Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.7% → 99.7%

Time: 11.9s

Alternatives: 15

Speedup: 1.4×

• 48.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.7% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0.0002\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0002)))
     (+ (* (* 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 <= 0.0002)) {
		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 <= 0.0002))
		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, 0.0002]], $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 2.0000000000000001e-4 < (-.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))) < 2.0000000000000001e-4

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 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. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. distribute-lft-out 99.9%

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

      8. 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0002\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 0.0002\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 (- INFINITY)) (not (<= t_1 0.0002)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* 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 <= 0.0002)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (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 <= 0.0002)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (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 <= 0.0002):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (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 <= 0.0002))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(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 <= 0.0002)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (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, 0.0002]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $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 2.0000000000000001e-4 < (-.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))) < 2.0000000000000001e-4

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0002\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0.0002\right):\\ \;\;\;\;t\_0 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0002)))
     (* t_0 J)
     (+ U (* l (* 2.0 (* J (cos (* K 0.5)))))))))

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 <= 0.0002)) {
		tmp = t_0 * J;
	} else {
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0002)) {
		tmp = t_0 * J;
	} else {
		tmp = U + (l * (2.0 * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0002):
		tmp = t_0 * J
	else:
		tmp = U + (l * (2.0 * (J * math.cos((K * 0.5)))))
	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 <= 0.0002))
		tmp = Float64(t_0 * J);
	else
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.0002)))
		tmp = t_0 * J;
	else
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	end
	tmp_2 = 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, 0.0002]], $MachinePrecision]], N[(t$95$0 * J), $MachinePrecision], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $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 2.0000000000000001e-4 < (-.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

   3. Taylor expanded in K around 0 79.6%

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

   4. Taylor expanded in J around inf 79.6%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.0000000000000001e-4

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 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. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. distribute-lft-out 99.9%

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

      8. 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%

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

   6. Taylor expanded in l around 0 99.7%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0002\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -36:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.0115:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+102}:\\ \;\;\;\;t\_1 + U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* J (* 0.3333333333333333 (pow l 3.0))))))
        (t_1 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -8.2e+139)
     t_0
     (if (<= l -36.0)
       t_1
       (if (<= l 0.0115)
         (+ U (* l (* 2.0 (* J (cos (* K 0.5))))))
         (if (<= l 3e+102) (+ t_1 U) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * (0.3333333333333333 * pow(l, 3.0))));
	double t_1 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -8.2e+139) {
		tmp = t_0;
	} else if (l <= -36.0) {
		tmp = t_1;
	} else if (l <= 0.0115) {
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	} else if (l <= 3e+102) {
		tmp = t_1 + 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) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * (j * (0.3333333333333333d0 * (l ** 3.0d0))))
    t_1 = (exp(l) - exp(-l)) * j
    if (l <= (-8.2d+139)) then
        tmp = t_0
    else if (l <= (-36.0d0)) then
        tmp = t_1
    else if (l <= 0.0115d0) then
        tmp = u + (l * (2.0d0 * (j * cos((k * 0.5d0)))))
    else if (l <= 3d+102) then
        tmp = t_1 + 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 = U + (Math.cos((K / 2.0)) * (J * (0.3333333333333333 * Math.pow(l, 3.0))));
	double t_1 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -8.2e+139) {
		tmp = t_0;
	} else if (l <= -36.0) {
		tmp = t_1;
	} else if (l <= 0.0115) {
		tmp = U + (l * (2.0 * (J * Math.cos((K * 0.5)))));
	} else if (l <= 3e+102) {
		tmp = t_1 + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * (0.3333333333333333 * math.pow(l, 3.0))))
	t_1 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -8.2e+139:
		tmp = t_0
	elif l <= -36.0:
		tmp = t_1
	elif l <= 0.0115:
		tmp = U + (l * (2.0 * (J * math.cos((K * 0.5)))))
	elif l <= 3e+102:
		tmp = t_1 + U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))))
	t_1 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -8.2e+139)
		tmp = t_0;
	elseif (l <= -36.0)
		tmp = t_1;
	elseif (l <= 0.0115)
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 3e+102)
		tmp = Float64(t_1 + U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * (0.3333333333333333 * (l ^ 3.0))));
	t_1 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -8.2e+139)
		tmp = t_0;
	elseif (l <= -36.0)
		tmp = t_1;
	elseif (l <= 0.0115)
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	elseif (l <= 3e+102)
		tmp = t_1 + U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -8.2e+139], t$95$0, If[LessEqual[l, -36.0], t$95$1, If[LessEqual[l, 0.0115], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3e+102], N[(t$95$1 + U), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -8.2000000000000004e139 or 2.9999999999999998e102 < 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 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 75.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot J\right) \cdot {\ell}^{3}} + U \]

      2. *-commutative 75.0%

         \[\leadsto \color{blue}{\left(J \cdot 0.3333333333333333\right)} \cdot {\ell}^{3} + U \]

      3. associate-*r* 75.0%

         \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   6. Simplified 100.0%

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

   if -8.2000000000000004e139 < l < -36

   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 87.5%

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

   4. Taylor expanded in J around inf 87.5%

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

   if -36 < l < 0.0115

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 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. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. distribute-lft-out 99.9%

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

      8. 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%

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

   6. Taylor expanded in l around 0 99.7%

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

   if 0.0115 < l < 2.9999999999999998e102

   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 83.3%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+139}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -36:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.0115:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t\_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -200:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 0.16:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t\_2 + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* J (* 0.3333333333333333 (pow l 3.0))))))
        (t_2 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -8.2e+139)
     t_1
     (if (<= l -200.0)
       t_2
       (if (<= l 0.16)
         (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
         (if (<= l 5.5e+102) (+ t_2 U) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (J * (0.3333333333333333 * pow(l, 3.0))));
	double t_2 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -8.2e+139) {
		tmp = t_1;
	} else if (l <= -200.0) {
		tmp = t_2;
	} else if (l <= 0.16) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else if (l <= 5.5e+102) {
		tmp = t_2 + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * (j * (0.3333333333333333d0 * (l ** 3.0d0))))
    t_2 = (exp(l) - exp(-l)) * j
    if (l <= (-8.2d+139)) then
        tmp = t_1
    else if (l <= (-200.0d0)) then
        tmp = t_2
    else if (l <= 0.16d0) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    else if (l <= 5.5d+102) then
        tmp = t_2 + u
    else
        tmp = t_1
    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 * (J * (0.3333333333333333 * Math.pow(l, 3.0))));
	double t_2 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -8.2e+139) {
		tmp = t_1;
	} else if (l <= -200.0) {
		tmp = t_2;
	} else if (l <= 0.16) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else if (l <= 5.5e+102) {
		tmp = t_2 + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (J * (0.3333333333333333 * math.pow(l, 3.0))))
	t_2 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -8.2e+139:
		tmp = t_1
	elif l <= -200.0:
		tmp = t_2
	elif l <= 0.16:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	elif l <= 5.5e+102:
		tmp = t_2 + U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))))
	t_2 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -8.2e+139)
		tmp = t_1;
	elseif (l <= -200.0)
		tmp = t_2;
	elseif (l <= 0.16)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	elseif (l <= 5.5e+102)
		tmp = Float64(t_2 + U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * (J * (0.3333333333333333 * (l ^ 3.0))));
	t_2 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -8.2e+139)
		tmp = t_1;
	elseif (l <= -200.0)
		tmp = t_2;
	elseif (l <= 0.16)
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	elseif (l <= 5.5e+102)
		tmp = t_2 + U;
	else
		tmp = t_1;
	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[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -8.2e+139], t$95$1, If[LessEqual[l, -200.0], t$95$2, If[LessEqual[l, 0.16], 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], If[LessEqual[l, 5.5e+102], N[(t$95$2 + U), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -8.2000000000000004e139 or 5.49999999999999981e102 < 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 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 75.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot J\right) \cdot {\ell}^{3}} + U \]

      2. *-commutative 75.0%

         \[\leadsto \color{blue}{\left(J \cdot 0.3333333333333333\right)} \cdot {\ell}^{3} + U \]

      3. associate-*r* 75.0%

         \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   6. Simplified 100.0%

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

   if -8.2000000000000004e139 < l < -200

   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 87.5%

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

   4. Taylor expanded in J around inf 87.5%

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

   if -200 < l < 0.160000000000000003

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 99.9%

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

   if 0.160000000000000003 < l < 5.49999999999999981e102

   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 83.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+139}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -200:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.16:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot {\ell}^{3}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := U + t\_1 \cdot \left(J \cdot t\_0\right)\\ t_3 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+139}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -310:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\ell \leq 0.19:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(t\_0 + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t\_3 + U\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (pow l 3.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (+ U (* t_1 (* J t_0))))
        (t_3 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -8.2e+139)
     t_2
     (if (<= l -310.0)
       t_3
       (if (<= l 0.19)
         (+ U (* t_1 (* J (+ t_0 (* l 2.0)))))
         (if (<= l 5.5e+102) (+ t_3 U) t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * pow(l, 3.0);
	double t_1 = cos((K / 2.0));
	double t_2 = U + (t_1 * (J * t_0));
	double t_3 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -8.2e+139) {
		tmp = t_2;
	} else if (l <= -310.0) {
		tmp = t_3;
	} else if (l <= 0.19) {
		tmp = U + (t_1 * (J * (t_0 + (l * 2.0))));
	} else if (l <= 5.5e+102) {
		tmp = t_3 + U;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (l ** 3.0d0)
    t_1 = cos((k / 2.0d0))
    t_2 = u + (t_1 * (j * t_0))
    t_3 = (exp(l) - exp(-l)) * j
    if (l <= (-8.2d+139)) then
        tmp = t_2
    else if (l <= (-310.0d0)) then
        tmp = t_3
    else if (l <= 0.19d0) then
        tmp = u + (t_1 * (j * (t_0 + (l * 2.0d0))))
    else if (l <= 5.5d+102) then
        tmp = t_3 + u
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * Math.pow(l, 3.0);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = U + (t_1 * (J * t_0));
	double t_3 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -8.2e+139) {
		tmp = t_2;
	} else if (l <= -310.0) {
		tmp = t_3;
	} else if (l <= 0.19) {
		tmp = U + (t_1 * (J * (t_0 + (l * 2.0))));
	} else if (l <= 5.5e+102) {
		tmp = t_3 + U;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * math.pow(l, 3.0)
	t_1 = math.cos((K / 2.0))
	t_2 = U + (t_1 * (J * t_0))
	t_3 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -8.2e+139:
		tmp = t_2
	elif l <= -310.0:
		tmp = t_3
	elif l <= 0.19:
		tmp = U + (t_1 * (J * (t_0 + (l * 2.0))))
	elif l <= 5.5e+102:
		tmp = t_3 + U
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * (l ^ 3.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(U + Float64(t_1 * Float64(J * t_0)))
	t_3 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -8.2e+139)
		tmp = t_2;
	elseif (l <= -310.0)
		tmp = t_3;
	elseif (l <= 0.19)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(t_0 + Float64(l * 2.0)))));
	elseif (l <= 5.5e+102)
		tmp = Float64(t_3 + U);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 * (l ^ 3.0);
	t_1 = cos((K / 2.0));
	t_2 = U + (t_1 * (J * t_0));
	t_3 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -8.2e+139)
		tmp = t_2;
	elseif (l <= -310.0)
		tmp = t_3;
	elseif (l <= 0.19)
		tmp = U + (t_1 * (J * (t_0 + (l * 2.0))));
	elseif (l <= 5.5e+102)
		tmp = t_3 + U;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(t$95$1 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -8.2e+139], t$95$2, If[LessEqual[l, -310.0], t$95$3, If[LessEqual[l, 0.19], N[(U + N[(t$95$1 * N[(J * N[(t$95$0 + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.5e+102], N[(t$95$3 + U), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -8.2000000000000004e139 or 5.49999999999999981e102 < 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 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 75.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot J\right) \cdot {\ell}^{3}} + U \]

      2. *-commutative 75.0%

         \[\leadsto \color{blue}{\left(J \cdot 0.3333333333333333\right)} \cdot {\ell}^{3} + U \]

      3. associate-*r* 75.0%

         \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   6. Simplified 100.0%

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

   if -8.2000000000000004e139 < l < -310

   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 87.5%

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

   4. Taylor expanded in J around inf 87.5%

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

   if -310 < l < 0.19

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 0.19 < l < 5.49999999999999981e102

   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 83.3%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+139}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -310:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.19:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + \cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ U (* (cos (* K 0.5)) (* l (* J 2.0))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = U + (cos((K * 0.5)) * (l * (J * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 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) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = u + (cos((k * 0.5d0)) * (l * (j * 2.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    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)) <= -0.005) {
		tmp = U + (Math.cos((K * 0.5)) * (l * (J * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = U + (math.cos((K * 0.5)) * (l * (J * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(U + Float64(cos(Float64(K * 0.5)) * Float64(l * Float64(J * 2.0))));
	else
		tmp = Float64(U + 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)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = U + (cos((K * 0.5)) * (l * (J * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(U + N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 80.5%

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

   3. Taylor expanded in l around 0 72.1%

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

      1. associate-*r* 72.1%

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

      2. associate-*r* 73.6%

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

   5. Simplified 73.6%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 82.7%

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

   3. Taylor expanded in l around 0 88.9%

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

   4. Taylor expanded in K around 0 84.7%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + \cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -360:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.012:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -360.0)
     t_0
     (if (<= l 0.012) (+ U (* l (* 2.0 (* J (cos (* K 0.5)))))) (+ t_0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -360.0) {
		tmp = t_0;
	} else if (l <= 0.012) {
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	} else {
		tmp = t_0 + 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) :: tmp
    t_0 = (exp(l) - exp(-l)) * j
    if (l <= (-360.0d0)) then
        tmp = t_0
    else if (l <= 0.012d0) then
        tmp = u + (l * (2.0d0 * (j * cos((k * 0.5d0)))))
    else
        tmp = t_0 + 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;
	double tmp;
	if (l <= -360.0) {
		tmp = t_0;
	} else if (l <= 0.012) {
		tmp = U + (l * (2.0 * (J * Math.cos((K * 0.5)))));
	} else {
		tmp = t_0 + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -360.0:
		tmp = t_0
	elif l <= 0.012:
		tmp = U + (l * (2.0 * (J * math.cos((K * 0.5)))))
	else:
		tmp = t_0 + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -360.0)
		tmp = t_0;
	elseif (l <= 0.012)
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(t_0 + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -360.0)
		tmp = t_0;
	elseif (l <= 0.012)
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	else
		tmp = t_0 + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -360.0], t$95$0, If[LessEqual[l, 0.012], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -360

   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 83.9%

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

   4. Taylor expanded in J around inf 83.9%

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

   if -360 < l < 0.012

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 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. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. distribute-lft-out 99.9%

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

      8. 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%

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

   6. Taylor expanded in l around 0 99.7%

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

   if 0.012 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 75.0%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -360:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.012:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1800:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 145000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* 0.3333333333333333 (pow l 3.0))))))
   (if (<= l -4.5e+60)
     t_0
     (if (<= l -1800.0)
       (pow U -4.0)
       (if (<= l 145000.0) (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (0.3333333333333333 * pow(l, 3.0)));
	double tmp;
	if (l <= -4.5e+60) {
		tmp = t_0;
	} else if (l <= -1800.0) {
		tmp = pow(U, -4.0);
	} else if (l <= 145000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (j * (0.3333333333333333d0 * (l ** 3.0d0)))
    if (l <= (-4.5d+60)) then
        tmp = t_0
    else if (l <= (-1800.0d0)) then
        tmp = u ** (-4.0d0)
    else if (l <= 145000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (0.3333333333333333 * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -4.5e+60) {
		tmp = t_0;
	} else if (l <= -1800.0) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 145000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (0.3333333333333333 * math.pow(l, 3.0)))
	tmp = 0
	if l <= -4.5e+60:
		tmp = t_0
	elif l <= -1800.0:
		tmp = math.pow(U, -4.0)
	elif l <= 145000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -4.5e+60)
		tmp = t_0;
	elseif (l <= -1800.0)
		tmp = U ^ -4.0;
	elseif (l <= 145000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (0.3333333333333333 * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -4.5e+60)
		tmp = t_0;
	elseif (l <= -1800.0)
		tmp = U ^ -4.0;
	elseif (l <= 145000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+60], t$95$0, If[LessEqual[l, -1800.0], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 145000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.50000000000000013e60 or 145000 < 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 84.6%

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

   4. Taylor expanded in K around 0 66.4%

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

   5. Taylor expanded in l around inf 66.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} + U \]
   6. Step-by-step derivation

      1. associate-*r* 66.4%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot J\right) \cdot {\ell}^{3}} + U \]

      2. *-commutative 66.4%

         \[\leadsto \color{blue}{\left(J \cdot 0.3333333333333333\right)} \cdot {\ell}^{3} + U \]

      3. associate-*r* 66.4%

         \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   7. Simplified 66.4%

      \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   if -4.50000000000000013e60 < l < -1800

   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 1.4%

      \[\leadsto \left(J \cdot \color{blue}{0.25}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if -1800 < l < 145000

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 99.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+60}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -1800:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 145000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -720:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 26000:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* 0.3333333333333333 (pow l 3.0))))))
   (if (<= l -3.8e+60)
     t_0
     (if (<= l -720.0)
       (pow U -4.0)
       (if (<= l 26000.0) (+ U (* l (* 2.0 (* J (cos (* K 0.5)))))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (0.3333333333333333 * pow(l, 3.0)));
	double tmp;
	if (l <= -3.8e+60) {
		tmp = t_0;
	} else if (l <= -720.0) {
		tmp = pow(U, -4.0);
	} else if (l <= 26000.0) {
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (j * (0.3333333333333333d0 * (l ** 3.0d0)))
    if (l <= (-3.8d+60)) then
        tmp = t_0
    else if (l <= (-720.0d0)) then
        tmp = u ** (-4.0d0)
    else if (l <= 26000.0d0) then
        tmp = u + (l * (2.0d0 * (j * cos((k * 0.5d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (0.3333333333333333 * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -3.8e+60) {
		tmp = t_0;
	} else if (l <= -720.0) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 26000.0) {
		tmp = U + (l * (2.0 * (J * Math.cos((K * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (0.3333333333333333 * math.pow(l, 3.0)))
	tmp = 0
	if l <= -3.8e+60:
		tmp = t_0
	elif l <= -720.0:
		tmp = math.pow(U, -4.0)
	elif l <= 26000.0:
		tmp = U + (l * (2.0 * (J * math.cos((K * 0.5)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -3.8e+60)
		tmp = t_0;
	elseif (l <= -720.0)
		tmp = U ^ -4.0;
	elseif (l <= 26000.0)
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(K * 0.5))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (0.3333333333333333 * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -3.8e+60)
		tmp = t_0;
	elseif (l <= -720.0)
		tmp = U ^ -4.0;
	elseif (l <= 26000.0)
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.8e+60], t$95$0, If[LessEqual[l, -720.0], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 26000.0], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.80000000000000009e60 or 26000 < 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 84.6%

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

   4. Taylor expanded in K around 0 66.4%

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

   5. Taylor expanded in l around inf 66.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} + U \]
   6. Step-by-step derivation

      1. associate-*r* 66.4%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot J\right) \cdot {\ell}^{3}} + U \]

      2. *-commutative 66.4%

         \[\leadsto \color{blue}{\left(J \cdot 0.3333333333333333\right)} \cdot {\ell}^{3} + U \]

      3. associate-*r* 66.4%

         \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   7. Simplified 66.4%

      \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   if -3.80000000000000009e60 < l < -720

   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 1.4%

      \[\leadsto \left(J \cdot \color{blue}{0.25}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if -720 < l < 26000

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 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. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. distribute-lft-out 99.9%

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

      8. 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%

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

   6. Taylor expanded in l around 0 99.7%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+60}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -720:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 26000:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 72.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+60}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -860:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-23}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* 0.3333333333333333 (pow l 3.0))))))
   (if (<= l -3.5e+60)
     t_0
     (if (<= l -860.0)
       (pow U -4.0)
       (if (<= l 1.3e-23) (+ U (* J (* l 2.0))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (0.3333333333333333 * pow(l, 3.0)));
	double tmp;
	if (l <= -3.5e+60) {
		tmp = t_0;
	} else if (l <= -860.0) {
		tmp = pow(U, -4.0);
	} else if (l <= 1.3e-23) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (j * (0.3333333333333333d0 * (l ** 3.0d0)))
    if (l <= (-3.5d+60)) then
        tmp = t_0
    else if (l <= (-860.0d0)) then
        tmp = u ** (-4.0d0)
    else if (l <= 1.3d-23) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (0.3333333333333333 * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -3.5e+60) {
		tmp = t_0;
	} else if (l <= -860.0) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 1.3e-23) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (0.3333333333333333 * math.pow(l, 3.0)))
	tmp = 0
	if l <= -3.5e+60:
		tmp = t_0
	elif l <= -860.0:
		tmp = math.pow(U, -4.0)
	elif l <= 1.3e-23:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -3.5e+60)
		tmp = t_0;
	elseif (l <= -860.0)
		tmp = U ^ -4.0;
	elseif (l <= 1.3e-23)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (0.3333333333333333 * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -3.5e+60)
		tmp = t_0;
	elseif (l <= -860.0)
		tmp = U ^ -4.0;
	elseif (l <= 1.3e-23)
		tmp = U + (J * (l * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.5e+60], t$95$0, If[LessEqual[l, -860.0], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 1.3e-23], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.5000000000000002e60 or 1.3e-23 < l

   1. Initial program 98.9%

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

   3. Taylor expanded in l around 0 85.4%

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

   4. Taylor expanded in K around 0 66.1%

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

   5. Taylor expanded in l around inf 66.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} + U \]
   6. Step-by-step derivation

      1. associate-*r* 66.1%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot J\right) \cdot {\ell}^{3}} + U \]

      2. *-commutative 66.1%

         \[\leadsto \color{blue}{\left(J \cdot 0.3333333333333333\right)} \cdot {\ell}^{3} + U \]

      3. associate-*r* 66.1%

         \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   7. Simplified 66.1%

      \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]

   if -3.5000000000000002e60 < l < -860

   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 1.4%

      \[\leadsto \left(J \cdot \color{blue}{0.25}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Applied egg-rr 43.8%

      \[\leadsto \color{blue}{{U}^{-4}} \]

   if -860 < l < 1.3e-23

   1. Initial program 69.0%

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

   3. Taylor expanded in l around 0 99.9%

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

   4. Taylor expanded in K around 0 87.0%

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

   5. Taylor expanded in l around 0 87.0%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+60}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -860:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-23}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.8% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-13} \lor \neg \left(\ell \leq 16500000000000\right):\\ \;\;\;\;U \cdot \left(U + -1\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.55e-13) (not (<= l 16500000000000.0))) (* U (+ U -1.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.55e-13) || !(l <= 16500000000000.0)) {
		tmp = U * (U + -1.0);
	} 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 <= (-1.55d-13)) .or. (.not. (l <= 16500000000000.0d0))) then
        tmp = u * (u + (-1.0d0))
    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 <= -1.55e-13) || !(l <= 16500000000000.0)) {
		tmp = U * (U + -1.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.55e-13) or not (l <= 16500000000000.0):
		tmp = U * (U + -1.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.55e-13) || !(l <= 16500000000000.0))
		tmp = Float64(U * Float64(U + -1.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.55e-13) || ~((l <= 16500000000000.0)))
		tmp = U * (U + -1.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.55e-13], N[Not[LessEqual[l, 16500000000000.0]], $MachinePrecision]], N[(U * N[(U + -1.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.55e-13 or 1.65e13 < l

   1. Initial program 99.4%

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

   4. Applied egg-rr 2.7%

      \[\leadsto \left(J \cdot \color{blue}{0.25}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Applied egg-rr 22.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(U, U, -U\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 22.9%

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

      2. neg-mul-1 22.9%

         \[\leadsto U \cdot U + \color{blue}{-1 \cdot U} \]

      3. distribute-rgt-out 22.9%

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

   8. Simplified 22.9%

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

   if -1.55e-13 < l < 1.65e13

   1. Initial program 69.7%

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

   4. Applied egg-rr 41.8%

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

   5. Taylor expanded in J around 0 68.4%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-13} \lor \neg \left(\ell \leq 16500000000000\right):\\ \;\;\;\;U \cdot \left(U + -1\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.8% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-13} \lor \neg \left(\ell \leq 20000000000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.55e-13) (not (<= l 20000000000000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.55e-13) || !(l <= 20000000000000.0)) {
		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 <= (-1.55d-13)) .or. (.not. (l <= 20000000000000.0d0))) 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 <= -1.55e-13) || !(l <= 20000000000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.55e-13) or not (l <= 20000000000000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.55e-13) || !(l <= 20000000000000.0))
		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 <= -1.55e-13) || ~((l <= 20000000000000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.55e-13], N[Not[LessEqual[l, 20000000000000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.55e-13 or 2e13 < l

   1. Initial program 99.4%

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

   4. Applied egg-rr 2.7%

      \[\leadsto \left(J \cdot \color{blue}{0.25}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Applied egg-rr 22.8%

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

   if -1.55e-13 < l < 2e13

   1. Initial program 69.7%

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

   4. Applied egg-rr 41.8%

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

   5. Taylor expanded in J around 0 68.4%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-13} \lor \neg \left(\ell \leq 20000000000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.0% 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 82.3%

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

3. Taylor expanded in l around 0 89.8%

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

4. Taylor expanded in K around 0 74.8%

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

5. Taylor expanded in l around 0 58.5%

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

6. Final simplification 58.5%

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

Alternative 15: 37.6% 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 82.3%

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

4. Applied egg-rr 25.3%

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

5. Taylor expanded in J around 0 40.5%

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

6. Final simplification 40.5%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))