (J * 2.0)) * (1.0 + (-0.125 * (K * K))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.02d0)) then
        tmp = u + ((l * (j * 2.0d0)) * (1.0d0 + ((-0.125d0) * (k * k))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.02) {
		tmp = U + ((l * (J * 2.0)) * (1.0 + (-0.125 * (K * K))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.02:
		tmp = U + ((l * (J * 2.0)) * (1.0 + (-0.125 * (K * K))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.02)
		tmp = Float64(U + Float64(Float64(l * Float64(J * 2.0)) * Float64(1.0 + Float64(-0.125 * Float64(K * K)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.02)
		tmp = U + ((l * (J * 2.0)) * (1.0 + (-0.125 * (K * K))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.02], N[(U + N[(N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 91.7%

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

   3. Taylor expanded in l around 0 58.2%

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

      1. associate-*r* 58.2%

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

   5. Simplified 58.2%

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

   6. Taylor expanded in K around 0 59.0%

      \[\leadsto \left(\left(2 \cdot J\right) \cdot \ell\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]
   7. Step-by-step derivation

      1. unpow2 59.0%

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

   8. Applied egg-rr 59.0%

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

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

   1. Initial program 90.8%

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

   3. Taylor expanded in l around 0 91.3%

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

      1. unpow2 91.3%

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

   5. Applied egg-rr 91.3%

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

   6. Taylor expanded in K around 0 87.9%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \left(\ell \cdot \left(J \cdot 2\right)\right) \cdot \left(1 + -0.125 \cdot \left(K \cdot K\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.0% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * 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 + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
end function

Java

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

Python

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

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in l around 0 89.0%

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

   1. unpow2 89.0%

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

5. Applied egg-rr 89.0%

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

6. Final simplification 89.0%

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

Alternative 12: 42.7% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(K \cdot \left(-K\right)\right)\\ \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+155}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{+16}:\\ \;\;\;\;J \cdot \frac{U}{J}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+14}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* K (- K)))))
   (if (<= l -1.65e+155)
     t_0
     (if (<= l -5.5e+16) (* J (/ U J)) (if (<= l 8e+14) U t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (K * -K);
	double tmp;
	if (l <= -1.65e+155) {
		tmp = t_0;
	} else if (l <= -5.5e+16) {
		tmp = J * (U / J);
	} else if (l <= 8e+14) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (k * -k)
    if (l <= (-1.65d+155)) then
        tmp = t_0
    else if (l <= (-5.5d+16)) then
        tmp = j * (u / j)
    else if (l <= 8d+14) then
        tmp = u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (K * -K);
	double tmp;
	if (l <= -1.65e+155) {
		tmp = t_0;
	} else if (l <= -5.5e+16) {
		tmp = J * (U / J);
	} else if (l <= 8e+14) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (K * -K)
	tmp = 0
	if l <= -1.65e+155:
		tmp = t_0
	elif l <= -5.5e+16:
		tmp = J * (U / J)
	elif l <= 8e+14:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(K * Float64(-K)))
	tmp = 0.0
	if (l <= -1.65e+155)
		tmp = t_0;
	elseif (l <= -5.5e+16)
		tmp = Float64(J * Float64(U / J));
	elseif (l <= 8e+14)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (K * -K);
	tmp = 0.0;
	if (l <= -1.65e+155)
		tmp = t_0;
	elseif (l <= -5.5e+16)
		tmp = J * (U / J);
	elseif (l <= 8e+14)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(K * (-K)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.65e+155], t$95$0, If[LessEqual[l, -5.5e+16], N[(J * N[(U / J), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8e+14], U, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.6499999999999999e155 or 8e14 < 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

   4. Applied egg-rr 3.2%

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

   5. Taylor expanded in K around 0 20.8%

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

      1. +-commutative 20.8%

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

      2. *-commutative 20.8%

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

      3. mul-1-neg 20.8%

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

      4. unsub-neg 20.8%

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

   7. Simplified 20.8%

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

   8. Taylor expanded in K around inf 19.8%

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

      1. associate-*r* 19.8%

         \[\leadsto \color{blue}{\left(-1 \cdot J\right) \cdot {K}^{2}} \]

      2. neg-mul-1 19.8%

         \[\leadsto \color{blue}{\left(-J\right)} \cdot {K}^{2} \]

      3. *-commutative 19.8%

         \[\leadsto \color{blue}{{K}^{2} \cdot \left(-J\right)} \]

   10. Simplified 19.8%

       \[\leadsto \color{blue}{{K}^{2} \cdot \left(-J\right)} \]
   11. Step-by-step derivation

       1. unpow2 32.7%

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

   12. Applied egg-rr 19.8%

       \[\leadsto \color{blue}{\left(K \cdot K\right)} \cdot \left(-J\right) \]

   if -1.6499999999999999e155 < l < -5.5e16

   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

   4. Applied egg-rr 2.8%

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

   5. Taylor expanded in J around inf 24.2%

      \[\leadsto \color{blue}{J \cdot \left(8 \cdot \cos \left(0.5 \cdot K\right) + \frac{U}{J}\right)} \]

   6. Taylor expanded in U around inf 24.5%

      \[\leadsto J \cdot \color{blue}{\frac{U}{J}} \]

   if -5.5e16 < l < 8e14

   1. Initial program 83.4%

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

   4. Applied egg-rr 50.4%

      \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in J around 0 76.0%

      \[\leadsto \color{blue}{U} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{+155}:\\ \;\;\;\;J \cdot \left(K \cdot \left(-K\right)\right)\\ \mathbf{elif}\;\ell \leq -5.5 \cdot 10^{+16}:\\ \;\;\;\;J \cdot \frac{U}{J}\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+14}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(K \cdot \left(-K\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.2% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+20} \lor \neg \left(\ell \leq 1.28 \cdot 10^{+14}\right):\\ \;\;\;\;J \cdot \frac{U}{J}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4.2e+20) (not (<= l 1.28e+14))) (* J (/ U J)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.2e+20) || !(l <= 1.28e+14)) {
		tmp = J * (U / J);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4.2d+20)) .or. (.not. (l <= 1.28d+14))) then
        tmp = j * (u / j)
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.2e+20) || !(l <= 1.28e+14)) {
		tmp = J * (U / J);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4.2e+20) or not (l <= 1.28e+14):
		tmp = J * (U / J)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4.2e+20) || !(l <= 1.28e+14))
		tmp = Float64(J * Float64(U / J));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4.2e+20) || ~((l <= 1.28e+14)))
		tmp = J * (U / J);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4.2e+20], N[Not[LessEqual[l, 1.28e+14]], $MachinePrecision]], N[(J * N[(U / J), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -4.2e20 or 1.28e14 < 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

   4. Applied egg-rr 3.2%

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

   5. Taylor expanded in J around inf 14.1%

      \[\leadsto \color{blue}{J \cdot \left(8 \cdot \cos \left(0.5 \cdot K\right) + \frac{U}{J}\right)} \]

   6. Taylor expanded in U around inf 13.5%

      \[\leadsto J \cdot \color{blue}{\frac{U}{J}} \]

   if -4.2e20 < l < 1.28e14

   1. Initial program 83.2%

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

   4. Applied egg-rr 50.7%

      \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in J around 0 76.5%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2 \cdot 10^{+20} \lor \neg \left(\ell \leq 1.28 \cdot 10^{+14}\right):\\ \;\;\;\;J \cdot \frac{U}{J}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.4% accurate, 24.0× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in l around 0 89.0%

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

   1. unpow2 89.0%

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

5. Applied egg-rr 89.0%

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

6. Taylor expanded in K around 0 74.8%

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

7. Final simplification 74.8%

   \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right) \]
8. Add Preprocessing

Alternative 15: 52.9% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;U \leq 2.6 \cdot 10^{+244}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \frac{U}{J}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= U 2.6e+244) (+ U (* l (* J 2.0))) (* J (/ U J))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (U <= 2.6e+244) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * (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 (u <= 2.6d+244) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = j * (u / j)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (U <= 2.6e+244) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * (U / J);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if U <= 2.6e+244:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = J * (U / J)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (U <= 2.6e+244)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(J * Float64(U / J));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (U <= 2.6e+244)
		tmp = U + (l * (J * 2.0));
	else
		tmp = J * (U / J);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[U, 2.6e+244], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(U / J), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if U < 2.6e244

   1. Initial program 90.6%

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

   3. Taylor expanded in l around 0 63.7%

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

      1. associate-*r* 63.7%

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

   5. Simplified 63.7%

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

   6. Taylor expanded in K around 0 54.9%

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

      1. associate-*r* 54.9%

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

   8. Simplified 54.9%

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

   if 2.6e244 < U

   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

   4. Applied egg-rr 30.8%

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

   5. Taylor expanded in J around inf 76.7%

      \[\leadsto \color{blue}{J \cdot \left(8 \cdot \cos \left(0.5 \cdot K\right) + \frac{U}{J}\right)} \]

   6. Taylor expanded in U around inf 76.7%

      \[\leadsto J \cdot \color{blue}{\frac{U}{J}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;U \leq 2.6 \cdot 10^{+244}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \frac{U}{J}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.6% 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 91.0%

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

4. Applied egg-rr 28.6%

   \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Taylor expanded in J around 0 42.1%

   \[\leadsto \color{blue}{U} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))