Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.1% → 99.5%

Time: 12.8s

Alternatives: 17

Speedup: 1.4×

• 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: 86.1% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* 2.0 (* J (log1p (expm1 (* l (cos (* K 0.5))))))) U))

C

double code(double J, double l, double K, double U) {
	return (2.0 * (J * log1p(expm1((l * cos((K * 0.5))))))) + U;
}

Java

public static double code(double J, double l, double K, double U) {
	return (2.0 * (J * Math.log1p(Math.expm1((l * Math.cos((K * 0.5))))))) + U;
}

Python

def code(J, l, K, U):
	return (2.0 * (J * math.log1p(math.expm1((l * math.cos((K * 0.5))))))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(2.0 * Float64(J * log1p(expm1(Float64(l * cos(Float64(K * 0.5))))))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(2.0 * N[(J * N[Log[1 + N[(Exp[N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in l around 0 65.8%

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

   1. log1p-expm1-u 98.7%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}\right) + U \]

   2. *-commutative 98.7%

      \[\leadsto 2 \cdot \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right)\right)\right) + U \]

5. Applied egg-rr 98.7%

   \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)}\right) + U \]
6. Add Preprocessing

Alternative 2: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+84} \lor \neg \left(\ell \leq -0.0126 \lor \neg \left(\ell \leq 2\right) \land \ell \leq 9.8 \cdot 10^{+89}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.8e+84)
         (not (or (<= l -0.0126) (and (not (<= l 2.0)) (<= l 9.8e+89)))))
   (+
    U
    (* J (* l (* (cos (* K 0.5)) (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
   (+ U (* J (- (exp l) (exp (- l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.8e+84) || !((l <= -0.0126) || (!(l <= 2.0) && (l <= 9.8e+89)))) {
		tmp = U + (J * (l * (cos((K * 0.5)) * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else {
		tmp = U + (J * (exp(l) - exp(-l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-6.8d+84)) .or. (.not. (l <= (-0.0126d0)) .or. (.not. (l <= 2.0d0)) .and. (l <= 9.8d+89))) then
        tmp = u + (j * (l * (cos((k * 0.5d0)) * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    else
        tmp = u + (j * (exp(l) - exp(-l)))
    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 <= -0.0126) || (!(l <= 2.0) && (l <= 9.8e+89)))) {
		tmp = U + (J * (l * (Math.cos((K * 0.5)) * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.8e+84) or not ((l <= -0.0126) or (not (l <= 2.0) and (l <= 9.8e+89))):
		tmp = U + (J * (l * (math.cos((K * 0.5)) * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	else:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.8e+84) || !((l <= -0.0126) || (!(l <= 2.0) && (l <= 9.8e+89))))
		tmp = Float64(U + Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -6.8e+84) || ~(((l <= -0.0126) || (~((l <= 2.0)) && (l <= 9.8e+89)))))
		tmp = U + (J * (l * (cos((K * 0.5)) * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	else
		tmp = U + (J * (exp(l) - exp(-l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.8e+84], N[Not[Or[LessEqual[l, -0.0126], And[N[Not[LessEqual[l, 2.0]], $MachinePrecision], LessEqual[l, 9.8e+89]]]], $MachinePrecision]], N[(U + N[(J * N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.7999999999999996e84 or -0.0126 < l < 2 or 9.79999999999999992e89 < l

   1. Initial program 79.5%

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

   3. Taylor expanded in l around 0 98.3%

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

   4. Taylor expanded in J around 0 98.2%

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

   if -6.7999999999999996e84 < l < -0.0126 or 2 < l < 9.79999999999999992e89

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 82.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+84} \lor \neg \left(\ell \leq -0.0126 \lor \neg \left(\ell \leq 2\right) \land \ell \leq 9.8 \cdot 10^{+89}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 2 + 0.3333333333333333 \cdot {\ell}^{2}\\ t_1 := U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot t\_0\right)\right)\\ t_2 := U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 2:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot t\_0\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+89}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))
        (t_1 (+ U (* J (* l (* (cos (* K 0.5)) t_0)))))
        (t_2 (+ U (* J (- (exp l) (exp (- l)))))))
   (if (<= l -6.8e+84)
     t_1
     (if (<= l -0.05)
       t_2
       (if (<= l 2.0)
         (+ U (* (* J (* l t_0)) (cos (/ K 2.0))))
         (if (<= l 8.5e+89) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 2.0 + (0.3333333333333333 * pow(l, 2.0));
	double t_1 = U + (J * (l * (cos((K * 0.5)) * t_0)));
	double t_2 = U + (J * (exp(l) - exp(-l)));
	double tmp;
	if (l <= -6.8e+84) {
		tmp = t_1;
	} else if (l <= -0.05) {
		tmp = t_2;
	} else if (l <= 2.0) {
		tmp = U + ((J * (l * t_0)) * cos((K / 2.0)));
	} else if (l <= 8.5e+89) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))
    t_1 = u + (j * (l * (cos((k * 0.5d0)) * t_0)))
    t_2 = u + (j * (exp(l) - exp(-l)))
    if (l <= (-6.8d+84)) then
        tmp = t_1
    else if (l <= (-0.05d0)) then
        tmp = t_2
    else if (l <= 2.0d0) then
        tmp = u + ((j * (l * t_0)) * cos((k / 2.0d0)))
    else if (l <= 8.5d+89) then
        tmp = t_2
    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 = 2.0 + (0.3333333333333333 * Math.pow(l, 2.0));
	double t_1 = U + (J * (l * (Math.cos((K * 0.5)) * t_0)));
	double t_2 = U + (J * (Math.exp(l) - Math.exp(-l)));
	double tmp;
	if (l <= -6.8e+84) {
		tmp = t_1;
	} else if (l <= -0.05) {
		tmp = t_2;
	} else if (l <= 2.0) {
		tmp = U + ((J * (l * t_0)) * Math.cos((K / 2.0)));
	} else if (l <= 8.5e+89) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 2.0 + (0.3333333333333333 * math.pow(l, 2.0))
	t_1 = U + (J * (l * (math.cos((K * 0.5)) * t_0)))
	t_2 = U + (J * (math.exp(l) - math.exp(-l)))
	tmp = 0
	if l <= -6.8e+84:
		tmp = t_1
	elif l <= -0.05:
		tmp = t_2
	elif l <= 2.0:
		tmp = U + ((J * (l * t_0)) * math.cos((K / 2.0)))
	elif l <= 8.5e+89:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))
	t_1 = Float64(U + Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) * t_0))))
	t_2 = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))))
	tmp = 0.0
	if (l <= -6.8e+84)
		tmp = t_1;
	elseif (l <= -0.05)
		tmp = t_2;
	elseif (l <= 2.0)
		tmp = Float64(U + Float64(Float64(J * Float64(l * t_0)) * cos(Float64(K / 2.0))));
	elseif (l <= 8.5e+89)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 2.0 + (0.3333333333333333 * (l ^ 2.0));
	t_1 = U + (J * (l * (cos((K * 0.5)) * t_0)));
	t_2 = U + (J * (exp(l) - exp(-l)));
	tmp = 0.0;
	if (l <= -6.8e+84)
		tmp = t_1;
	elseif (l <= -0.05)
		tmp = t_2;
	elseif (l <= 2.0)
		tmp = U + ((J * (l * t_0)) * cos((K / 2.0)));
	elseif (l <= 8.5e+89)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(J * N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.8e+84], t$95$1, If[LessEqual[l, -0.05], t$95$2, If[LessEqual[l, 2.0], N[(U + N[(N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.5e+89], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.7999999999999996e84 or 8.50000000000000045e89 < 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 97.0%

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

   4. Taylor expanded in J around 0 97.0%

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

   if -6.7999999999999996e84 < l < -0.050000000000000003 or 2 < l < 8.50000000000000045e89

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 82.9%

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

   if -0.050000000000000003 < l < 2

   1. Initial program 63.5%

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

   3. Taylor expanded in l around 0 99.3%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.8 \cdot 10^{+84}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.05:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{+89}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.997:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.997)
   (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.997) {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.997d0) then
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.997) {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.997:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.997)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.997)
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.997], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.996999999999999997

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0 65.4%

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

      1. *-commutative 65.4%

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

      2. associate-*r* 65.5%

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

      3. associate-*l* 65.5%

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

      4. *-commutative 65.5%

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

      5. *-commutative 65.5%

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

      6. associate-*l* 65.5%

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

      7. *-commutative 65.5%

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

   5. Simplified 65.5%

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

   if 0.996999999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.2%

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

   3. Taylor expanded in l around 0 84.2%

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

   4. Taylor expanded in K around 0 84.1%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.997:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.0007 \lor \neg \left(\ell \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.0007) (not (<= l 1.9e-6)))
   (+ U (* J (- (exp l) (exp (- l)))))
   (+ U (* (cos (/ K 2.0)) (* l (* 2.0 J))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0007) || !(l <= 1.9e-6)) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.0007d0)) .or. (.not. (l <= 1.9d-6))) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else
        tmp = u + (cos((k / 2.0d0)) * (l * (2.0d0 * j)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0007) || !(l <= 1.9e-6)) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (l * (2.0 * J)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.0007) or not (l <= 1.9e-6):
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = U + (math.cos((K / 2.0)) * (l * (2.0 * J)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.0007) || !(l <= 1.9e-6))
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(2.0 * J))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.0007) || ~((l <= 1.9e-6)))
		tmp = U + (J * (exp(l) - exp(-l)));
	else
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.0007], N[Not[LessEqual[l, 1.9e-6]], $MachinePrecision]], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.99999999999999993e-4 or 1.9e-6 < l

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 72.5%

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

   if -6.99999999999999993e-4 < l < 1.9e-6

   1. Initial program 62.4%

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

   3. Taylor expanded in l around 0 99.7%

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

      1. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.0007 \lor \neg \left(\ell \leq 1.9 \cdot 10^{-6}\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -1.1 \cdot 10^{+24}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 1900000000:\\ \;\;\;\;\mathsf{fma}\left(J, 2 \cdot \ell, U\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{+82}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
   (if (<= l -1.7e+70)
     t_0
     (if (<= l -1.1e+24)
       (* U (- U -4.0))
       (if (<= l 1900000000.0)
         (fma J (* 2.0 l) U)
         (if (<= l 8.4e+82) (pow U -4.0) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -1.7e+70) {
		tmp = t_0;
	} else if (l <= -1.1e+24) {
		tmp = U * (U - -4.0);
	} else if (l <= 1900000000.0) {
		tmp = fma(J, (2.0 * l), U);
	} else if (l <= 8.4e+82) {
		tmp = pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -1.7e+70)
		tmp = t_0;
	elseif (l <= -1.1e+24)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 1900000000.0)
		tmp = fma(J, Float64(2.0 * l), U);
	elseif (l <= 8.4e+82)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.7e+70], t$95$0, If[LessEqual[l, -1.1e+24], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1900000000.0], N[(J * N[(2.0 * l), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 8.4e+82], N[Power[U, -4.0], $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.7e70 or 8.4000000000000001e82 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 94.2%

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

   4. Taylor expanded in K around 0 66.5%

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

   5. Taylor expanded in l around inf 66.5%

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

      1. *-commutative 66.5%

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

      2. *-commutative 66.5%

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

      3. associate-*r* 66.5%

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

   7. Simplified 66.5%

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

   if -1.7e70 < l < -1.10000000000000001e24

   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 43.0%

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

   if -1.10000000000000001e24 < l < 1.9e9

   1. Initial program 66.0%

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

   3. Taylor expanded in l around 0 92.2%

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

   4. Taylor expanded in K around 0 75.3%

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

      1. +-commutative 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*r* 75.3%

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

      4. fma-define 75.3%

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

      5. *-commutative 75.3%

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

   6. Simplified 75.3%

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

   if 1.9e9 < l < 8.4000000000000001e82

   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 44.6%

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

Alternative 7: 75.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \frac{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \ell\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 5e-5)
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))
   (* U (+ 1.0 (* 2.0 (/ (* (cos (* K 0.5)) (* J l)) U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-5) {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	} else {
		tmp = U * (1.0 + (2.0 * ((cos((K * 0.5)) * (J * l)) / U)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 5d-5) then
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    else
        tmp = u * (1.0d0 + (2.0d0 * ((cos((k * 0.5d0)) * (j * l)) / u)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 5e-5) {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	} else {
		tmp = U * (1.0 + (2.0 * ((Math.cos((K * 0.5)) * (J * l)) / U)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 5e-5:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	else:
		tmp = U * (1.0 + (2.0 * ((math.cos((K * 0.5)) * (J * l)) / U)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 5e-5)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(Float64(cos(Float64(K * 0.5)) * Float64(J * l)) / U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 5e-5)
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	else
		tmp = U * (1.0 + (2.0 * ((cos((K * 0.5)) * (J * l)) / U)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 5e-5], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(2.0 * N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * l), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 5.00000000000000024e-5

   1. Initial program 83.4%

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

   3. Taylor expanded in l around 0 84.0%

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

   4. Taylor expanded in K around 0 72.0%

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

   if 5.00000000000000024e-5 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 80.4%

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

   3. Taylor expanded in l around 0 64.2%

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

   4. Taylor expanded in U around inf 65.9%

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

      1. associate-*r* 65.9%

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

   6. Simplified 65.9%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \frac{\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \ell\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1900000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
   (if (<= l -6e+94)
     t_0
     (if (<= l 1900000000.0)
       (+ U (* (cos (/ K 2.0)) (* l (* 2.0 J))))
       (if (<= l 9.5e+76) (pow U -4.0) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -6e+94) {
		tmp = t_0;
	} else if (l <= 1900000000.0) {
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	} else if (l <= 9.5e+76) {
		tmp = pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l ** 3.0d0) * (j * 0.3333333333333333d0)
    if (l <= (-6d+94)) then
        tmp = t_0
    else if (l <= 1900000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * (2.0d0 * j)))
    else if (l <= 9.5d+76) then
        tmp = u ** (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -6e+94) {
		tmp = t_0;
	} else if (l <= 1900000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (2.0 * J)));
	} else if (l <= 9.5e+76) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * (J * 0.3333333333333333)
	tmp = 0
	if l <= -6e+94:
		tmp = t_0
	elif l <= 1900000000.0:
		tmp = U + (math.cos((K / 2.0)) * (l * (2.0 * J)))
	elif l <= 9.5e+76:
		tmp = math.pow(U, -4.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -6e+94)
		tmp = t_0;
	elseif (l <= 1900000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(2.0 * J))));
	elseif (l <= 9.5e+76)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * (J * 0.3333333333333333);
	tmp = 0.0;
	if (l <= -6e+94)
		tmp = t_0;
	elseif (l <= 1900000000.0)
		tmp = U + (cos((K / 2.0)) * (l * (2.0 * J)));
	elseif (l <= 9.5e+76)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6e+94], t$95$0, If[LessEqual[l, 1900000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.5e+76], N[Power[U, -4.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.0000000000000001e94 or 9.5000000000000003e76 < 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 96.9%

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

   4. Taylor expanded in K around 0 68.6%

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

   5. Taylor expanded in l around inf 68.5%

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

      1. *-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-*r* 68.5%

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

   7. Simplified 68.5%

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

   if -6.0000000000000001e94 < l < 1.9e9

   1. Initial program 70.1%

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

   3. Taylor expanded in l around 0 83.5%

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

      1. associate-*r* 83.5%

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

   5. Simplified 83.5%

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

   if 1.9e9 < l < 9.5000000000000003e76

   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 44.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+94}:\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq 1900000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(2 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{+76}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3900000000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+77}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
   (if (<= l -6.6e+94)
     t_0
     (if (<= l 3900000000.0)
       (+ U (* l (* J (* 2.0 (cos (* K 0.5))))))
       (if (<= l 1.45e+77) (pow U -4.0) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -6.6e+94) {
		tmp = t_0;
	} else if (l <= 3900000000.0) {
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	} else if (l <= 1.45e+77) {
		tmp = pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l ** 3.0d0) * (j * 0.3333333333333333d0)
    if (l <= (-6.6d+94)) then
        tmp = t_0
    else if (l <= 3900000000.0d0) then
        tmp = u + (l * (j * (2.0d0 * cos((k * 0.5d0)))))
    else if (l <= 1.45d+77) then
        tmp = u ** (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -6.6e+94) {
		tmp = t_0;
	} else if (l <= 3900000000.0) {
		tmp = U + (l * (J * (2.0 * Math.cos((K * 0.5)))));
	} else if (l <= 1.45e+77) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * (J * 0.3333333333333333)
	tmp = 0
	if l <= -6.6e+94:
		tmp = t_0
	elif l <= 3900000000.0:
		tmp = U + (l * (J * (2.0 * math.cos((K * 0.5)))))
	elif l <= 1.45e+77:
		tmp = math.pow(U, -4.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -6.6e+94)
		tmp = t_0;
	elseif (l <= 3900000000.0)
		tmp = Float64(U + Float64(l * Float64(J * Float64(2.0 * cos(Float64(K * 0.5))))));
	elseif (l <= 1.45e+77)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * (J * 0.3333333333333333);
	tmp = 0.0;
	if (l <= -6.6e+94)
		tmp = t_0;
	elseif (l <= 3900000000.0)
		tmp = U + (l * (J * (2.0 * cos((K * 0.5)))));
	elseif (l <= 1.45e+77)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.6e+94], t$95$0, If[LessEqual[l, 3900000000.0], N[(U + N[(l * N[(J * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.45e+77], N[Power[U, -4.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.6e94 or 1.4500000000000001e77 < 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 96.9%

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

   4. Taylor expanded in K around 0 68.6%

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

   5. Taylor expanded in l around inf 68.5%

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

      1. *-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-*r* 68.5%

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

   7. Simplified 68.5%

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

   if -6.6e94 < l < 3.9e9

   1. Initial program 70.1%

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

   3. Taylor expanded in l around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-*r* 83.5%

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

      3. associate-*l* 83.5%

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

      4. *-commutative 83.5%

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

      5. *-commutative 83.5%

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

      6. associate-*l* 83.5%

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

      7. *-commutative 83.5%

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

   5. Simplified 83.5%

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

   if 3.9e9 < l < 1.4500000000000001e77

   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 44.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \cdot 10^{+94}:\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq 3900000000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{+77}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -6 \cdot 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1900000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+76}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (pow l 3.0) (* J 0.3333333333333333))))
   (if (<= l -6e+94)
     t_0
     (if (<= l 1900000000.0)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 1.25e+76) (pow U -4.0) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -6e+94) {
		tmp = t_0;
	} else if (l <= 1900000000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 1.25e+76) {
		tmp = pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l ** 3.0d0) * (j * 0.3333333333333333d0)
    if (l <= (-6d+94)) then
        tmp = t_0
    else if (l <= 1900000000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 1.25d+76) then
        tmp = u ** (-4.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.pow(l, 3.0) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -6e+94) {
		tmp = t_0;
	} else if (l <= 1900000000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 1.25e+76) {
		tmp = Math.pow(U, -4.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.pow(l, 3.0) * (J * 0.3333333333333333)
	tmp = 0
	if l <= -6e+94:
		tmp = t_0
	elif l <= 1900000000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 1.25e+76:
		tmp = math.pow(U, -4.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64((l ^ 3.0) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -6e+94)
		tmp = t_0;
	elseif (l <= 1900000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 1.25e+76)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (l ^ 3.0) * (J * 0.3333333333333333);
	tmp = 0.0;
	if (l <= -6e+94)
		tmp = t_0;
	elseif (l <= 1900000000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 1.25e+76)
		tmp = U ^ -4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6e+94], t$95$0, If[LessEqual[l, 1900000000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.25e+76], N[Power[U, -4.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.0000000000000001e94 or 1.24999999999999998e76 < 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 96.9%

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

   4. Taylor expanded in K around 0 68.6%

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

   5. Taylor expanded in l around inf 68.5%

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

      1. *-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. associate-*r* 68.5%

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

   7. Simplified 68.5%

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

   if -6.0000000000000001e94 < l < 1.9e9

   1. Initial program 70.1%

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

   3. Taylor expanded in l around 0 83.4%

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

   if 1.9e9 < l < 1.24999999999999998e76

   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 44.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+94}:\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq 1900000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.25 \cdot 10^{+76}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in l around 0 65.8%

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

4. Taylor expanded in K around 0 53.0%

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

   1. +-commutative 53.0%

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

   2. *-commutative 53.0%

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

   3. associate-*r* 53.0%

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

   4. fma-define 53.0%

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

   5. *-commutative 53.0%

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

6. Simplified 53.0%

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

Alternative 12: 42.6% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+151}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{+23} \lor \neg \left(\ell \leq 1.95\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3.9e+151)
   (- -4.0 (* U U))
   (if (or (<= l -7.8e+23) (not (<= l 1.95))) (* U (- U -4.0)) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.9e+151) {
		tmp = -4.0 - (U * U);
	} else if ((l <= -7.8e+23) || !(l <= 1.95)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-3.9d+151)) then
        tmp = (-4.0d0) - (u * u)
    else if ((l <= (-7.8d+23)) .or. (.not. (l <= 1.95d0))) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3.9e+151) {
		tmp = -4.0 - (U * U);
	} else if ((l <= -7.8e+23) || !(l <= 1.95)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3.9e+151:
		tmp = -4.0 - (U * U)
	elif (l <= -7.8e+23) or not (l <= 1.95):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3.9e+151)
		tmp = Float64(-4.0 - Float64(U * U));
	elseif ((l <= -7.8e+23) || !(l <= 1.95))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -3.9e+151)
		tmp = -4.0 - (U * U);
	elseif ((l <= -7.8e+23) || ~((l <= 1.95)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3.9e+151], N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -7.8e+23], N[Not[LessEqual[l, 1.95]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]]

Derivation

1. Split input into 3 regimes

2. if l < -3.89999999999999976e151

   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-define 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. Add Preprocessing

   6. Applied egg-rr 22.4%

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

      1. cancel-sign-sub-inv 22.4%

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

   8. Simplified 22.4%

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

   if -3.89999999999999976e151 < l < -7.8000000000000001e23 or 1.94999999999999996 < 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 22.2%

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

   if -7.8000000000000001e23 < l < 1.94999999999999996

   1. Initial program 65.0%

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

   3. Taylor expanded in J around 0 57.7%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.9 \cdot 10^{+151}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{+23} \lor \neg \left(\ell \leq 1.95\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.6% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+24} \lor \neg \left(\ell \leq 1.95\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.85e+24) (not (<= l 1.95))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.85e+24) || !(l <= 1.95)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.85d+24)) .or. (.not. (l <= 1.95d0))) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.85e+24) || !(l <= 1.95)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.85e+24) or not (l <= 1.95):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.85e+24) || !(l <= 1.95))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.85e+24) || ~((l <= 1.95)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.85e+24], N[Not[LessEqual[l, 1.95]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.85e24 or 1.94999999999999996 < 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 19.1%

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

   if -1.85e24 < l < 1.94999999999999996

   1. Initial program 65.0%

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

   3. Taylor expanded in J around 0 57.7%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{+24} \lor \neg \left(\ell \leq 1.95\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.3% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.1 \cdot 10^{-7} \lor \neg \left(\ell \leq 1.95\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -9.1e-7) (not (<= l 1.95))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -9.1e-7) || !(l <= 1.95)) {
		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 <= (-9.1d-7)) .or. (.not. (l <= 1.95d0))) 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 <= -9.1e-7) || !(l <= 1.95)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -9.1e-7) or not (l <= 1.95):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -9.1e-7) || !(l <= 1.95))
		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 <= -9.1e-7) || ~((l <= 1.95)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -9.1e-7], N[Not[LessEqual[l, 1.95]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -9.0999999999999997e-7 or 1.94999999999999996 < l

   1. Initial program 99.8%

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

   4. Applied egg-rr 18.2%

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

   if -9.0999999999999997e-7 < l < 1.94999999999999996

   1. Initial program 63.1%

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

   3. Taylor expanded in J around 0 61.0%

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

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.1 \cdot 10^{-7} \lor \neg \left(\ell \leq 1.95\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 54.8% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in l around 0 65.8%

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

4. Taylor expanded in K around 0 53.0%

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

   1. +-commutative 53.0%

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

   2. *-commutative 53.0%

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

   3. associate-*r* 53.0%

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

   4. *-commutative 53.0%

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

6. Simplified 53.0%

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

7. Final simplification 53.0%

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

Alternative 16: 37.2% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

3. Taylor expanded in J around 0 29.6%

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

Alternative 17: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.7%

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

4. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

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

6. Simplified 2.8%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024094 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))