Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.8%

Time: 17.2s

Alternatives: 14

Speedup: 1.5×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0.04\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot t_0\right)\right) + \left(0.016666666666666666 \cdot \left(J \cdot \left(t_0 \cdot {\ell}^{5}\right)\right) + \left(0.3333333333333333 \cdot \left(J \cdot \left(t_0 \cdot {\ell}^{3}\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot t_0\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.04)))
     (+ (* (* t_1 J) (cos (/ K 2.0))) U)
     (+
      U
      (+
       (* 0.0003968253968253968 (* J (* (pow l 7.0) t_0)))
       (+
        (* 0.016666666666666666 (* J (* t_0 (pow l 5.0))))
        (+
         (* 0.3333333333333333 (* J (* t_0 (pow l 3.0))))
         (* 2.0 (* J (* l t_0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.04)) {
		tmp = ((t_1 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + ((0.0003968253968253968 * (J * (pow(l, 7.0) * t_0))) + ((0.016666666666666666 * (J * (t_0 * pow(l, 5.0)))) + ((0.3333333333333333 * (J * (t_0 * pow(l, 3.0)))) + (2.0 * (J * (l * t_0))))));
	}
	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 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.04)) {
		tmp = ((t_1 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + ((0.0003968253968253968 * (J * (Math.pow(l, 7.0) * t_0))) + ((0.016666666666666666 * (J * (t_0 * Math.pow(l, 5.0)))) + ((0.3333333333333333 * (J * (t_0 * Math.pow(l, 3.0)))) + (2.0 * (J * (l * t_0))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.04):
		tmp = ((t_1 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + ((0.0003968253968253968 * (J * (math.pow(l, 7.0) * t_0))) + ((0.016666666666666666 * (J * (t_0 * math.pow(l, 5.0)))) + ((0.3333333333333333 * (J * (t_0 * math.pow(l, 3.0)))) + (2.0 * (J * (l * t_0))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 0.04))
		tmp = Float64(Float64(Float64(t_1 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(Float64(0.0003968253968253968 * Float64(J * Float64((l ^ 7.0) * t_0))) + Float64(Float64(0.016666666666666666 * Float64(J * Float64(t_0 * (l ^ 5.0)))) + Float64(Float64(0.3333333333333333 * Float64(J * Float64(t_0 * (l ^ 3.0)))) + Float64(2.0 * Float64(J * Float64(l * t_0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 0.04)))
		tmp = ((t_1 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + ((0.0003968253968253968 * (J * ((l ^ 7.0) * t_0))) + ((0.016666666666666666 * (J * (t_0 * (l ^ 5.0)))) + ((0.3333333333333333 * (J * (t_0 * (l ^ 3.0)))) + (2.0 * (J * (l * t_0))))));
	end
	tmp_2 = 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[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.04]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(0.0003968253968253968 * N[(J * N[(N[Power[l, 7.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[(J * N[(t$95$0 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(J * N[(t$95$0 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(J * N[(l * t$95$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 0.0400000000000000008 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0400000000000000008

   1. Initial program 78.6%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + \left(0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.04\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot \cos \left(K \cdot 0.5\right)\right)\right) + \left(0.016666666666666666 \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{5}\right)\right) + \left(0.3333333333333333 \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{3}\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.04\right):\\ \;\;\;\;\left(t_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.04)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       J
       (*
        (cos (* K 0.5))
        (+
         (* 0.016666666666666666 (pow l 5.0))
         (+ (* 0.3333333333333333 (pow l 3.0)) (* l 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 <= 0.04)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (cos((K * 0.5)) * ((0.016666666666666666 * pow(l, 5.0)) + ((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.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.04)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (Math.cos((K * 0.5)) * ((0.016666666666666666 * Math.pow(l, 5.0)) + ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)))));
	}
	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 <= 0.04):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (J * (math.cos((K * 0.5)) * ((0.016666666666666666 * math.pow(l, 5.0)) + ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 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 <= 0.04))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + 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 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.04)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (J * (cos((K * 0.5)) * ((0.016666666666666666 * (l ^ 5.0)) + ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)))));
	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, 0.04]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $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 0.0400000000000000008 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0400000000000000008

   1. Initial program 78.6%

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

      1. associate-*l* 78.6%

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

      2. fma-def 78.6%

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

   3. Simplified 78.6%

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

   4. Taylor expanded in l around 0 99.9%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   5. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{U + J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.04\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\left(t_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       J
       (* (cos (* K 0.5)) (fma 2.0 l (* 0.3333333333333333 (pow l 3.0)))))))))

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 <= 0.0)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (cos((K * 0.5)) * fma(2.0, l, (0.3333333333333333 * pow(l, 3.0)))));
	}
	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 <= 0.0))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * fma(2.0, l, Float64(0.3333333333333333 * (l ^ 3.0))))));
	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, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 * l + N[(0.3333333333333333 * N[Power[l, 3.0], $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 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 78.3%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \left(0.3333333333333333 \cdot \color{blue}{\left(\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)} + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right) + U \]

      2. associate-*r* 99.9%

         \[\leadsto \left(\color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot J} + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right) + U \]

      3. *-commutative 99.9%

         \[\leadsto \left(\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot J + 2 \cdot \color{blue}{\left(\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 99.9%

         \[\leadsto \left(\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot J + \color{blue}{\left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 99.9%

         \[\leadsto \color{blue}{J \cdot \left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right) + 2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

      6. +-commutative 99.9%

         \[\leadsto J \cdot \color{blue}{\left(2 \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

      7. associate-*r* 99.9%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      8. associate-*r* 99.9%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      9. distribute-rgt-out 99.9%

         \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      10. +-commutative 99.9%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      11. *-commutative 99.9%

          \[\leadsto J \cdot \color{blue}{\left(\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      12. +-commutative 99.9%

          \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

      13. fma-def 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \mathsf{fma}\left(2, \ell, 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 0.0)))
     (+ (* (* 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 <= 0.0)) {
		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 <= 0.0)) {
		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 <= 0.0):
		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 <= 0.0))
		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 <= 0.0)))
		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, 0.0]], $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 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 78.3%

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

   2. Taylor expanded in l around 0 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 5: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{5}\right) \cdot \left(J \cdot 0.016666666666666666\right)\\ \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -850:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 0.35:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U))
        (t_1 (* (* (cos (* K 0.5)) (pow l 5.0)) (* J 0.016666666666666666))))
   (if (<= l -1.1e+58)
     t_1
     (if (<= l -850.0)
       t_0
       (if (<= l 0.35)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 2.45e+61) t_0 t_1))))))

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)) * pow(l, 5.0)) * (J * 0.016666666666666666);
	double tmp;
	if (l <= -1.1e+58) {
		tmp = t_1;
	} else if (l <= -850.0) {
		tmp = t_0;
	} else if (l <= 0.35) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2.45e+61) {
		tmp = t_0;
	} 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) :: tmp
    t_0 = ((exp(l) - exp(-l)) * j) + u
    t_1 = (cos((k * 0.5d0)) * (l ** 5.0d0)) * (j * 0.016666666666666666d0)
    if (l <= (-1.1d+58)) then
        tmp = t_1
    else if (l <= (-850.0d0)) then
        tmp = t_0
    else if (l <= 0.35d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 2.45d+61) then
        tmp = t_0
    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.exp(l) - Math.exp(-l)) * J) + U;
	double t_1 = (Math.cos((K * 0.5)) * Math.pow(l, 5.0)) * (J * 0.016666666666666666);
	double tmp;
	if (l <= -1.1e+58) {
		tmp = t_1;
	} else if (l <= -850.0) {
		tmp = t_0;
	} else if (l <= 0.35) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2.45e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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)) * math.pow(l, 5.0)) * (J * 0.016666666666666666)
	tmp = 0
	if l <= -1.1e+58:
		tmp = t_1
	elif l <= -850.0:
		tmp = t_0
	elif l <= 0.35:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 2.45e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = Float64(Float64(cos(Float64(K * 0.5)) * (l ^ 5.0)) * Float64(J * 0.016666666666666666))
	tmp = 0.0
	if (l <= -1.1e+58)
		tmp = t_1;
	elseif (l <= -850.0)
		tmp = t_0;
	elseif (l <= 0.35)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 2.45e+61)
		tmp = t_0;
	else
		tmp = t_1;
	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)) * (l ^ 5.0)) * (J * 0.016666666666666666);
	tmp = 0.0;
	if (l <= -1.1e+58)
		tmp = t_1;
	elseif (l <= -850.0)
		tmp = t_0;
	elseif (l <= 0.35)
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 2.45e+61)
		tmp = t_0;
	else
		tmp = t_1;
	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[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.1e+58], t$95$1, If[LessEqual[l, -850.0], t$95$0, If[LessEqual[l, 0.35], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.45e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.1e58 or 2.45000000000000013e61 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in l around 0 99.1%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   5. Taylor expanded in l around inf 99.1%

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

      1. *-commutative 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l* 99.1%

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

   7. Simplified 99.1%

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

   if -1.1e58 < l < -850 or 0.34999999999999998 < l < 2.45000000000000013e61

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 74.2%

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

   if -850 < l < 0.34999999999999998

   1. Initial program 78.8%

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

   2. Taylor expanded in l around 0 98.8%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+58}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{5}\right) \cdot \left(J \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;\ell \leq -850:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.35:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.45 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{5}\right) \cdot \left(J \cdot 0.016666666666666666\right)\\ \end{array} \]

Alternative 6: 97.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := \left(t_1 \cdot {\ell}^{5}\right) \cdot \left(J \cdot 0.016666666666666666\right)\\ \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+58}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -850:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 0.0145:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t_1\right)\right)\\ \mathbf{elif}\;\ell \leq 4.45 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U))
        (t_1 (cos (* K 0.5)))
        (t_2 (* (* t_1 (pow l 5.0)) (* J 0.016666666666666666))))
   (if (<= l -1.02e+58)
     t_2
     (if (<= l -850.0)
       t_0
       (if (<= l 0.0145)
         (+ U (* 2.0 (* J (* l t_1))))
         (if (<= l 4.45e+61) t_0 t_2))))))

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 t_2 = (t_1 * pow(l, 5.0)) * (J * 0.016666666666666666);
	double tmp;
	if (l <= -1.02e+58) {
		tmp = t_2;
	} else if (l <= -850.0) {
		tmp = t_0;
	} else if (l <= 0.0145) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 4.45e+61) {
		tmp = t_0;
	} 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) :: tmp
    t_0 = ((exp(l) - exp(-l)) * j) + u
    t_1 = cos((k * 0.5d0))
    t_2 = (t_1 * (l ** 5.0d0)) * (j * 0.016666666666666666d0)
    if (l <= (-1.02d+58)) then
        tmp = t_2
    else if (l <= (-850.0d0)) then
        tmp = t_0
    else if (l <= 0.0145d0) then
        tmp = u + (2.0d0 * (j * (l * t_1)))
    else if (l <= 4.45d+61) then
        tmp = t_0
    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 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double t_1 = Math.cos((K * 0.5));
	double t_2 = (t_1 * Math.pow(l, 5.0)) * (J * 0.016666666666666666);
	double tmp;
	if (l <= -1.02e+58) {
		tmp = t_2;
	} else if (l <= -850.0) {
		tmp = t_0;
	} else if (l <= 0.0145) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 4.45e+61) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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))
	t_2 = (t_1 * math.pow(l, 5.0)) * (J * 0.016666666666666666)
	tmp = 0
	if l <= -1.02e+58:
		tmp = t_2
	elif l <= -850.0:
		tmp = t_0
	elif l <= 0.0145:
		tmp = U + (2.0 * (J * (l * t_1)))
	elif l <= 4.45e+61:
		tmp = t_0
	else:
		tmp = t_2
	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))
	t_2 = Float64(Float64(t_1 * (l ^ 5.0)) * Float64(J * 0.016666666666666666))
	tmp = 0.0
	if (l <= -1.02e+58)
		tmp = t_2;
	elseif (l <= -850.0)
		tmp = t_0;
	elseif (l <= 0.0145)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_1))));
	elseif (l <= 4.45e+61)
		tmp = t_0;
	else
		tmp = t_2;
	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));
	t_2 = (t_1 * (l ^ 5.0)) * (J * 0.016666666666666666);
	tmp = 0.0;
	if (l <= -1.02e+58)
		tmp = t_2;
	elseif (l <= -850.0)
		tmp = t_0;
	elseif (l <= 0.0145)
		tmp = U + (2.0 * (J * (l * t_1)));
	elseif (l <= 4.45e+61)
		tmp = t_0;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.02e+58], t$95$2, If[LessEqual[l, -850.0], t$95$0, If[LessEqual[l, 0.0145], N[(U + N[(2.0 * N[(J * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.45e+61], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.02000000000000005e58 or 4.45000000000000002e61 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in l around 0 99.1%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   5. Taylor expanded in l around inf 99.1%

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

      1. *-commutative 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l* 99.1%

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

   7. Simplified 99.1%

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

   if -1.02000000000000005e58 < l < -850 or 0.0145000000000000007 < l < 4.45000000000000002e61

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 74.2%

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

   if -850 < l < 0.0145000000000000007

   1. Initial program 78.8%

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

   2. Taylor expanded in l around 0 98.3%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.02 \cdot 10^{+58}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{5}\right) \cdot \left(J \cdot 0.016666666666666666\right)\\ \mathbf{elif}\;\ell \leq -850:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.0145:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.45 \cdot 10^{+61}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{5}\right) \cdot \left(J \cdot 0.016666666666666666\right)\\ \end{array} \]

Alternative 7: 87.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -850 \lor \neg \left(\ell \leq 0.0128\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -850.0) (not (<= l 0.0128)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -850.0) || !(l <= 0.0128)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	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 <= (-850.0d0)) .or. (.not. (l <= 0.0128d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -850.0) || !(l <= 0.0128)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -850.0) or not (l <= 0.0128):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -850.0) || !(l <= 0.0128))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -850.0) || ~((l <= 0.0128)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -850.0], N[Not[LessEqual[l, 0.0128]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -850 or 0.0128000000000000006 < 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. Taylor expanded in K around 0 74.1%

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

   if -850 < l < 0.0128000000000000006

   1. Initial program 78.8%

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

   2. Taylor expanded in l around 0 98.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -850 \lor \neg \left(\ell \leq 0.0128\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 8: 64.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{+33}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 980 \lor \neg \left(\ell \leq 1.7 \cdot 10^{+226}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))
   (if (<= l -5.2e+164)
     t_0
     (if (<= l -9.5e+33)
       (pow U -3.0)
       (if (or (<= l 980.0) (not (<= l 1.7e+226))) t_0 (log1p (expm1 U)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l * cos((K * 0.5)))));
	double tmp;
	if (l <= -5.2e+164) {
		tmp = t_0;
	} else if (l <= -9.5e+33) {
		tmp = pow(U, -3.0);
	} else if ((l <= 980.0) || !(l <= 1.7e+226)) {
		tmp = t_0;
	} else {
		tmp = log1p(expm1(U));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	double tmp;
	if (l <= -5.2e+164) {
		tmp = t_0;
	} else if (l <= -9.5e+33) {
		tmp = Math.pow(U, -3.0);
	} else if ((l <= 980.0) || !(l <= 1.7e+226)) {
		tmp = t_0;
	} else {
		tmp = Math.log1p(Math.expm1(U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	tmp = 0
	if l <= -5.2e+164:
		tmp = t_0
	elif l <= -9.5e+33:
		tmp = math.pow(U, -3.0)
	elif (l <= 980.0) or not (l <= 1.7e+226):
		tmp = t_0
	else:
		tmp = math.log1p(math.expm1(U))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))))
	tmp = 0.0
	if (l <= -5.2e+164)
		tmp = t_0;
	elseif (l <= -9.5e+33)
		tmp = U ^ -3.0;
	elseif ((l <= 980.0) || !(l <= 1.7e+226))
		tmp = t_0;
	else
		tmp = log1p(expm1(U));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.2e+164], t$95$0, If[LessEqual[l, -9.5e+33], N[Power[U, -3.0], $MachinePrecision], If[Or[LessEqual[l, 980.0], N[Not[LessEqual[l, 1.7e+226]], $MachinePrecision]], t$95$0, N[Log[1 + N[(Exp[U] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.1999999999999998e164 or -9.5000000000000003e33 < l < 980 or 1.69999999999999989e226 < l

   1. Initial program 84.9%

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

   2. Taylor expanded in l around 0 84.7%

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

   if -5.1999999999999998e164 < l < -9.5000000000000003e33

   1. Initial program 100.0%

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

   3. Applied egg-rr 35.6%

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

   if 980 < l < 1.69999999999999989e226

   1. Initial program 100.0%

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

   3. Applied egg-rr 37.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{+164}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{+33}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq 980 \lor \neg \left(\ell \leq 1.7 \cdot 10^{+226}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \end{array} \]

Alternative 9: 43.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 - U \cdot U\\ \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{+40}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq -820:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 7200000:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+119} \lor \neg \left(\ell \leq 8.2 \cdot 10^{+174}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;{U}^{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- -4.0 (* U U))))
   (if (<= l -2.05e+104)
     t_0
     (if (<= l -2.5e+40)
       (pow U -3.0)
       (if (<= l -820.0)
         t_0
         (if (<= l 7200000.0)
           U
           (if (or (<= l 2.9e+119) (not (<= l 8.2e+174)))
             (* U U)
             (pow U -3.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = -4.0 - (U * U);
	double tmp;
	if (l <= -2.05e+104) {
		tmp = t_0;
	} else if (l <= -2.5e+40) {
		tmp = pow(U, -3.0);
	} else if (l <= -820.0) {
		tmp = t_0;
	} else if (l <= 7200000.0) {
		tmp = U;
	} else if ((l <= 2.9e+119) || !(l <= 8.2e+174)) {
		tmp = U * U;
	} else {
		tmp = pow(U, -3.0);
	}
	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 = (-4.0d0) - (u * u)
    if (l <= (-2.05d+104)) then
        tmp = t_0
    else if (l <= (-2.5d+40)) then
        tmp = u ** (-3.0d0)
    else if (l <= (-820.0d0)) then
        tmp = t_0
    else if (l <= 7200000.0d0) then
        tmp = u
    else if ((l <= 2.9d+119) .or. (.not. (l <= 8.2d+174))) then
        tmp = u * u
    else
        tmp = u ** (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = -4.0 - (U * U);
	double tmp;
	if (l <= -2.05e+104) {
		tmp = t_0;
	} else if (l <= -2.5e+40) {
		tmp = Math.pow(U, -3.0);
	} else if (l <= -820.0) {
		tmp = t_0;
	} else if (l <= 7200000.0) {
		tmp = U;
	} else if ((l <= 2.9e+119) || !(l <= 8.2e+174)) {
		tmp = U * U;
	} else {
		tmp = Math.pow(U, -3.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = -4.0 - (U * U)
	tmp = 0
	if l <= -2.05e+104:
		tmp = t_0
	elif l <= -2.5e+40:
		tmp = math.pow(U, -3.0)
	elif l <= -820.0:
		tmp = t_0
	elif l <= 7200000.0:
		tmp = U
	elif (l <= 2.9e+119) or not (l <= 8.2e+174):
		tmp = U * U
	else:
		tmp = math.pow(U, -3.0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(-4.0 - Float64(U * U))
	tmp = 0.0
	if (l <= -2.05e+104)
		tmp = t_0;
	elseif (l <= -2.5e+40)
		tmp = U ^ -3.0;
	elseif (l <= -820.0)
		tmp = t_0;
	elseif (l <= 7200000.0)
		tmp = U;
	elseif ((l <= 2.9e+119) || !(l <= 8.2e+174))
		tmp = Float64(U * U);
	else
		tmp = U ^ -3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = -4.0 - (U * U);
	tmp = 0.0;
	if (l <= -2.05e+104)
		tmp = t_0;
	elseif (l <= -2.5e+40)
		tmp = U ^ -3.0;
	elseif (l <= -820.0)
		tmp = t_0;
	elseif (l <= 7200000.0)
		tmp = U;
	elseif ((l <= 2.9e+119) || ~((l <= 8.2e+174)))
		tmp = U * U;
	else
		tmp = U ^ -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.05e+104], t$95$0, If[LessEqual[l, -2.5e+40], N[Power[U, -3.0], $MachinePrecision], If[LessEqual[l, -820.0], t$95$0, If[LessEqual[l, 7200000.0], U, If[Or[LessEqual[l, 2.9e+119], N[Not[LessEqual[l, 8.2e+174]], $MachinePrecision]], N[(U * U), $MachinePrecision], N[Power[U, -3.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.04999999999999992e104 or -2.50000000000000002e40 < l < -820

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 29.6%

      \[\leadsto \color{blue}{-4 + \left(-U\right) \cdot U} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 29.6%

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

   7. Simplified 29.6%

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

   if -2.04999999999999992e104 < l < -2.50000000000000002e40 or 2.90000000000000007e119 < l < 8.20000000000000061e174

   1. Initial program 100.0%

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

   3. Applied egg-rr 42.4%

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

   if -820 < l < 7.2e6

   1. Initial program 79.0%

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

   2. Taylor expanded in J around 0 76.1%

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

   if 7.2e6 < l < 2.90000000000000007e119 or 8.20000000000000061e174 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{U \cdot U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+104}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq -2.5 \cdot 10^{+40}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{elif}\;\ell \leq -820:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq 7200000:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 2.9 \cdot 10^{+119} \lor \neg \left(\ell \leq 8.2 \cdot 10^{+174}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;{U}^{-3}\\ \end{array} \]

Alternative 10: 63.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

2. Taylor expanded in l around 0 61.6%

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

3. Final simplification 61.6%

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

Alternative 11: 44.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 - U \cdot U\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.55 \cdot 10^{+26}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq -800:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 7200000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- -4.0 (* U U))))
   (if (<= l -4.8e+104)
     t_0
     (if (<= l -1.55e+26)
       (pow U -4.0)
       (if (<= l -800.0) t_0 (if (<= l 7200000.0) U (* U U)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = -4.0 - (U * U);
	double tmp;
	if (l <= -4.8e+104) {
		tmp = t_0;
	} else if (l <= -1.55e+26) {
		tmp = pow(U, -4.0);
	} else if (l <= -800.0) {
		tmp = t_0;
	} else if (l <= 7200000.0) {
		tmp = U;
	} else {
		tmp = U * 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 = (-4.0d0) - (u * u)
    if (l <= (-4.8d+104)) then
        tmp = t_0
    else if (l <= (-1.55d+26)) then
        tmp = u ** (-4.0d0)
    else if (l <= (-800.0d0)) then
        tmp = t_0
    else if (l <= 7200000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = -4.0 - (U * U);
	double tmp;
	if (l <= -4.8e+104) {
		tmp = t_0;
	} else if (l <= -1.55e+26) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= -800.0) {
		tmp = t_0;
	} else if (l <= 7200000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = -4.0 - (U * U)
	tmp = 0
	if l <= -4.8e+104:
		tmp = t_0
	elif l <= -1.55e+26:
		tmp = math.pow(U, -4.0)
	elif l <= -800.0:
		tmp = t_0
	elif l <= 7200000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(-4.0 - Float64(U * U))
	tmp = 0.0
	if (l <= -4.8e+104)
		tmp = t_0;
	elseif (l <= -1.55e+26)
		tmp = U ^ -4.0;
	elseif (l <= -800.0)
		tmp = t_0;
	elseif (l <= 7200000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = -4.0 - (U * U);
	tmp = 0.0;
	if (l <= -4.8e+104)
		tmp = t_0;
	elseif (l <= -1.55e+26)
		tmp = U ^ -4.0;
	elseif (l <= -800.0)
		tmp = t_0;
	elseif (l <= 7200000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.8e+104], t$95$0, If[LessEqual[l, -1.55e+26], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, -800.0], t$95$0, If[LessEqual[l, 7200000.0], U, N[(U * U), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.8e104 or -1.55e26 < l < -800

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 30.3%

      \[\leadsto \color{blue}{-4 + \left(-U\right) \cdot U} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 30.3%

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

   7. Simplified 30.3%

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

   if -4.8e104 < l < -1.55e26

   1. Initial program 100.0%

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

   3. Applied egg-rr 39.6%

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

   if -800 < l < 7.2e6

   1. Initial program 79.0%

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

   2. Taylor expanded in J around 0 76.1%

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

   if 7.2e6 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{U \cdot U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+104}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq -1.55 \cdot 10^{+26}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq -800:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq 7200000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 12: 43.1% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+84} \lor \neg \left(\ell \leq 7200000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.8e+84) (not (<= l 7200000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.8e+84) || !(l <= 7200000.0)) {
		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 <= (-6.8d+84)) .or. (.not. (l <= 7200000.0d0))) 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 <= -6.8e+84) || !(l <= 7200000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.8e+84) or not (l <= 7200000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.8e+84) || !(l <= 7200000.0))
		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 <= -6.8e+84) || ~((l <= 7200000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.8e+84], N[Not[LessEqual[l, 7200000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -6.7999999999999996e84 or 7.2e6 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 20.5%

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

   if -6.7999999999999996e84 < l < 7.2e6

   1. Initial program 81.1%

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

   2. Taylor expanded in J around 0 68.4%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+84} \lor \neg \left(\ell \leq 7200000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 43.5% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -620:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq 7200000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -620.0) (- -4.0 (* U U)) (if (<= l 7200000.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -620.0) {
		tmp = -4.0 - (U * U);
	} else if (l <= 7200000.0) {
		tmp = U;
	} else {
		tmp = U * 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 <= (-620.0d0)) then
        tmp = (-4.0d0) - (u * u)
    else if (l <= 7200000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -620.0) {
		tmp = -4.0 - (U * U);
	} else if (l <= 7200000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -620.0:
		tmp = -4.0 - (U * U)
	elif l <= 7200000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -620.0)
		tmp = Float64(-4.0 - Float64(U * U));
	elseif (l <= 7200000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -620.0)
		tmp = -4.0 - (U * U);
	elseif (l <= 7200000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -620.0], N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7200000.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -620

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 25.3%

      \[\leadsto \color{blue}{-4 + \left(-U\right) \cdot U} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv 25.3%

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

   7. Simplified 25.3%

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

   if -620 < l < 7.2e6

   1. Initial program 79.0%

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

   2. Taylor expanded in J around 0 76.1%

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

   if 7.2e6 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 24.1%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -620:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq 7200000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 14: 37.8% 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 90.0%

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

2. Taylor expanded in J around 0 37.6%

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

3. Final simplification 37.6%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023321 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))