Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.6%

Time: 12.9s

Alternatives: 14

Speedup: 1.4×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. J = 3.037118322456653e+94
  2. l = 1.9680373771648034e-23
  3. K = -1.4713660546846983e-55
  4. U = 2.5859466830668957e-275

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

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

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 2e-10)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (* 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 <= 2e-10)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (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 <= 2e-10)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (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 <= 2e-10):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (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 <= 2e-10))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * 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 <= 2e-10)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (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, 2e-10]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * 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 2.00000000000000007e-10 < (-.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))) < 2.00000000000000007e-10

   1. Initial program 71.3%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

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

Alternative 2: 78.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.48:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;t\_0 \leq -0.01:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.48)
     (+ U (* 2.0 (* J (+ l (* -0.125 (* l (pow K 2.0)))))))
     (if (<= t_0 -0.01)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (+ U (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.48) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * pow(K, 2.0))))));
	} else if (t_0 <= -0.01) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (l * J)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.48d0)) then
        tmp = u + (2.0d0 * (j * (l + ((-0.125d0) * (l * (k ** 2.0d0))))))
    else if (t_0 <= (-0.01d0)) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + ((0.3333333333333333d0 * (j * (l ** 3.0d0))) + (2.0d0 * (l * j)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.48) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * Math.pow(K, 2.0))))));
	} else if (t_0 <= -0.01) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.48:
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * math.pow(K, 2.0))))))
	elif t_0 <= -0.01:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.48)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * (K ^ 2.0)))))));
	elseif (t_0 <= -0.01)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.48)
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K ^ 2.0))))));
	elseif (t_0 <= -0.01)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.48], N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.47999999999999998

   1. Initial program 90.2%

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

   3. Taylor expanded in l around 0 37.8%

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

   4. Taylor expanded in K around 0 64.5%

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

   if -0.47999999999999998 < (cos.f64 (/.f64 K 2)) < -0.0100000000000000002

   1. Initial program 89.8%

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

   3. Taylor expanded in l around 0 74.9%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0 87.8%

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

      1. +-commutative 87.8%

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

      2. associate-*r* 87.8%

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

      3. associate-*r* 87.8%

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

      4. associate-*r* 87.8%

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

      5. associate-*r* 87.8%

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

      6. distribute-rgt-out 87.8%

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

      7. associate-*r* 87.8%

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

      8. *-commutative 87.8%

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

      9. *-commutative 87.8%

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

   5. Simplified 87.8%

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

   6. Taylor expanded in K around 0 83.8%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.48:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot {K}^{2}\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -122:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 58000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+102}:\\ \;\;\;\;t\_1 + U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (* J 0.3333333333333333) (* (pow l 3.0) (cos (* K 0.5))))))
        (t_1 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -5.7e+102)
     t_0
     (if (<= l -122.0)
       t_1
       (if (<= l 58000000000.0)
         (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
         (if (<= l 1.25e+102) (+ t_1 U) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * 0.3333333333333333) * (pow(l, 3.0) * cos((K * 0.5))));
	double t_1 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -5.7e+102) {
		tmp = t_0;
	} else if (l <= -122.0) {
		tmp = t_1;
	} else if (l <= 58000000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.25e+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 + ((j * 0.3333333333333333d0) * ((l ** 3.0d0) * cos((k * 0.5d0))))
    t_1 = (exp(l) - exp(-l)) * j
    if (l <= (-5.7d+102)) then
        tmp = t_0
    else if (l <= (-122.0d0)) then
        tmp = t_1
    else if (l <= 58000000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else if (l <= 1.25d+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 + ((J * 0.3333333333333333) * (Math.pow(l, 3.0) * Math.cos((K * 0.5))));
	double t_1 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -5.7e+102) {
		tmp = t_0;
	} else if (l <= -122.0) {
		tmp = t_1;
	} else if (l <= 58000000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.25e+102) {
		tmp = t_1 + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * 0.3333333333333333) * (math.pow(l, 3.0) * math.cos((K * 0.5))))
	t_1 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -5.7e+102:
		tmp = t_0
	elif l <= -122.0:
		tmp = t_1
	elif l <= 58000000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	elif l <= 1.25e+102:
		tmp = t_1 + U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * 0.3333333333333333) * Float64((l ^ 3.0) * cos(Float64(K * 0.5)))))
	t_1 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -5.7e+102)
		tmp = t_0;
	elseif (l <= -122.0)
		tmp = t_1;
	elseif (l <= 58000000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif (l <= 1.25e+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 + ((J * 0.3333333333333333) * ((l ^ 3.0) * cos((K * 0.5))));
	t_1 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -5.7e+102)
		tmp = t_0;
	elseif (l <= -122.0)
		tmp = t_1;
	elseif (l <= 58000000000.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	elseif (l <= 1.25e+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[(J * 0.3333333333333333), $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * N[Cos[N[(K * 0.5), $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, -5.7e+102], t$95$0, If[LessEqual[l, -122.0], t$95$1, If[LessEqual[l, 58000000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+102], N[(t$95$1 + U), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.6999999999999999e102 or 1.25e102 < 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%

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

      1. +-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*r* 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

      6. distribute-rgt-out 100.0%

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

      7. associate-*r* 100.0%

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

      8. *-commutative 100.0%

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

      9. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

   8. Simplified 100.0%

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

   if -5.6999999999999999e102 < l < -122

   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 81.8%

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

   4. Taylor expanded in J around inf 81.8%

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

   if -122 < l < 5.8e10

   1. Initial program 71.7%

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

   3. Taylor expanded in l around 0 99.1%

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

   if 5.8e10 < l < 1.25e102

   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 72.7%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+102}:\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -122:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 58000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -122:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 58000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -122.0)
     t_0
     (if (<= l 58000000000.0)
       (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
       (+ 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 <= -122.0) {
		tmp = t_0;
	} else if (l <= 58000000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} 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 <= (-122.0d0)) then
        tmp = t_0
    else if (l <= 58000000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    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 <= -122.0) {
		tmp = t_0;
	} else if (l <= 58000000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} 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 <= -122.0:
		tmp = t_0
	elif l <= 58000000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	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 <= -122.0)
		tmp = t_0;
	elseif (l <= 58000000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	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 <= -122.0)
		tmp = t_0;
	elseif (l <= 58000000000.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	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, -122.0], t$95$0, If[LessEqual[l, 58000000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -122

   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 71.9%

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

   4. Taylor expanded in J around inf 71.9%

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

   if -122 < l < 5.8e10

   1. Initial program 71.7%

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

   3. Taylor expanded in l around 0 99.1%

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

   if 5.8e10 < 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 68.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -122:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 58000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -122 \lor \neg \left(\ell \leq 58000000000\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -122.0) (not (<= l 58000000000.0)))
   (* (- (exp l) (exp (- l))) J)
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -122.0) || !(l <= 58000000000.0)) {
		tmp = (exp(l) - exp(-l)) * J;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (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 ((l <= (-122.0d0)) .or. (.not. (l <= 58000000000.0d0))) then
        tmp = (exp(l) - exp(-l)) * j
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (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 ((l <= -122.0) || !(l <= 58000000000.0)) {
		tmp = (Math.exp(l) - Math.exp(-l)) * J;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -122.0) or not (l <= 58000000000.0):
		tmp = (math.exp(l) - math.exp(-l)) * J
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -122.0) || !(l <= 58000000000.0))
		tmp = Float64(Float64(exp(l) - exp(Float64(-l))) * J);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -122.0) || ~((l <= 58000000000.0)))
		tmp = (exp(l) - exp(-l)) * J;
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -122.0], N[Not[LessEqual[l, 58000000000.0]], $MachinePrecision]], N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -122 or 5.8e10 < 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 70.1%

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

   4. Taylor expanded in J around inf 70.1%

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

   if -122 < l < 5.8e10

   1. Initial program 71.7%

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

   3. Taylor expanded in l around 0 99.1%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -122 \lor \neg \left(\ell \leq 58000000000\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+284} \lor \neg \left(\ell \leq -3 \cdot 10^{+20} \lor \neg \left(\ell \leq 1.3 \cdot 10^{+69}\right) \land \ell \leq 1.55 \cdot 10^{+179}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.4e+284)
         (not (or (<= l -3e+20) (and (not (<= l 1.3e+69)) (<= l 1.55e+179)))))
   (+ U (* l (* J 2.0)))
   (pow U -4.0)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.4e+284) || !((l <= -3e+20) || (!(l <= 1.3e+69) && (l <= 1.55e+179)))) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = pow(U, -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.4d+284)) .or. (.not. (l <= (-3d+20)) .or. (.not. (l <= 1.3d+69)) .and. (l <= 1.55d+179))) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u ** (-4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.4e+284) || !((l <= -3e+20) || (!(l <= 1.3e+69) && (l <= 1.55e+179)))) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = Math.pow(U, -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.4e+284) or not ((l <= -3e+20) or (not (l <= 1.3e+69) and (l <= 1.55e+179))):
		tmp = U + (l * (J * 2.0))
	else:
		tmp = math.pow(U, -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.4e+284) || !((l <= -3e+20) || (!(l <= 1.3e+69) && (l <= 1.55e+179))))
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = U ^ -4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.4e+284) || ~(((l <= -3e+20) || (~((l <= 1.3e+69)) && (l <= 1.55e+179)))))
		tmp = U + (l * (J * 2.0));
	else
		tmp = U ^ -4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.4e+284], N[Not[Or[LessEqual[l, -3e+20], And[N[Not[LessEqual[l, 1.3e+69]], $MachinePrecision], LessEqual[l, 1.55e+179]]]], $MachinePrecision]], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -4.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.4000000000000001e284 or -3e20 < l < 1.3000000000000001e69 or 1.55e179 < l

   1. Initial program 80.9%

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

   3. Taylor expanded in l around 0 77.7%

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

   4. Taylor expanded in K around 0 67.6%

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

      1. associate-*r* 67.6%

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

      2. *-commutative 67.6%

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

      3. *-commutative 67.6%

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

   6. Simplified 67.6%

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

   if -2.4000000000000001e284 < l < -3e20 or 1.3000000000000001e69 < l < 1.55e179

   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 33.4%

      \[\leadsto \color{blue}{{U}^{-4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+284} \lor \neg \left(\ell \leq -3 \cdot 10^{+20} \lor \neg \left(\ell \leq 1.3 \cdot 10^{+69}\right) \land \ell \leq 1.55 \cdot 10^{+179}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-4}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+283}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{+20}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+178}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* l (* J 2.0)))))
   (if (<= l -1e+283)
     t_0
     (if (<= l -3e+20)
       (pow U -4.0)
       (if (<= l 1.2e+71)
         (fma l (* J 2.0) U)
         (if (<= l 7e+178) (pow U -4.0) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (l * (J * 2.0));
	double tmp;
	if (l <= -1e+283) {
		tmp = t_0;
	} else if (l <= -3e+20) {
		tmp = pow(U, -4.0);
	} else if (l <= 1.2e+71) {
		tmp = fma(l, (J * 2.0), U);
	} else if (l <= 7e+178) {
		tmp = pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(l * Float64(J * 2.0)))
	tmp = 0.0
	if (l <= -1e+283)
		tmp = t_0;
	elseif (l <= -3e+20)
		tmp = U ^ -4.0;
	elseif (l <= 1.2e+71)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif (l <= 7e+178)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+283], t$95$0, If[LessEqual[l, -3e+20], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 1.2e+71], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 7e+178], N[Power[U, -4.0], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999955e282 or 7.00000000000000001e178 < 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 44.3%

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

   4. Taylor expanded in K around 0 36.4%

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

      1. associate-*r* 36.4%

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

      2. *-commutative 36.4%

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

      3. *-commutative 36.4%

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

   6. Simplified 36.4%

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

   if -9.99999999999999955e282 < l < -3e20 or 1.1999999999999999e71 < l < 7.00000000000000001e178

   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 33.4%

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

   if -3e20 < l < 1.1999999999999999e71

   1. Initial program 75.1%

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

   3. Taylor expanded in l around 0 87.8%

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

   4. Taylor expanded in K around 0 77.1%

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

      1. +-commutative 77.1%

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

      2. associate-*r* 77.1%

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

      3. *-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. fma-define 77.1%

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

   6. Simplified 77.1%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+283}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{+20}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{+178}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -18000 \lor \neg \left(\ell \leq 2.2 \cdot 10^{+78}\right):\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -18000.0) (not (<= l 2.2e+78)))
   (+ U (* (* J 0.3333333333333333) (pow l 3.0)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -18000.0) || !(l <= 2.2e+78)) {
		tmp = U + ((J * 0.3333333333333333) * pow(l, 3.0));
	} 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 <= (-18000.0d0)) .or. (.not. (l <= 2.2d+78))) then
        tmp = u + ((j * 0.3333333333333333d0) * (l ** 3.0d0))
    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 <= -18000.0) || !(l <= 2.2e+78)) {
		tmp = U + ((J * 0.3333333333333333) * Math.pow(l, 3.0));
	} 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 <= -18000.0) or not (l <= 2.2e+78):
		tmp = U + ((J * 0.3333333333333333) * math.pow(l, 3.0))
	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 <= -18000.0) || !(l <= 2.2e+78))
		tmp = Float64(U + Float64(Float64(J * 0.3333333333333333) * (l ^ 3.0)));
	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 <= -18000.0) || ~((l <= 2.2e+78)))
		tmp = U + ((J * 0.3333333333333333) * (l ^ 3.0));
	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, -18000.0], N[Not[LessEqual[l, 2.2e+78]], $MachinePrecision]], N[(U + N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $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 < -18000 or 2.20000000000000014e78 < 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 84.5%

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

      1. +-commutative 84.5%

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

      2. associate-*r* 84.5%

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

      3. associate-*r* 84.5%

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

      4. associate-*r* 84.5%

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

      5. associate-*r* 84.5%

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

      6. distribute-rgt-out 84.5%

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

      7. associate-*r* 84.5%

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

      8. *-commutative 84.5%

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

      9. *-commutative 84.5%

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

   5. Simplified 84.5%

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

   6. Taylor expanded in l around inf 84.5%

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

      1. associate-*r* 84.5%

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

      2. *-commutative 84.5%

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

      3. *-commutative 84.5%

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

   8. Simplified 84.5%

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

   9. Taylor expanded in K around 0 58.7%

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

   if -18000 < l < 2.20000000000000014e78

   1. Initial program 74.5%

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

   3. Taylor expanded in l around 0 89.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -18000 \lor \neg \left(\ell \leq 2.2 \cdot 10^{+78}\right):\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -120000 \lor \neg \left(\ell \leq 2.25 \cdot 10^{+78}\right):\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -120000.0) (not (<= l 2.25e+78)))
   (+ U (* (* J 0.3333333333333333) (pow l 3.0)))
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -120000.0) || !(l <= 2.25e+78)) {
		tmp = U + ((J * 0.3333333333333333) * pow(l, 3.0));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (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 ((l <= (-120000.0d0)) .or. (.not. (l <= 2.25d+78))) then
        tmp = u + ((j * 0.3333333333333333d0) * (l ** 3.0d0))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (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 ((l <= -120000.0) || !(l <= 2.25e+78)) {
		tmp = U + ((J * 0.3333333333333333) * Math.pow(l, 3.0));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -120000.0) or not (l <= 2.25e+78):
		tmp = U + ((J * 0.3333333333333333) * math.pow(l, 3.0))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -120000.0) || !(l <= 2.25e+78))
		tmp = Float64(U + Float64(Float64(J * 0.3333333333333333) * (l ^ 3.0)));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -120000.0) || ~((l <= 2.25e+78)))
		tmp = U + ((J * 0.3333333333333333) * (l ^ 3.0));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -120000.0], N[Not[LessEqual[l, 2.25e+78]], $MachinePrecision]], N[(U + N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.2e5 or 2.25e78 < 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 84.5%

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

      1. +-commutative 84.5%

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

      2. associate-*r* 84.5%

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

      3. associate-*r* 84.5%

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

      4. associate-*r* 84.5%

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

      5. associate-*r* 84.5%

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

      6. distribute-rgt-out 84.5%

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

      7. associate-*r* 84.5%

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

      8. *-commutative 84.5%

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

      9. *-commutative 84.5%

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

   5. Simplified 84.5%

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

   6. Taylor expanded in l around inf 84.5%

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

      1. associate-*r* 84.5%

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

      2. *-commutative 84.5%

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

      3. *-commutative 84.5%

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

   8. Simplified 84.5%

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

   9. Taylor expanded in K around 0 58.7%

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

   if -1.2e5 < l < 2.25e78

   1. Initial program 74.5%

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

   3. Taylor expanded in l around 0 89.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -120000 \lor \neg \left(\ell \leq 2.25 \cdot 10^{+78}\right):\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 0.00017\right):\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.5) (not (<= l 0.00017)))
   (+ U (* (* J 0.3333333333333333) (pow l 3.0)))
   (fma l (* J 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5) || !(l <= 0.00017)) {
		tmp = U + ((J * 0.3333333333333333) * pow(l, 3.0));
	} else {
		tmp = fma(l, (J * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.5) || !(l <= 0.00017))
		tmp = Float64(U + Float64(Float64(J * 0.3333333333333333) * (l ^ 3.0)));
	else
		tmp = fma(l, Float64(J * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.5], N[Not[LessEqual[l, 0.00017]], $MachinePrecision]], N[(U + N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.5 or 1.7e-4 < 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 76.6%

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

      1. +-commutative 76.6%

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

      2. associate-*r* 76.6%

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

      3. associate-*r* 76.6%

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

      4. associate-*r* 76.6%

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

      5. associate-*r* 76.6%

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

      6. distribute-rgt-out 76.6%

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

      7. associate-*r* 76.6%

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

      8. *-commutative 76.6%

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

      9. *-commutative 76.6%

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

   5. Simplified 76.6%

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

   6. Taylor expanded in l around inf 76.6%

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

      1. associate-*r* 76.6%

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

      2. *-commutative 76.6%

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

      3. *-commutative 76.6%

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

   8. Simplified 76.6%

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

   9. Taylor expanded in K around 0 53.4%

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

   if -2.5 < l < 1.7e-4

   1. Initial program 71.3%

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

   3. Taylor expanded in l around 0 99.9%

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

   4. Taylor expanded in K around 0 87.6%

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

      1. +-commutative 87.6%

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

      2. associate-*r* 87.6%

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

      3. *-commutative 87.6%

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

      4. *-commutative 87.6%

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

      5. fma-define 87.6%

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

   6. Simplified 87.6%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 0.00017\right):\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot {\ell}^{3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.1% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -230000 \lor \neg \left(\ell \leq 780\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -230000.0) (not (<= l 780.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -230000.0) || !(l <= 780.0)) {
		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 <= (-230000.0d0)) .or. (.not. (l <= 780.0d0))) 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 <= -230000.0) || !(l <= 780.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -230000.0) or not (l <= 780.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -230000.0) || !(l <= 780.0))
		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 <= -230000.0) || ~((l <= 780.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -230000.0], N[Not[LessEqual[l, 780.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -2.3e5 or 780 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 16.8%

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

   if -2.3e5 < l < 780

   1. Initial program 71.7%

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

   3. Taylor expanded in J around 0 70.2%

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

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -230000 \lor \neg \left(\ell \leq 780\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.0% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in l around 0 60.0%

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

4. Taylor expanded in K around 0 50.5%

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

   1. associate-*r* 50.5%

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

   2. *-commutative 50.5%

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

   3. *-commutative 50.5%

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

6. Simplified 50.5%

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

7. Final simplification 50.5%

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

Alternative 13: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

4. Applied egg-rr 2.7%

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

   1. *-inverses 2.7%

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

6. Simplified 2.7%

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

7. Final simplification 2.7%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 14: 36.6% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Taylor expanded in J around 0 34.0%

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

4. Final simplification 34.0%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))