Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.8% → 99.9%

Time: 12.2s

Alternatives: 13

Speedup: 1.3×

• 50.2% 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: 86.8% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;t\_0 \leq 0.04:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;U + t\_1 \cdot \left(t\_0 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 -0.2)
     (+ (* t_0 (* J (cos (* 0.5 K)))) U)
     (if (<= t_0 0.04)
       (+
        U
        (*
         (*
          J
          (*
           l
           (+
            2.0
            (*
             (pow l 2.0)
             (+
              0.3333333333333333
              (*
               (pow l 2.0)
               (+
                0.016666666666666666
                (* (pow l 2.0) 0.0003968253968253968))))))))
         t_1))
       (+ U (* t_1 (* t_0 J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (t_0 * (J * cos((0.5 * K)))) + U;
	} else if (t_0 <= 0.04) {
		tmp = U + ((J * (l * (2.0 + (pow(l, 2.0) * (0.3333333333333333 + (pow(l, 2.0) * (0.016666666666666666 + (pow(l, 2.0) * 0.0003968253968253968)))))))) * t_1);
	} else {
		tmp = U + (t_1 * (t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    t_1 = cos((k / 2.0d0))
    if (t_0 <= (-0.2d0)) then
        tmp = (t_0 * (j * cos((0.5d0 * k)))) + u
    else if (t_0 <= 0.04d0) then
        tmp = u + ((j * (l * (2.0d0 + ((l ** 2.0d0) * (0.3333333333333333d0 + ((l ** 2.0d0) * (0.016666666666666666d0 + ((l ** 2.0d0) * 0.0003968253968253968d0)))))))) * t_1)
    else
        tmp = u + (t_1 * (t_0 * j))
    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);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (t_0 * (J * Math.cos((0.5 * K)))) + U;
	} else if (t_0 <= 0.04) {
		tmp = U + ((J * (l * (2.0 + (Math.pow(l, 2.0) * (0.3333333333333333 + (Math.pow(l, 2.0) * (0.016666666666666666 + (Math.pow(l, 2.0) * 0.0003968253968253968)))))))) * t_1);
	} else {
		tmp = U + (t_1 * (t_0 * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.2:
		tmp = (t_0 * (J * math.cos((0.5 * K)))) + U
	elif t_0 <= 0.04:
		tmp = U + ((J * (l * (2.0 + (math.pow(l, 2.0) * (0.3333333333333333 + (math.pow(l, 2.0) * (0.016666666666666666 + (math.pow(l, 2.0) * 0.0003968253968253968)))))))) * t_1)
	else:
		tmp = U + (t_1 * (t_0 * J))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(Float64(t_0 * Float64(J * cos(Float64(0.5 * K)))) + U);
	elseif (t_0 <= 0.04)
		tmp = Float64(U + Float64(Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * Float64(0.3333333333333333 + Float64((l ^ 2.0) * Float64(0.016666666666666666 + Float64((l ^ 2.0) * 0.0003968253968253968)))))))) * t_1));
	else
		tmp = Float64(U + Float64(t_1 * Float64(t_0 * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = (t_0 * (J * cos((0.5 * K)))) + U;
	elseif (t_0 <= 0.04)
		tmp = U + ((J * (l * (2.0 + ((l ^ 2.0) * (0.3333333333333333 + ((l ^ 2.0) * (0.016666666666666666 + ((l ^ 2.0) * 0.0003968253968253968)))))))) * t_1);
	else
		tmp = U + (t_1 * (t_0 * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(N[(t$95$0 * N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.04], N[(U + N[(N[(J * N[(l * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[Power[l, 2.0], $MachinePrecision] * N[(0.016666666666666666 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$1 * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.20000000000000001

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0400000000000000008

   1. Initial program 68.3%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

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

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.2:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 0.04:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;t\_0 \leq 0.04:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_1 \cdot \left(t\_0 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 -0.2)
     (+ (* t_0 (* J (cos (* 0.5 K)))) U)
     (if (<= t_0 0.04)
       (+
        U
        (*
         t_1
         (*
          J
          (*
           l
           (+
            2.0
            (*
             (pow l 2.0)
             (+ 0.3333333333333333 (* (pow l 2.0) 0.016666666666666666))))))))
       (+ U (* t_1 (* t_0 J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (t_0 * (J * cos((0.5 * K)))) + U;
	} else if (t_0 <= 0.04) {
		tmp = U + (t_1 * (J * (l * (2.0 + (pow(l, 2.0) * (0.3333333333333333 + (pow(l, 2.0) * 0.016666666666666666)))))));
	} else {
		tmp = U + (t_1 * (t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    t_1 = cos((k / 2.0d0))
    if (t_0 <= (-0.2d0)) then
        tmp = (t_0 * (j * cos((0.5d0 * k)))) + u
    else if (t_0 <= 0.04d0) then
        tmp = u + (t_1 * (j * (l * (2.0d0 + ((l ** 2.0d0) * (0.3333333333333333d0 + ((l ** 2.0d0) * 0.016666666666666666d0)))))))
    else
        tmp = u + (t_1 * (t_0 * j))
    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);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (t_0 * (J * Math.cos((0.5 * K)))) + U;
	} else if (t_0 <= 0.04) {
		tmp = U + (t_1 * (J * (l * (2.0 + (Math.pow(l, 2.0) * (0.3333333333333333 + (Math.pow(l, 2.0) * 0.016666666666666666)))))));
	} else {
		tmp = U + (t_1 * (t_0 * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.2:
		tmp = (t_0 * (J * math.cos((0.5 * K)))) + U
	elif t_0 <= 0.04:
		tmp = U + (t_1 * (J * (l * (2.0 + (math.pow(l, 2.0) * (0.3333333333333333 + (math.pow(l, 2.0) * 0.016666666666666666)))))))
	else:
		tmp = U + (t_1 * (t_0 * J))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(Float64(t_0 * Float64(J * cos(Float64(0.5 * K)))) + U);
	elseif (t_0 <= 0.04)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * Float64(0.3333333333333333 + Float64((l ^ 2.0) * 0.016666666666666666))))))));
	else
		tmp = Float64(U + Float64(t_1 * Float64(t_0 * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = (t_0 * (J * cos((0.5 * K)))) + U;
	elseif (t_0 <= 0.04)
		tmp = U + (t_1 * (J * (l * (2.0 + ((l ^ 2.0) * (0.3333333333333333 + ((l ^ 2.0) * 0.016666666666666666)))))));
	else
		tmp = U + (t_1 * (t_0 * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(N[(t$95$0 * N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 0.04], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$1 * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.20000000000000001

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0400000000000000008

   1. Initial program 68.3%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

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

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.2:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 0.04:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -0.2 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(t\_0 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -0.2) (not (<= t_0 5e-16)))
     (+ U (* (cos (/ K 2.0)) (* t_0 J)))
     (+
      U
      (*
       l
       (*
        (* J (cos (* 0.5 K)))
        (+ 2.0 (* (pow l 2.0) 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 5e-16)) {
		tmp = U + (cos((K / 2.0)) * (t_0 * J));
	} else {
		tmp = U + (l * ((J * cos((0.5 * K))) * (2.0 + (pow(l, 2.0) * 0.3333333333333333))));
	}
	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)
    if ((t_0 <= (-0.2d0)) .or. (.not. (t_0 <= 5d-16))) then
        tmp = u + (cos((k / 2.0d0)) * (t_0 * j))
    else
        tmp = u + (l * ((j * cos((0.5d0 * k))) * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0))))
    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);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 5e-16)) {
		tmp = U + (Math.cos((K / 2.0)) * (t_0 * J));
	} else {
		tmp = U + (l * ((J * Math.cos((0.5 * K))) * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -0.2) or not (t_0 <= 5e-16):
		tmp = U + (math.cos((K / 2.0)) * (t_0 * J))
	else:
		tmp = U + (l * ((J * math.cos((0.5 * K))) * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 5e-16))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(t_0 * J)));
	else
		tmp = Float64(U + Float64(l * Float64(Float64(J * cos(Float64(0.5 * K))) * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -0.2) || ~((t_0 <= 5e-16)))
		tmp = U + (cos((K / 2.0)) * (t_0 * J));
	else
		tmp = U + (l * ((J * cos((0.5 * K))) * (2.0 + ((l ^ 2.0) * 0.3333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 5e-16]], $MachinePrecision]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.20000000000000001 or 5.0000000000000004e-16 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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

   if -0.20000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000004e-16

   1. Initial program 67.3%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. add-cube-cbrt 99.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)} \cdot \sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. pow3 99.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right)}^{3}} \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \ell\right)}\right)}^{3}} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*l* 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l* 99.9%

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

      8. distribute-lft-out 99.9%

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

      9. *-commutative 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.2 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-16}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := J \cdot \cos \left(0.5 \cdot K\right)\\ \mathbf{if}\;t\_0 \leq -0.2:\\ \;\;\;\;t\_0 \cdot t\_1 + U\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-16}:\\ \;\;\;\;U + \ell \cdot \left(t\_1 \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(t\_0 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (* J (cos (* 0.5 K)))))
   (if (<= t_0 -0.2)
     (+ (* t_0 t_1) U)
     (if (<= t_0 5e-16)
       (+ U (* l (* t_1 (+ 2.0 (* (pow l 2.0) 0.3333333333333333)))))
       (+ U (* (cos (/ K 2.0)) (* t_0 J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = J * cos((0.5 * K));
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (t_0 * t_1) + U;
	} else if (t_0 <= 5e-16) {
		tmp = U + (l * (t_1 * (2.0 + (pow(l, 2.0) * 0.3333333333333333))));
	} else {
		tmp = U + (cos((K / 2.0)) * (t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    t_1 = j * cos((0.5d0 * k))
    if (t_0 <= (-0.2d0)) then
        tmp = (t_0 * t_1) + u
    else if (t_0 <= 5d-16) then
        tmp = u + (l * (t_1 * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0))))
    else
        tmp = u + (cos((k / 2.0d0)) * (t_0 * j))
    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);
	double t_1 = J * Math.cos((0.5 * K));
	double tmp;
	if (t_0 <= -0.2) {
		tmp = (t_0 * t_1) + U;
	} else if (t_0 <= 5e-16) {
		tmp = U + (l * (t_1 * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333))));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (t_0 * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = J * math.cos((0.5 * K))
	tmp = 0
	if t_0 <= -0.2:
		tmp = (t_0 * t_1) + U
	elif t_0 <= 5e-16:
		tmp = U + (l * (t_1 * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333))))
	else:
		tmp = U + (math.cos((K / 2.0)) * (t_0 * J))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = Float64(J * cos(Float64(0.5 * K)))
	tmp = 0.0
	if (t_0 <= -0.2)
		tmp = Float64(Float64(t_0 * t_1) + U);
	elseif (t_0 <= 5e-16)
		tmp = Float64(U + Float64(l * Float64(t_1 * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333)))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(t_0 * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = J * cos((0.5 * K));
	tmp = 0.0;
	if (t_0 <= -0.2)
		tmp = (t_0 * t_1) + U;
	elseif (t_0 <= 5e-16)
		tmp = U + (l * (t_1 * (2.0 + ((l ^ 2.0) * 0.3333333333333333))));
	else
		tmp = U + (cos((K / 2.0)) * (t_0 * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.2], N[(N[(t$95$0 * t$95$1), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$0, 5e-16], N[(U + N[(l * N[(t$95$1 * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.20000000000000001

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000004e-16

   1. Initial program 67.3%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. add-cube-cbrt 99.3%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)} \cdot \sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. pow3 99.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right)}^{3}} \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \ell\right)}\right)}^{3}} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*l* 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l* 99.9%

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

      8. distribute-lft-out 99.9%

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

      9. *-commutative 99.9%

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

   10. Simplified 99.9%

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

   if 5.0000000000000004e-16 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.2:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-16}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-48} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+210}\right):\\ \;\;\;\;U + t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (or (<= t_0 -5e-48) (not (<= t_0 2e+210)))
     (+ U t_0)
     (+ U (* 2.0 (* J (* l (cos (* 0.5 K)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if ((t_0 <= -5e-48) || !(t_0 <= 2e+210)) {
		tmp = U + t_0;
	} else {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	}
	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 ((t_0 <= (-5d-48)) .or. (.not. (t_0 <= 2d+210))) then
        tmp = u + t_0
    else
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    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 ((t_0 <= -5e-48) || !(t_0 <= 2e+210)) {
		tmp = U + t_0;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if (t_0 <= -5e-48) or not (t_0 <= 2e+210):
		tmp = U + t_0
	else:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if ((t_0 <= -5e-48) || !(t_0 <= 2e+210))
		tmp = Float64(U + t_0);
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if ((t_0 <= -5e-48) || ~((t_0 <= 2e+210)))
		tmp = U + t_0;
	else
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	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[Or[LessEqual[t$95$0, -5e-48], N[Not[LessEqual[t$95$0, 2e+210]], $MachinePrecision]], N[(U + t$95$0), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -4.9999999999999999e-48 or 1.99999999999999985e210 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 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

   3. Taylor expanded in K around 0 78.7%

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

   if -4.9999999999999999e-48 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 1.99999999999999985e210

   1. Initial program 68.5%

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

   3. Taylor expanded in l around 0 99.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq -5 \cdot 10^{-48} \lor \neg \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq 2 \cdot 10^{+210}\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -175:\\ \;\;\;\;U + t\_1 \cdot J\\ \mathbf{elif}\;\ell \leq 0.18:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+43}:\\ \;\;\;\;U + t\_1 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* (pow l 7.0) (* J 0.0003968253968253968)))))
        (t_1 (- (exp l) (exp (- l)))))
   (if (<= l -3.4e+24)
     t_0
     (if (<= l -175.0)
       (+ U (* t_1 J))
       (if (<= l 0.18)
         (+ U (* 2.0 (* J (* l (cos (* 0.5 K))))))
         (if (<= l 5.6e+43) (+ U (* t_1 -0.5)) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 7.0) * (J * 0.0003968253968253968)));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if (l <= -3.4e+24) {
		tmp = t_0;
	} else if (l <= -175.0) {
		tmp = U + (t_1 * J);
	} else if (l <= 0.18) {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	} else if (l <= 5.6e+43) {
		tmp = U + (t_1 * -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * ((l ** 7.0d0) * (j * 0.0003968253968253968d0)))
    t_1 = exp(l) - exp(-l)
    if (l <= (-3.4d+24)) then
        tmp = t_0
    else if (l <= (-175.0d0)) then
        tmp = u + (t_1 * j)
    else if (l <= 0.18d0) then
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    else if (l <= 5.6d+43) then
        tmp = u + (t_1 * (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (Math.pow(l, 7.0) * (J * 0.0003968253968253968)));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if (l <= -3.4e+24) {
		tmp = t_0;
	} else if (l <= -175.0) {
		tmp = U + (t_1 * J);
	} else if (l <= 0.18) {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	} else if (l <= 5.6e+43) {
		tmp = U + (t_1 * -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (math.pow(l, 7.0) * (J * 0.0003968253968253968)))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if l <= -3.4e+24:
		tmp = t_0
	elif l <= -175.0:
		tmp = U + (t_1 * J)
	elif l <= 0.18:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	elif l <= 5.6e+43:
		tmp = U + (t_1 * -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968))))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (l <= -3.4e+24)
		tmp = t_0;
	elseif (l <= -175.0)
		tmp = Float64(U + Float64(t_1 * J));
	elseif (l <= 0.18)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	elseif (l <= 5.6e+43)
		tmp = Float64(U + Float64(t_1 * -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * ((l ^ 7.0) * (J * 0.0003968253968253968)));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if (l <= -3.4e+24)
		tmp = t_0;
	elseif (l <= -175.0)
		tmp = U + (t_1 * J);
	elseif (l <= 0.18)
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	elseif (l <= 5.6e+43)
		tmp = U + (t_1 * -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.4e+24], t$95$0, If[LessEqual[l, -175.0], N[(U + N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.18], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+43], N[(U + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.4000000000000001e24 or 5.60000000000000038e43 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 98.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 98.4%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{7}\right) \cdot 0.0003968253968253968\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 98.4%

         \[\leadsto \left(\color{blue}{\left({\ell}^{7} \cdot J\right)} \cdot 0.0003968253968253968\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*r* 98.4%

         \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 98.4%

      \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -3.4000000000000001e24 < l < -175

   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 0 100.0%

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

   if -175 < l < 0.17999999999999999

   1. Initial program 68.8%

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

   3. Taylor expanded in l around 0 98.7%

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

   if 0.17999999999999999 < l < 5.60000000000000038e43

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 87.5%

      \[\leadsto \left(e^{\ell} - e^{-\ell}\right) \cdot \color{blue}{-0.5} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{elif}\;\ell \leq -175:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.18:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+43}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -175:\\ \;\;\;\;U + t\_1 \cdot J\\ \mathbf{elif}\;\ell \leq 0.18:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq 1.14 \cdot 10^{+42}:\\ \;\;\;\;U + t\_1 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* (pow l 7.0) (* J 0.0003968253968253968)))))
        (t_1 (- (exp l) (exp (- l)))))
   (if (<= l -3.4e+24)
     t_0
     (if (<= l -175.0)
       (+ U (* t_1 J))
       (if (<= l 0.18)
         (+
          U
          (*
           l
           (*
            (* J (cos (* 0.5 K)))
            (+ 2.0 (* (pow l 2.0) 0.3333333333333333)))))
         (if (<= l 1.14e+42) (+ U (* t_1 -0.5)) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (pow(l, 7.0) * (J * 0.0003968253968253968)));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if (l <= -3.4e+24) {
		tmp = t_0;
	} else if (l <= -175.0) {
		tmp = U + (t_1 * J);
	} else if (l <= 0.18) {
		tmp = U + (l * ((J * cos((0.5 * K))) * (2.0 + (pow(l, 2.0) * 0.3333333333333333))));
	} else if (l <= 1.14e+42) {
		tmp = U + (t_1 * -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * ((l ** 7.0d0) * (j * 0.0003968253968253968d0)))
    t_1 = exp(l) - exp(-l)
    if (l <= (-3.4d+24)) then
        tmp = t_0
    else if (l <= (-175.0d0)) then
        tmp = u + (t_1 * j)
    else if (l <= 0.18d0) then
        tmp = u + (l * ((j * cos((0.5d0 * k))) * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0))))
    else if (l <= 1.14d+42) then
        tmp = u + (t_1 * (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (Math.pow(l, 7.0) * (J * 0.0003968253968253968)));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if (l <= -3.4e+24) {
		tmp = t_0;
	} else if (l <= -175.0) {
		tmp = U + (t_1 * J);
	} else if (l <= 0.18) {
		tmp = U + (l * ((J * Math.cos((0.5 * K))) * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333))));
	} else if (l <= 1.14e+42) {
		tmp = U + (t_1 * -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (math.pow(l, 7.0) * (J * 0.0003968253968253968)))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if l <= -3.4e+24:
		tmp = t_0
	elif l <= -175.0:
		tmp = U + (t_1 * J)
	elif l <= 0.18:
		tmp = U + (l * ((J * math.cos((0.5 * K))) * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333))))
	elif l <= 1.14e+42:
		tmp = U + (t_1 * -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968))))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (l <= -3.4e+24)
		tmp = t_0;
	elseif (l <= -175.0)
		tmp = Float64(U + Float64(t_1 * J));
	elseif (l <= 0.18)
		tmp = Float64(U + Float64(l * Float64(Float64(J * cos(Float64(0.5 * K))) * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333)))));
	elseif (l <= 1.14e+42)
		tmp = Float64(U + Float64(t_1 * -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * ((l ^ 7.0) * (J * 0.0003968253968253968)));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if (l <= -3.4e+24)
		tmp = t_0;
	elseif (l <= -175.0)
		tmp = U + (t_1 * J);
	elseif (l <= 0.18)
		tmp = U + (l * ((J * cos((0.5 * K))) * (2.0 + ((l ^ 2.0) * 0.3333333333333333))));
	elseif (l <= 1.14e+42)
		tmp = U + (t_1 * -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.4e+24], t$95$0, If[LessEqual[l, -175.0], N[(U + N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.18], N[(U + N[(l * N[(N[(J * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.14e+42], N[(U + N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.4000000000000001e24 or 1.14000000000000004e42 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 98.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 98.4%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 98.4%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{7}\right) \cdot 0.0003968253968253968\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 98.4%

         \[\leadsto \left(\color{blue}{\left({\ell}^{7} \cdot J\right)} \cdot 0.0003968253968253968\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*r* 98.4%

         \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 98.4%

      \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -3.4000000000000001e24 < l < -175

   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 0 100.0%

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

   if -175 < l < 0.17999999999999999

   1. Initial program 68.8%

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

   3. Taylor expanded in l around 0 99.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. add-cube-cbrt 99.2%

         \[\leadsto \color{blue}{\left(\left(\sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)} \cdot \sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. pow3 99.3%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right)}^{3}} \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, \mathsf{fma}\left({\ell}^{2}, 0.0003968253968253968, 0.016666666666666666\right), 0.3333333333333333\right), 2\right) \cdot \left(J \cdot \ell\right)}\right)}^{3}} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Taylor expanded in l around 0 99.3%

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

      1. +-commutative 99.3%

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

      2. *-commutative 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-*r* 99.3%

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

      5. associate-*l* 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*l* 99.3%

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

      8. distribute-lft-out 99.3%

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

      9. *-commutative 99.3%

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

   10. Simplified 99.3%

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

   if 0.17999999999999999 < l < 1.14000000000000004e42

   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 J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   7. Applied egg-rr 87.5%

      \[\leadsto \left(e^{\ell} - e^{-\ell}\right) \cdot \color{blue}{-0.5} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+24}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{elif}\;\ell \leq -175:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.18:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq 1.14 \cdot 10^{+42}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.19:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.19)
   (+ U (* 2.0 (* J (* l (cos (* 0.5 K))))))
   (+ U (* J (* l (+ 2.0 (* (pow l 2.0) 0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.19) {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	} else {
		tmp = U + (J * (l * (2.0 + (pow(l, 2.0) * 0.3333333333333333))));
	}
	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)) <= 0.19d0) then
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    else
        tmp = u + (j * (l * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0))))
    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)) <= 0.19) {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	} else {
		tmp = U + (J * (l * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.19:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	else:
		tmp = U + (J * (l * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.19)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.19)
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	else
		tmp = U + (J * (l * (2.0 + ((l ^ 2.0) * 0.3333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.19], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.19

   1. Initial program 81.5%

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

   3. Taylor expanded in l around 0 63.2%

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

   if 0.19 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 85.7%

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

   3. Taylor expanded in K around 0 85.3%

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

   4. Taylor expanded in l around 0 83.0%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.19:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

3. Taylor expanded in l around 0 63.4%

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

4. Final simplification 63.4%

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

Alternative 10: 41.9% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -145000 \lor \neg \left(\ell \leq 9 \cdot 10^{+30}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -145000.0) (not (<= l 9e+30))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -145000.0) || !(l <= 9e+30)) {
		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 <= (-145000.0d0)) .or. (.not. (l <= 9d+30))) 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 <= -145000.0) || !(l <= 9e+30)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -145000.0) or not (l <= 9e+30):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -145000.0) || !(l <= 9e+30))
		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 <= -145000.0) || ~((l <= 9e+30)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -145000.0], N[Not[LessEqual[l, 9e+30]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -145000 or 8.9999999999999999e30 < l

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-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 13.7%

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

   if -145000 < l < 8.9999999999999999e30

   1. Initial program 70.2%

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

      1. associate-*l* 70.2%

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

      2. fma-define 70.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in J around 0 63.9%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -145000 \lor \neg \left(\ell \leq 9 \cdot 10^{+30}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.7% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.7%

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

3. Taylor expanded in K around 0 73.5%

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

4. Taylor expanded in l around 0 53.8%

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

5. Final simplification 53.8%

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

Alternative 12: 2.8% 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 84.7%

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

   1. associate-*l* 84.8%

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

   2. fma-define 84.8%

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

3. Simplified 84.8%

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

6. Applied egg-rr 2.5%

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

   1. *-inverses 2.5%

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

8. Simplified 2.5%

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

9. Final simplification 2.5%

   \[\leadsto 1 \]
10. Add Preprocessing

Alternative 13: 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 84.7%

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

   1. associate-*l* 84.8%

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

   2. fma-define 84.8%

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

3. Simplified 84.8%

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

5. Taylor expanded in J around 0 33.7%

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

6. Final simplification 33.7%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024112 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))