Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.0% → 99.7%

Time: 11.6s

Alternatives: 18

Speedup: 1.3×

• 49.6% 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: 86.0% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0.001\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.001)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       l
       (* (* J (cos (* K 0.5))) (fma 0.3333333333333333 (pow l 2.0) 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.001)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (l * ((J * cos((K * 0.5))) * fma(0.3333333333333333, pow(l, 2.0), 2.0)));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.001))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(l * Float64(Float64(J * cos(Float64(K * 0.5))) * fma(0.3333333333333333, (l ^ 2.0), 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.001]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(l * N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 1e-3 < (-.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 -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1e-3

   1. Initial program 75.2%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

      7. *-commutative 99.9%

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

      8. distribute-lft-out 99.9%

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

      9. fma-define 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\ell \cdot \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.001\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 0.001\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

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 (- INFINITY)) (not (<= t_1 0.001)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 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 <= -((double) INFINITY)) || !(t_1 <= 0.001)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.001)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 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 <= -math.inf) or not (t_1 <= 0.001):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 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 <= Float64(-Inf)) || !(t_1 <= 0.001))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 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 <= -Inf) || ~((t_1 <= 0.001)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 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, (-Infinity)], N[Not[LessEqual[t$95$1, 0.001]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 1e-3 < (-.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 -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1e-3

   1. Initial program 75.2%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.001\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{5} \cdot \left(J \cdot \left(t\_0 \cdot 0.016666666666666666\right)\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -600:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 400000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 5.0) (* J (* t_0 0.016666666666666666))))))
   (if (<= l -1.9e+47)
     t_1
     (if (<= l -600.0)
       (+ (* (- (exp l) (exp (- l))) J) U)
       (if (<= l 400000.0)
         (+ U (* 2.0 (* J (* l t_0))))
         (if (<= l 2.5e+43) (log1p (expm1 (- (/ -4.0 U) U))) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (pow(l, 5.0) * (J * (t_0 * 0.016666666666666666)));
	double tmp;
	if (l <= -1.9e+47) {
		tmp = t_1;
	} else if (l <= -600.0) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else if (l <= 400000.0) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 2.5e+43) {
		tmp = log1p(expm1(((-4.0 / U) - U)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (Math.pow(l, 5.0) * (J * (t_0 * 0.016666666666666666)));
	double tmp;
	if (l <= -1.9e+47) {
		tmp = t_1;
	} else if (l <= -600.0) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else if (l <= 400000.0) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 2.5e+43) {
		tmp = Math.log1p(Math.expm1(((-4.0 / U) - U)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (math.pow(l, 5.0) * (J * (t_0 * 0.016666666666666666)))
	tmp = 0
	if l <= -1.9e+47:
		tmp = t_1
	elif l <= -600.0:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	elif l <= 400000.0:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif l <= 2.5e+43:
		tmp = math.log1p(math.expm1(((-4.0 / U) - U)))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64((l ^ 5.0) * Float64(J * Float64(t_0 * 0.016666666666666666))))
	tmp = 0.0
	if (l <= -1.9e+47)
		tmp = t_1;
	elseif (l <= -600.0)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	elseif (l <= 400000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif (l <= 2.5e+43)
		tmp = log1p(expm1(Float64(Float64(-4.0 / U) - U)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[Power[l, 5.0], $MachinePrecision] * N[(J * N[(t$95$0 * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9e+47], t$95$1, If[LessEqual[l, -600.0], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 400000.0], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5e+43], N[Log[1 + N[(Exp[N[(N[(-4.0 / U), $MachinePrecision] - U), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.9000000000000002e47 or 2.5000000000000002e43 < 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 98.2%

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

      1. *-commutative 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in l around inf 98.2%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto 0.016666666666666666 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{5}\right)}\right) + U \]

      2. associate-*r* 98.2%

         \[\leadsto 0.016666666666666666 \cdot \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot {\ell}^{5}\right)} + U \]

      3. associate-*l* 98.2%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot {\ell}^{5}} + U \]

      4. *-commutative 98.2%

         \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

      5. associate-*r* 98.2%

         \[\leadsto {\ell}^{5} \cdot \color{blue}{\left(\left(0.016666666666666666 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      6. *-commutative 98.2%

         \[\leadsto {\ell}^{5} \cdot \left(\color{blue}{\left(J \cdot 0.016666666666666666\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

      7. associate-*l* 98.2%

         \[\leadsto {\ell}^{5} \cdot \color{blue}{\left(J \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(J \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if -1.9000000000000002e47 < l < -600

   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 90.9%

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

   if -600 < l < 4e5

   1. Initial program 75.8%

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

   3. Taylor expanded in l around 0 98.0%

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

   if 4e5 < l < 2.5000000000000002e43

   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 6.4%

      \[\leadsto \color{blue}{\frac{-4}{U} - U} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+47}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq -600:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 400000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + {\ell}^{5} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.016666666666666666\right)\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -600:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 400000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+41}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (pow l 5.0) (* J (* (cos (* K 0.5)) 0.016666666666666666))))))
   (if (<= l -1.9e+47)
     t_0
     (if (<= l -600.0)
       (+ (* (- (exp l) (exp (- l))) J) U)
       (if (<= l 400000.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
         (if (<= l 1e+41) (log1p (expm1 (- (/ -4.0 U) U))) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (pow(l, 5.0) * (J * (cos((K * 0.5)) * 0.016666666666666666)));
	double tmp;
	if (l <= -1.9e+47) {
		tmp = t_0;
	} else if (l <= -600.0) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else if (l <= 400000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else if (l <= 1e+41) {
		tmp = log1p(expm1(((-4.0 / U) - U)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.pow(l, 5.0) * (J * (Math.cos((K * 0.5)) * 0.016666666666666666)));
	double tmp;
	if (l <= -1.9e+47) {
		tmp = t_0;
	} else if (l <= -600.0) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else if (l <= 400000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else if (l <= 1e+41) {
		tmp = Math.log1p(Math.expm1(((-4.0 / U) - U)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.pow(l, 5.0) * (J * (math.cos((K * 0.5)) * 0.016666666666666666)))
	tmp = 0
	if l <= -1.9e+47:
		tmp = t_0
	elif l <= -600.0:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	elif l <= 400000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	elif l <= 1e+41:
		tmp = math.log1p(math.expm1(((-4.0 / U) - U)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64((l ^ 5.0) * Float64(J * Float64(cos(Float64(K * 0.5)) * 0.016666666666666666))))
	tmp = 0.0
	if (l <= -1.9e+47)
		tmp = t_0;
	elseif (l <= -600.0)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	elseif (l <= 400000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	elseif (l <= 1e+41)
		tmp = log1p(expm1(Float64(Float64(-4.0 / U) - U)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Power[l, 5.0], $MachinePrecision] * N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9e+47], t$95$0, If[LessEqual[l, -600.0], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 400000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1e+41], N[Log[1 + N[(Exp[N[(N[(-4.0 / U), $MachinePrecision] - U), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.9000000000000002e47 or 1.00000000000000001e41 < 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 98.2%

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

      1. *-commutative 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in l around inf 98.2%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 98.2%

         \[\leadsto 0.016666666666666666 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{5}\right)}\right) + U \]

      2. associate-*r* 98.2%

         \[\leadsto 0.016666666666666666 \cdot \color{blue}{\left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot {\ell}^{5}\right)} + U \]

      3. associate-*l* 98.2%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot {\ell}^{5}} + U \]

      4. *-commutative 98.2%

         \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

      5. associate-*r* 98.2%

         \[\leadsto {\ell}^{5} \cdot \color{blue}{\left(\left(0.016666666666666666 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      6. *-commutative 98.2%

         \[\leadsto {\ell}^{5} \cdot \left(\color{blue}{\left(J \cdot 0.016666666666666666\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

      7. associate-*l* 98.2%

         \[\leadsto {\ell}^{5} \cdot \color{blue}{\left(J \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   8. Simplified 98.2%

      \[\leadsto \color{blue}{{\ell}^{5} \cdot \left(J \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if -1.9000000000000002e47 < l < -600

   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 90.9%

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

   if -600 < l < 4e5

   1. Initial program 75.8%

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

   3. Taylor expanded in l around 0 98.4%

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

   if 4e5 < l < 1.00000000000000001e41

   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 6.4%

      \[\leadsto \color{blue}{\frac{-4}{U} - U} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 100.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+47}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq -600:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 400000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 10^{+41}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U \cdot \left(1 + 2 \cdot \left(\ell \cdot \left(J \cdot \frac{t\_0}{U}\right)\right)\right)\\ t_2 := \mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)\\ \mathbf{if}\;\ell \leq -1.28 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -3700000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 400000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{+116}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+160}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+217}:\\ \;\;\;\;U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (* U (+ 1.0 (* 2.0 (* l (* J (/ t_0 U)))))))
        (t_2 (log1p (expm1 (- (/ -4.0 U) U)))))
   (if (<= l -1.28e+132)
     t_1
     (if (<= l -3700000000000.0)
       t_2
       (if (<= l 400000.0)
         (+ U (* 2.0 (* J (* l t_0))))
         (if (<= l 5.1e+116)
           t_2
           (if (<= l 1.9e+160)
             (pow U -3.0)
             (if (<= l 1.1e+217)
               (* U (+ 1.0 (* (* J 2.0) (/ l U))))
               t_1))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U * (1.0 + (2.0 * (l * (J * (t_0 / U)))));
	double t_2 = log1p(expm1(((-4.0 / U) - U)));
	double tmp;
	if (l <= -1.28e+132) {
		tmp = t_1;
	} else if (l <= -3700000000000.0) {
		tmp = t_2;
	} else if (l <= 400000.0) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 5.1e+116) {
		tmp = t_2;
	} else if (l <= 1.9e+160) {
		tmp = pow(U, -3.0);
	} else if (l <= 1.1e+217) {
		tmp = U * (1.0 + ((J * 2.0) * (l / U)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U * (1.0 + (2.0 * (l * (J * (t_0 / U)))));
	double t_2 = Math.log1p(Math.expm1(((-4.0 / U) - U)));
	double tmp;
	if (l <= -1.28e+132) {
		tmp = t_1;
	} else if (l <= -3700000000000.0) {
		tmp = t_2;
	} else if (l <= 400000.0) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 5.1e+116) {
		tmp = t_2;
	} else if (l <= 1.9e+160) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= 1.1e+217) {
		tmp = U * (1.0 + ((J * 2.0) * (l / U)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U * (1.0 + (2.0 * (l * (J * (t_0 / U)))))
	t_2 = math.log1p(math.expm1(((-4.0 / U) - U)))
	tmp = 0
	if l <= -1.28e+132:
		tmp = t_1
	elif l <= -3700000000000.0:
		tmp = t_2
	elif l <= 400000.0:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif l <= 5.1e+116:
		tmp = t_2
	elif l <= 1.9e+160:
		tmp = math.pow(U, -3.0)
	elif l <= 1.1e+217:
		tmp = U * (1.0 + ((J * 2.0) * (l / U)))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(l * Float64(J * Float64(t_0 / U))))))
	t_2 = log1p(expm1(Float64(Float64(-4.0 / U) - U)))
	tmp = 0.0
	if (l <= -1.28e+132)
		tmp = t_1;
	elseif (l <= -3700000000000.0)
		tmp = t_2;
	elseif (l <= 400000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif (l <= 5.1e+116)
		tmp = t_2;
	elseif (l <= 1.9e+160)
		tmp = U ^ -3.0;
	elseif (l <= 1.1e+217)
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * 2.0) * Float64(l / U))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U * N[(1.0 + N[(2.0 * N[(l * N[(J * N[(t$95$0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[1 + N[(Exp[N[(N[(-4.0 / U), $MachinePrecision] - U), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.28e+132], t$95$1, If[LessEqual[l, -3700000000000.0], t$95$2, If[LessEqual[l, 400000.0], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.1e+116], t$95$2, If[LessEqual[l, 1.9e+160], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, 1.1e+217], N[(U * N[(1.0 + N[(N[(J * 2.0), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.2800000000000001e132 or 1.1e217 < 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 55.0%

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

      1. associate-*r* 55.0%

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

   5. Simplified 55.0%

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

   6. Taylor expanded in U around inf 68.8%

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

      1. associate-*r* 68.8%

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

   8. Simplified 68.8%

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

      1. associate-/l* 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-*l* 68.8%

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

   10. Applied egg-rr 68.8%

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

   if -1.2800000000000001e132 < l < -3.7e12 or 4e5 < l < 5.09999999999999999e116

   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 3.3%

      \[\leadsto \color{blue}{\frac{-4}{U} - U} \]
   5. Step-by-step derivation

      1. log1p-expm1-u 64.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)} \]

   6. Applied egg-rr 64.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)} \]

   if -3.7e12 < l < 4e5

   1. Initial program 76.1%

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

   3. Taylor expanded in l around 0 96.6%

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

   if 5.09999999999999999e116 < l < 1.90000000000000006e160

   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 75.7%

      \[\leadsto \color{blue}{{U}^{-3}} \]

   if 1.90000000000000006e160 < l < 1.1e217

   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 24.7%

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

      1. associate-*r* 24.7%

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

   5. Simplified 24.7%

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

   6. Taylor expanded in U around inf 42.8%

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

      1. associate-*r* 42.8%

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

   8. Simplified 42.8%

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

   9. Taylor expanded in K around 0 42.8%

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

       1. associate-/l* 62.2%

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

       2. associate-*r* 62.2%

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

       3. *-commutative 62.2%

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

   11. Simplified 62.2%

       \[\leadsto U \cdot \left(1 + \color{blue}{\left(J \cdot 2\right) \cdot \frac{\ell}{U}}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.28 \cdot 10^{+132}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(\ell \cdot \left(J \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -3700000000000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)\\ \mathbf{elif}\;\ell \leq 400000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-4}{U} - U\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+160}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+217}:\\ \;\;\;\;U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(\ell \cdot \left(J \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -780:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+250}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(\ell \cdot \left(J \cdot \frac{t\_1}{U}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U)) (t_1 (cos (* K 0.5))))
   (if (<= l -780.0)
     t_0
     (if (<= l 7.5e-6)
       (+ U (* 2.0 (* J (* l t_1))))
       (if (<= l 1.3e+250) t_0 (* U (+ 1.0 (* 2.0 (* l (* J (/ t_1 U)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -780.0) {
		tmp = t_0;
	} else if (l <= 7.5e-6) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 1.3e+250) {
		tmp = t_0;
	} else {
		tmp = U * (1.0 + (2.0 * (l * (J * (t_1 / 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) - exp(-l)) * j) + u
    t_1 = cos((k * 0.5d0))
    if (l <= (-780.0d0)) then
        tmp = t_0
    else if (l <= 7.5d-6) then
        tmp = u + (2.0d0 * (j * (l * t_1)))
    else if (l <= 1.3d+250) then
        tmp = t_0
    else
        tmp = u * (1.0d0 + (2.0d0 * (l * (j * (t_1 / 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) - Math.exp(-l)) * J) + U;
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -780.0) {
		tmp = t_0;
	} else if (l <= 7.5e-6) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 1.3e+250) {
		tmp = t_0;
	} else {
		tmp = U * (1.0 + (2.0 * (l * (J * (t_1 / U)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = ((math.exp(l) - math.exp(-l)) * J) + U
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -780.0:
		tmp = t_0
	elif l <= 7.5e-6:
		tmp = U + (2.0 * (J * (l * t_1)))
	elif l <= 1.3e+250:
		tmp = t_0
	else:
		tmp = U * (1.0 + (2.0 * (l * (J * (t_1 / U)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -780.0)
		tmp = t_0;
	elseif (l <= 7.5e-6)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_1))));
	elseif (l <= 1.3e+250)
		tmp = t_0;
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(l * Float64(J * Float64(t_1 / U))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((exp(l) - exp(-l)) * J) + U;
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -780.0)
		tmp = t_0;
	elseif (l <= 7.5e-6)
		tmp = U + (2.0 * (J * (l * t_1)));
	elseif (l <= 1.3e+250)
		tmp = t_0;
	else
		tmp = U * (1.0 + (2.0 * (l * (J * (t_1 / U)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -780.0], t$95$0, If[LessEqual[l, 7.5e-6], N[(U + N[(2.0 * N[(J * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.3e+250], t$95$0, N[(U * N[(1.0 + N[(2.0 * N[(l * N[(J * N[(t$95$1 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -780 or 7.50000000000000019e-6 < l < 1.30000000000000006e250

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 76.6%

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

   if -780 < l < 7.50000000000000019e-6

   1. Initial program 74.9%

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

   3. Taylor expanded in l around 0 99.8%

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

   if 1.30000000000000006e250 < 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 74.7%

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

      1. associate-*r* 74.7%

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

   5. Simplified 74.7%

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

   6. Taylor expanded in U around inf 87.3%

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

      1. associate-*r* 87.3%

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

   8. Simplified 87.3%

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

      1. associate-/l* 87.3%

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

      2. *-commutative 87.3%

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

      3. associate-*l* 87.3%

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

   10. Applied egg-rr 87.3%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -780:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 7.5 \cdot 10^{-6}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{+250}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(\ell \cdot \left(J \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.38:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.38)
   (* J (+ (* 2.0 (* l (cos (* K 0.5)))) (/ U J)))
   (* U (+ 1.0 (* (* J 2.0) (/ l U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.38) {
		tmp = J * ((2.0 * (l * cos((K * 0.5)))) + (U / J));
	} else {
		tmp = U * (1.0 + ((J * 2.0) * (l / 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 (cos((k / 2.0d0)) <= 0.38d0) then
        tmp = j * ((2.0d0 * (l * cos((k * 0.5d0)))) + (u / j))
    else
        tmp = u * (1.0d0 + ((j * 2.0d0) * (l / u)))
    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)) <= 0.38) {
		tmp = J * ((2.0 * (l * Math.cos((K * 0.5)))) + (U / J));
	} else {
		tmp = U * (1.0 + ((J * 2.0) * (l / U)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.38:
		tmp = J * ((2.0 * (l * math.cos((K * 0.5)))) + (U / J))
	else:
		tmp = U * (1.0 + ((J * 2.0) * (l / U)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.38)
		tmp = Float64(J * Float64(Float64(2.0 * Float64(l * cos(Float64(K * 0.5)))) + Float64(U / J)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * 2.0) * Float64(l / U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.38)
		tmp = J * ((2.0 * (l * cos((K * 0.5)))) + (U / J));
	else
		tmp = U * (1.0 + ((J * 2.0) * (l / U)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.38], N[(J * N[(N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * 2.0), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.38

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0 71.3%

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

      1. associate-*r* 71.3%

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

   5. Simplified 71.3%

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

   6. Taylor expanded in J around inf 71.4%

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

   if 0.38 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.5%

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

   3. Taylor expanded in l around 0 62.6%

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

      1. associate-*r* 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in U around inf 68.1%

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

      1. associate-*r* 68.1%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in K around 0 65.9%

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

       1. associate-/l* 69.3%

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

       2. associate-*r* 69.3%

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

       3. *-commutative 69.3%

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

   11. Simplified 69.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.38:\\ \;\;\;\;J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right) + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.38:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.38)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (* U (+ 1.0 (* (* J 2.0) (/ l U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.38) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U * (1.0 + ((J * 2.0) * (l / 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 (cos((k / 2.0d0)) <= 0.38d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u * (1.0d0 + ((j * 2.0d0) * (l / u)))
    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)) <= 0.38) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U * (1.0 + ((J * 2.0) * (l / U)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.38:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U * (1.0 + ((J * 2.0) * (l / U)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.38)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * 2.0) * Float64(l / U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.38)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U * (1.0 + ((J * 2.0) * (l / U)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.38], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * 2.0), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.38

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0 71.3%

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

   if 0.38 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.5%

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

   3. Taylor expanded in l around 0 62.6%

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

      1. associate-*r* 62.6%

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

   5. Simplified 62.6%

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

   6. Taylor expanded in U around inf 68.1%

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

      1. associate-*r* 68.1%

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

   8. Simplified 68.1%

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

   9. Taylor expanded in K around 0 65.9%

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

       1. associate-/l* 69.3%

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

       2. associate-*r* 69.3%

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

       3. *-commutative 69.3%

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

   11. Simplified 69.3%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.38:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.008:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.008)
   (+ U (* (* J -0.25) (* l (pow K 2.0))))
   (* U (+ 1.0 (* (* J 2.0) (/ l U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.008) {
		tmp = U + ((J * -0.25) * (l * pow(K, 2.0)));
	} else {
		tmp = U * (1.0 + ((J * 2.0) * (l / 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 (cos((k / 2.0d0)) <= (-0.008d0)) then
        tmp = u + ((j * (-0.25d0)) * (l * (k ** 2.0d0)))
    else
        tmp = u * (1.0d0 + ((j * 2.0d0) * (l / u)))
    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)) <= -0.008) {
		tmp = U + ((J * -0.25) * (l * Math.pow(K, 2.0)));
	} else {
		tmp = U * (1.0 + ((J * 2.0) * (l / U)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.008:
		tmp = U + ((J * -0.25) * (l * math.pow(K, 2.0)))
	else:
		tmp = U * (1.0 + ((J * 2.0) * (l / U)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.008)
		tmp = Float64(U + Float64(Float64(J * -0.25) * Float64(l * (K ^ 2.0))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * 2.0) * Float64(l / U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.008)
		tmp = U + ((J * -0.25) * (l * (K ^ 2.0)));
	else
		tmp = U * (1.0 + ((J * 2.0) * (l / U)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.008], N[(U + N[(N[(J * -0.25), $MachinePrecision] * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * 2.0), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0080000000000000002

   1. Initial program 89.9%

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

   3. Taylor expanded in l around 0 70.9%

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

      1. associate-*r* 70.9%

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

   5. Simplified 70.9%

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

   6. Taylor expanded in K around 0 41.0%

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

   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%

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

      3. *-commutative 60.7%

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

   9. Simplified 60.7%

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

   if -0.0080000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.8%

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

   3. Taylor expanded in l around 0 63.8%

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

      1. associate-*r* 63.8%

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

   5. Simplified 63.8%

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

   6. Taylor expanded in U around inf 69.1%

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

      1. associate-*r* 69.1%

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

   8. Simplified 69.1%

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

   9. Taylor expanded in K around 0 64.5%

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

       1. associate-/l* 68.1%

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

       2. associate-*r* 68.1%

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

       3. *-commutative 68.1%

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

   11. Simplified 68.1%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.008:\\ \;\;\;\;U + \left(J \cdot -0.25\right) \cdot \left(\ell \cdot {K}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + 2 \cdot \frac{\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot J\right)}{U}\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (* U (+ 1.0 (* 2.0 (/ (* (cos (* K 0.5)) (* l J)) U)))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * ((cos((K * 0.5)) * (l * J)) / U)));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u * (1.0d0 + (2.0d0 * ((cos((k * 0.5d0)) * (l * j)) / u)))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * ((Math.cos((K * 0.5)) * (l * J)) / U)));
}

Python

def code(J, l, K, U):
	return U * (1.0 + (2.0 * ((math.cos((K * 0.5)) * (l * J)) / U)))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(2.0 * Float64(Float64(cos(Float64(K * 0.5)) * Float64(l * J)) / U))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + (2.0 * ((cos((K * 0.5)) * (l * J)) / U)));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(2.0 * N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * J), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 65.6%

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

   1. associate-*r* 65.6%

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

5. Simplified 65.6%

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

6. Taylor expanded in U around inf 70.3%

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

   1. associate-*r* 70.3%

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

8. Simplified 70.3%

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

9. Final simplification 70.3%

   \[\leadsto U \cdot \left(1 + 2 \cdot \frac{\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot J\right)}{U}\right) \]
10. Add Preprocessing

Alternative 11: 41.4% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+37} \lor \neg \left(\ell \leq 400000\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.2e+37) (not (<= l 400000.0))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.2e+37) || !(l <= 400000.0)) {
		tmp = U * (U - -4.0);
	} 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 <= (-5.2d+37)) .or. (.not. (l <= 400000.0d0))) then
        tmp = u * (u - (-4.0d0))
    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 <= -5.2e+37) || !(l <= 400000.0)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5.2e+37) or not (l <= 400000.0):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.2e+37) || !(l <= 400000.0))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -5.2e+37) || ~((l <= 400000.0)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.2e+37], N[Not[LessEqual[l, 400000.0]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -5.1999999999999998e37 or 4e5 < 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 16.4%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if -5.1999999999999998e37 < l < 4e5

   1. Initial program 77.3%

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

   3. Taylor expanded in J around 0 68.6%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+37} \lor \neg \left(\ell \leq 400000\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 45.4% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7.3 \cdot 10^{+15} \lor \neg \left(\ell \leq 450\right):\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -7.3e+15) (not (<= l 450.0))) (* l (* J 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7.3e+15) || !(l <= 450.0)) {
		tmp = l * (J * 2.0);
	} 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 <= (-7.3d+15)) .or. (.not. (l <= 450.0d0))) then
        tmp = l * (j * 2.0d0)
    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 <= -7.3e+15) || !(l <= 450.0)) {
		tmp = l * (J * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -7.3e+15) or not (l <= 450.0):
		tmp = l * (J * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -7.3e+15) || !(l <= 450.0))
		tmp = Float64(l * Float64(J * 2.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -7.3e+15) || ~((l <= 450.0)))
		tmp = l * (J * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -7.3e+15], N[Not[LessEqual[l, 450.0]], $MachinePrecision]], N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -7.3e15 or 450 < 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 32.1%

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

      1. associate-*r* 32.1%

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

   5. Simplified 32.1%

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

   6. Taylor expanded in U around inf 45.0%

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

      1. associate-*r* 45.0%

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

   8. Simplified 45.0%

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

   9. Taylor expanded in U around 0 32.3%

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

       1. associate-*r* 32.3%

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

   11. Simplified 32.3%

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

   12. Taylor expanded in K around 0 21.2%

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

       1. *-commutative 21.2%

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

       2. *-commutative 21.2%

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

       3. associate-*r* 21.2%

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

   14. Simplified 21.2%

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

   if -7.3e15 < l < 450

   1. Initial program 76.1%

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

   3. Taylor expanded in J around 0 72.1%

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

4. Final simplification 47.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.3 \cdot 10^{+15} \lor \neg \left(\ell \leq 450\right):\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.4% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+37} \lor \neg \left(\ell \leq 400000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.9e+37) (not (<= l 400000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.9e+37) || !(l <= 400000.0)) {
		tmp = U * U;
	} 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 <= (-2.9d+37)) .or. (.not. (l <= 400000.0d0))) then
        tmp = u * u
    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 <= -2.9e+37) || !(l <= 400000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.9e+37) or not (l <= 400000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.9e+37) || !(l <= 400000.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.9e+37) || ~((l <= 400000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.9e+37], N[Not[LessEqual[l, 400000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -2.89999999999999978e37 or 4e5 < 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 16.4%

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

   if -2.89999999999999978e37 < l < 4e5

   1. Initial program 77.3%

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

   3. Taylor expanded in J around 0 68.6%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+37} \lor \neg \left(\ell \leq 400000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.9% accurate, 28.4× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + 2 \cdot \frac{\ell \cdot J}{U}\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* U (+ 1.0 (* 2.0 (/ (* l J) U)))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * ((l * J) / U)));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u * (1.0d0 + (2.0d0 * ((l * j) / u)))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * ((l * J) / U)));
}

Python

def code(J, l, K, U):
	return U * (1.0 + (2.0 * ((l * J) / U)))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(2.0 * Float64(Float64(l * J) / U))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + (2.0 * ((l * J) / U)));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(2.0 * N[(N[(l * J), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 65.6%

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

   1. associate-*r* 65.6%

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

5. Simplified 65.6%

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

6. Taylor expanded in U around inf 70.3%

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

   1. associate-*r* 70.3%

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

8. Simplified 70.3%

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

9. Taylor expanded in K around 0 58.2%

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

10. Final simplification 58.2%

    \[\leadsto U \cdot \left(1 + 2 \cdot \frac{\ell \cdot J}{U}\right) \]
11. Add Preprocessing

Alternative 15: 59.5% accurate, 28.4× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* U (+ 1.0 (* (* J 2.0) (/ l U)))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + ((J * 2.0) * (l / U)));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u * (1.0d0 + ((j * 2.0d0) * (l / u)))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + ((J * 2.0) * (l / U)));
}

Python

def code(J, l, K, U):
	return U * (1.0 + ((J * 2.0) * (l / U)))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(Float64(J * 2.0) * Float64(l / U))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + ((J * 2.0) * (l / U)));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(N[(J * 2.0), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 65.6%

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

   1. associate-*r* 65.6%

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

5. Simplified 65.6%

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

6. Taylor expanded in U around inf 70.3%

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

   1. associate-*r* 70.3%

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

8. Simplified 70.3%

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

9. Taylor expanded in K around 0 58.2%

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

    1. associate-/l* 60.8%

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

    2. associate-*r* 60.8%

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

    3. *-commutative 60.8%

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

11. Simplified 60.8%

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

12. Final simplification 60.8%

    \[\leadsto U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right) \]
13. Add Preprocessing

Alternative 16: 53.3% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 65.6%

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

   1. associate-*r* 65.6%

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

5. Simplified 65.6%

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

6. Taylor expanded in K around 0 53.9%

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

   1. *-commutative 53.9%

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

   2. associate-*r* 53.9%

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

8. Simplified 53.9%

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

9. Final simplification 53.9%

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

Alternative 17: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 1.0)

C

double code(double J, double l, double K, double U) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 1.0;
}

Python

def code(J, l, K, U):
	return 1.0

Julia

function code(J, l, K, U)
	return 1.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 1.0;
end

Wolfram

code[J_, l_, K_, U_] := 1.0

Derivation

1. Initial program 87.6%

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

4. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{\frac{U}{U}} \]
5. Step-by-step derivation

   1. *-inverses 2.7%

      \[\leadsto \color{blue}{1} \]

6. Simplified 2.7%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 2.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 18: 36.4% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

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

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in J around 0 38.4%

   \[\leadsto \color{blue}{U} \]

4. Final simplification 38.4%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024079 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))