Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.5%

Time: 12.1s

Alternatives: 17

Speedup: 2.5×

• 49.8% 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: 86.2% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

3. Taylor expanded in l around 0 60.3%

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

   1. log1p-expm1-u 99.1%

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

   2. *-commutative 99.1%

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

5. Applied egg-rr 99.1%

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

Alternative 2: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 0.047:\\ \;\;\;\;U + t\_1 \cdot \left(\ell \cdot \left(J \cdot 2 + {\ell}^{2} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+61}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -125.0)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (<= l 0.047)
       (+
        U
        (*
         t_1
         (*
          l
          (+
           (* J 2.0)
           (*
            (pow l 2.0)
            (+
             (* 0.016666666666666666 (* J (* l l)))
             (* J 0.3333333333333333)))))))
       (if (<= l 1.08e+61)
         (+ U (* J (- (exp l) t_0)))
         (+
          U
          (* (* J 0.016666666666666666) (* (pow l 5.0) (cos (* K -0.5))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -125.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 0.047) {
		tmp = U + (t_1 * (l * ((J * 2.0) + (pow(l, 2.0) * ((0.016666666666666666 * (J * (l * l))) + (J * 0.3333333333333333))))));
	} else if (l <= 1.08e+61) {
		tmp = U + (J * (exp(l) - t_0));
	} else {
		tmp = U + ((J * 0.016666666666666666) * (pow(l, 5.0) * cos((K * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-125.0d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if (l <= 0.047d0) then
        tmp = u + (t_1 * (l * ((j * 2.0d0) + ((l ** 2.0d0) * ((0.016666666666666666d0 * (j * (l * l))) + (j * 0.3333333333333333d0))))))
    else if (l <= 1.08d+61) then
        tmp = u + (j * (exp(l) - t_0))
    else
        tmp = u + ((j * 0.016666666666666666d0) * ((l ** 5.0d0) * cos((k * (-0.5d0)))))
    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);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -125.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 0.047) {
		tmp = U + (t_1 * (l * ((J * 2.0) + (Math.pow(l, 2.0) * ((0.016666666666666666 * (J * (l * l))) + (J * 0.3333333333333333))))));
	} else if (l <= 1.08e+61) {
		tmp = U + (J * (Math.exp(l) - t_0));
	} else {
		tmp = U + ((J * 0.016666666666666666) * (Math.pow(l, 5.0) * Math.cos((K * -0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -125.0:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif l <= 0.047:
		tmp = U + (t_1 * (l * ((J * 2.0) + (math.pow(l, 2.0) * ((0.016666666666666666 * (J * (l * l))) + (J * 0.3333333333333333))))))
	elif l <= 1.08e+61:
		tmp = U + (J * (math.exp(l) - t_0))
	else:
		tmp = U + ((J * 0.016666666666666666) * (math.pow(l, 5.0) * math.cos((K * -0.5))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -125.0)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif (l <= 0.047)
		tmp = Float64(U + Float64(t_1 * Float64(l * Float64(Float64(J * 2.0) + Float64((l ^ 2.0) * Float64(Float64(0.016666666666666666 * Float64(J * Float64(l * l))) + Float64(J * 0.3333333333333333)))))));
	elseif (l <= 1.08e+61)
		tmp = Float64(U + Float64(J * Float64(exp(l) - t_0)));
	else
		tmp = Float64(U + Float64(Float64(J * 0.016666666666666666) * Float64((l ^ 5.0) * cos(Float64(K * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -125.0)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif (l <= 0.047)
		tmp = U + (t_1 * (l * ((J * 2.0) + ((l ^ 2.0) * ((0.016666666666666666 * (J * (l * l))) + (J * 0.3333333333333333))))));
	elseif (l <= 1.08e+61)
		tmp = U + (J * (exp(l) - t_0));
	else
		tmp = U + ((J * 0.016666666666666666) * ((l ^ 5.0) * cos((K * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -125.0], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.047], N[(U + N[(t$95$1 * N[(l * N[(N[(J * 2.0), $MachinePrecision] + N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[(0.016666666666666666 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.08e+61], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * 0.016666666666666666), $MachinePrecision] * N[(N[Power[l, 5.0], $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -125

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -125 < l < 0.047

   1. Initial program 74.8%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 0.047 < l < 1.08e61

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 75.0%

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

   if 1.08e61 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. cos-neg 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 0.047:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2 + {\ell}^{2} \cdot \left(0.016666666666666666 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+61}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+133} \lor \neg \left(\ell \leq -0.0019 \lor \neg \left(\ell \leq 0.035\right) \land \ell \leq 2.2 \cdot 10^{+88}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3.3e+133)
         (not (or (<= l -0.0019) (and (not (<= l 0.035)) (<= l 2.2e+88)))))
   (+ U (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333))))))
   (+ U (* J (- (exp l) (exp (- l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.3e+133) || !((l <= -0.0019) || (!(l <= 0.035) && (l <= 2.2e+88)))) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} 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 <= (-3.3d+133)) .or. (.not. (l <= (-0.0019d0)) .or. (.not. (l <= 0.035d0)) .and. (l <= 2.2d+88))) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0)))))
    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 <= -3.3e+133) || !((l <= -0.0019) || (!(l <= 0.035) && (l <= 2.2e+88)))) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -3.3e+133) or not ((l <= -0.0019) or (not (l <= 0.035) and (l <= 2.2e+88))):
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))))
	else:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -3.3e+133) || !((l <= -0.0019) || (!(l <= 0.035) && (l <= 2.2e+88))))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))))));
	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 <= -3.3e+133) || ~(((l <= -0.0019) || (~((l <= 0.035)) && (l <= 2.2e+88)))))
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	else
		tmp = U + (J * (exp(l) - exp(-l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3.3e+133], N[Not[Or[LessEqual[l, -0.0019], And[N[Not[LessEqual[l, 0.035]], $MachinePrecision], LessEqual[l, 2.2e+88]]]], $MachinePrecision]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $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 < -3.3e133 or -0.0019 < l < 0.035000000000000003 or 2.20000000000000009e88 < l

   1. Initial program 85.0%

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

   3. Taylor expanded in l around 0 99.5%

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

      1. unpow2 98.6%

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

   5. Applied egg-rr 99.5%

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

   if -3.3e133 < l < -0.0019 or 0.035000000000000003 < l < 2.20000000000000009e88

   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 82.1%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+133} \lor \neg \left(\ell \leq -0.0019 \lor \neg \left(\ell \leq 0.035\right) \land \ell \leq 2.2 \cdot 10^{+88}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 0.058:\\ \;\;\;\;U + t\_1 \cdot \left(\ell \cdot \left(J \cdot 2 + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+61}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -125.0)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (<= l 0.058)
       (+
        U
        (* t_1 (* l (+ (* J 2.0) (* 0.3333333333333333 (* J (pow l 2.0)))))))
       (if (<= l 4.3e+61)
         (+ U (* J (- (exp l) t_0)))
         (+
          U
          (* (* J 0.016666666666666666) (* (pow l 5.0) (cos (* K -0.5))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -125.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 0.058) {
		tmp = U + (t_1 * (l * ((J * 2.0) + (0.3333333333333333 * (J * pow(l, 2.0))))));
	} else if (l <= 4.3e+61) {
		tmp = U + (J * (exp(l) - t_0));
	} else {
		tmp = U + ((J * 0.016666666666666666) * (pow(l, 5.0) * cos((K * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-125.0d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if (l <= 0.058d0) then
        tmp = u + (t_1 * (l * ((j * 2.0d0) + (0.3333333333333333d0 * (j * (l ** 2.0d0))))))
    else if (l <= 4.3d+61) then
        tmp = u + (j * (exp(l) - t_0))
    else
        tmp = u + ((j * 0.016666666666666666d0) * ((l ** 5.0d0) * cos((k * (-0.5d0)))))
    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);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -125.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 0.058) {
		tmp = U + (t_1 * (l * ((J * 2.0) + (0.3333333333333333 * (J * Math.pow(l, 2.0))))));
	} else if (l <= 4.3e+61) {
		tmp = U + (J * (Math.exp(l) - t_0));
	} else {
		tmp = U + ((J * 0.016666666666666666) * (Math.pow(l, 5.0) * Math.cos((K * -0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -125.0:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif l <= 0.058:
		tmp = U + (t_1 * (l * ((J * 2.0) + (0.3333333333333333 * (J * math.pow(l, 2.0))))))
	elif l <= 4.3e+61:
		tmp = U + (J * (math.exp(l) - t_0))
	else:
		tmp = U + ((J * 0.016666666666666666) * (math.pow(l, 5.0) * math.cos((K * -0.5))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -125.0)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif (l <= 0.058)
		tmp = Float64(U + Float64(t_1 * Float64(l * Float64(Float64(J * 2.0) + Float64(0.3333333333333333 * Float64(J * (l ^ 2.0)))))));
	elseif (l <= 4.3e+61)
		tmp = Float64(U + Float64(J * Float64(exp(l) - t_0)));
	else
		tmp = Float64(U + Float64(Float64(J * 0.016666666666666666) * Float64((l ^ 5.0) * cos(Float64(K * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -125.0)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif (l <= 0.058)
		tmp = U + (t_1 * (l * ((J * 2.0) + (0.3333333333333333 * (J * (l ^ 2.0))))));
	elseif (l <= 4.3e+61)
		tmp = U + (J * (exp(l) - t_0));
	else
		tmp = U + ((J * 0.016666666666666666) * ((l ^ 5.0) * cos((K * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -125.0], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.058], N[(U + N[(t$95$1 * N[(l * N[(N[(J * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.3e+61], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * 0.016666666666666666), $MachinePrecision] * N[(N[Power[l, 5.0], $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -125

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -125 < l < 0.0580000000000000029

   1. Initial program 74.8%

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

   3. Taylor expanded in l around 0 99.8%

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

   if 0.0580000000000000029 < l < 4.3000000000000001e61

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 75.0%

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

   if 4.3000000000000001e61 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. cos-neg 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 0.058:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2 + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+61}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 0.06:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+58}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -125.0)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (<= l 0.06)
       (+ U (* t_1 (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333))))))
       (if (<= l 2e+58)
         (+ U (* J (- (exp l) t_0)))
         (+
          U
          (* (* J 0.016666666666666666) (* (pow l 5.0) (cos (* K -0.5))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -125.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 0.06) {
		tmp = U + (t_1 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else if (l <= 2e+58) {
		tmp = U + (J * (exp(l) - t_0));
	} else {
		tmp = U + ((J * 0.016666666666666666) * (pow(l, 5.0) * cos((K * -0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-125.0d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if (l <= 0.06d0) then
        tmp = u + (t_1 * (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0)))))
    else if (l <= 2d+58) then
        tmp = u + (j * (exp(l) - t_0))
    else
        tmp = u + ((j * 0.016666666666666666d0) * ((l ** 5.0d0) * cos((k * (-0.5d0)))))
    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);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -125.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 0.06) {
		tmp = U + (t_1 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else if (l <= 2e+58) {
		tmp = U + (J * (Math.exp(l) - t_0));
	} else {
		tmp = U + ((J * 0.016666666666666666) * (Math.pow(l, 5.0) * Math.cos((K * -0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -125.0:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif l <= 0.06:
		tmp = U + (t_1 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))))
	elif l <= 2e+58:
		tmp = U + (J * (math.exp(l) - t_0))
	else:
		tmp = U + ((J * 0.016666666666666666) * (math.pow(l, 5.0) * math.cos((K * -0.5))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -125.0)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif (l <= 0.06)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))))));
	elseif (l <= 2e+58)
		tmp = Float64(U + Float64(J * Float64(exp(l) - t_0)));
	else
		tmp = Float64(U + Float64(Float64(J * 0.016666666666666666) * Float64((l ^ 5.0) * cos(Float64(K * -0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -125.0)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif (l <= 0.06)
		tmp = U + (t_1 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	elseif (l <= 2e+58)
		tmp = U + (J * (exp(l) - t_0));
	else
		tmp = U + ((J * 0.016666666666666666) * ((l ^ 5.0) * cos((K * -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -125.0], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.06], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e+58], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * 0.016666666666666666), $MachinePrecision] * N[(N[Power[l, 5.0], $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -125

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -125 < l < 0.059999999999999998

   1. Initial program 74.8%

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

   3. Taylor expanded in l around 0 99.8%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.8%

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

   if 0.059999999999999998 < l < 1.99999999999999989e58

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 75.0%

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

   if 1.99999999999999989e58 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. metadata-eval 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. cos-neg 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 0.06:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+58}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 9.4 \cdot 10^{+58}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (* (* J 0.016666666666666666) (* (pow l 5.0) (cos (* K -0.5)))))))
   (if (<= l -125.0)
     t_0
     (if (<= l 0.056)
       (+
        U
        (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333))))))
       (if (<= l 9.4e+58) (+ U (* J (- (exp l) (exp (- l))))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * 0.016666666666666666) * (pow(l, 5.0) * cos((K * -0.5))));
	double tmp;
	if (l <= -125.0) {
		tmp = t_0;
	} else if (l <= 0.056) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else if (l <= 9.4e+58) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((j * 0.016666666666666666d0) * ((l ** 5.0d0) * cos((k * (-0.5d0)))))
    if (l <= (-125.0d0)) then
        tmp = t_0
    else if (l <= 0.056d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0)))))
    else if (l <= 9.4d+58) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * 0.016666666666666666) * (Math.pow(l, 5.0) * Math.cos((K * -0.5))));
	double tmp;
	if (l <= -125.0) {
		tmp = t_0;
	} else if (l <= 0.056) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else if (l <= 9.4e+58) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * 0.016666666666666666) * (math.pow(l, 5.0) * math.cos((K * -0.5))))
	tmp = 0
	if l <= -125.0:
		tmp = t_0
	elif l <= 0.056:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))))
	elif l <= 9.4e+58:
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * 0.016666666666666666) * Float64((l ^ 5.0) * cos(Float64(K * -0.5)))))
	tmp = 0.0
	if (l <= -125.0)
		tmp = t_0;
	elseif (l <= 0.056)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))))));
	elseif (l <= 9.4e+58)
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * 0.016666666666666666) * ((l ^ 5.0) * cos((K * -0.5))));
	tmp = 0.0;
	if (l <= -125.0)
		tmp = t_0;
	elseif (l <= 0.056)
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	elseif (l <= 9.4e+58)
		tmp = U + (J * (exp(l) - exp(-l)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * 0.016666666666666666), $MachinePrecision] * N[(N[Power[l, 5.0], $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -125.0], t$95$0, If[LessEqual[l, 0.056], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9.4e+58], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -125 or 9.39999999999999944e58 < 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 93.6%

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

      1. *-commutative 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in l around inf 93.6%

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

      1. associate-*r* 93.6%

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

      2. *-commutative 93.6%

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

      3. *-commutative 93.6%

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

      4. metadata-eval 93.6%

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

      5. distribute-rgt-neg-in 93.6%

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

      6. cos-neg 93.6%

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

   8. Simplified 93.6%

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

   if -125 < l < 0.0560000000000000012

   1. Initial program 74.8%

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

   3. Taylor expanded in l around 0 99.8%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.8%

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

   if 0.0560000000000000012 < l < 9.39999999999999944e58

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 75.0%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 0.056:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 9.4 \cdot 10^{+58}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot 0.016666666666666666\right) \cdot \left({\ell}^{5} \cdot \cos \left(K \cdot -0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.11:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.11)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.11) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.11d0) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else
        tmp = u + (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.11) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.11:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.11)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.11)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.11], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.110000000000000001

   1. Initial program 84.9%

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

   3. Taylor expanded in l around 0 63.6%

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

   if 0.110000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.9%

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

   3. Taylor expanded in l around 0 87.5%

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

   4. Taylor expanded in K around 0 86.0%

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

      1. associate-*r* 84.5%

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

      2. *-commutative 84.5%

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

      3. distribute-lft-out 84.5%

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

      4. associate-*r* 84.5%

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

      5. associate-*r* 86.0%

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

      6. distribute-lft-out 86.0%

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

      7. fma-define 86.0%

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

      8. associate-*r* 86.0%

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

      9. unpow2 86.0%

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

      10. cube-mult 86.0%

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

      11. *-commutative 86.0%

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

   6. Simplified 86.0%

      \[\leadsto \color{blue}{J \cdot \mathsf{fma}\left(\ell, 2, 0.3333333333333333 \cdot {\ell}^{3}\right)} + U \]
   7. Step-by-step derivation

      1. fma-undefine 86.0%

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

   8. Applied egg-rr 86.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.11:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.11:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.11)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (* l (+ 2.0 (* (pow l 2.0) 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.11) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.11d0) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    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 t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.11) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.11:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.11)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	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)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.11)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + ((l ^ 2.0) * 0.3333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.11], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $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))) < 0.110000000000000001

   1. Initial program 84.9%

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

   3. Taylor expanded in l around 0 63.6%

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

   if 0.110000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.9%

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

   3. Taylor expanded in l around 0 87.5%

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

   4. Taylor expanded in K around 0 86.0%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.11:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+133} \lor \neg \left(\ell \leq -125\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3.3e+133) (not (<= l -125.0)))
   (+ U (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333))))))
   (+ U (* J (- 27.0 (exp (- l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.3e+133) || !(l <= -125.0)) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else {
		tmp = U + (J * (27.0 - 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 <= (-3.3d+133)) .or. (.not. (l <= (-125.0d0)))) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0)))))
    else
        tmp = u + (j * (27.0d0 - 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 <= -3.3e+133) || !(l <= -125.0)) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -3.3e+133) or not (l <= -125.0):
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))))
	else:
		tmp = U + (J * (27.0 - math.exp(-l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -3.3e+133) || !(l <= -125.0))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))))));
	else
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -3.3e+133) || ~((l <= -125.0)))
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	else
		tmp = U + (J * (27.0 - exp(-l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3.3e+133], N[Not[LessEqual[l, -125.0]], $MachinePrecision]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.3e133 or -125 < l

   1. Initial program 86.5%

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

   3. Taylor expanded in l around 0 92.0%

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

      1. unpow2 93.7%

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

   5. Applied egg-rr 92.0%

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

   if -3.3e133 < l < -125

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 85.2%

      \[\leadsto \color{blue}{J \cdot \left(27 - e^{-\ell}\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+133} \lor \neg \left(\ell \leq -125\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 80000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -125.0)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 80000000000.0)
     (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
     (*
      U
      (+ 1.0 (/ (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0)))) U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -125.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 80000000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U * (1.0 + ((J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0)))) / U));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-125.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 80000000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else
        tmp = u * (1.0d0 + ((j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0)))) / u))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -125.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 80000000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U * (1.0 + ((J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0)))) / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -125.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 80000000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = U * (1.0 + ((J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0)))) / U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -125.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 80000000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0)))) / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -125.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 80000000000.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	else
		tmp = U * (1.0 + ((J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0)))) / U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -125.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 80000000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -125

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 76.1%

      \[\leadsto \color{blue}{J \cdot \left(27 - e^{-\ell}\right)} + U \]

   if -125 < l < 8e10

   1. Initial program 75.6%

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

   3. Taylor expanded in l around 0 95.9%

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

   if 8e10 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 76.8%

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

   4. Taylor expanded in K around 0 57.4%

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

      1. associate-*r* 54.4%

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

      2. *-commutative 54.4%

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

      3. distribute-lft-out 54.4%

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

      4. associate-*r* 54.4%

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

      5. associate-*r* 57.4%

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

      6. distribute-lft-out 57.4%

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

      7. fma-define 57.4%

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

      8. associate-*r* 57.4%

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

      9. unpow2 57.4%

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

      10. cube-mult 57.4%

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

      11. *-commutative 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in U around inf 60.4%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}{U}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 80000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 95000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -125.0)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 95000000000.0)
     (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
     (+ U (* 0.3333333333333333 (* J (pow l 3.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -125.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 95000000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-125.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 95000000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -125.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 95000000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -125.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 95000000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -125.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 95000000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -125.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 95000000000.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -125.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 95000000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -125

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 76.1%

      \[\leadsto \color{blue}{J \cdot \left(27 - e^{-\ell}\right)} + U \]

   if -125 < l < 9.5e10

   1. Initial program 75.6%

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

   3. Taylor expanded in l around 0 95.9%

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

   if 9.5e10 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 76.8%

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

   4. Taylor expanded in K around 0 57.4%

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

      1. associate-*r* 54.4%

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

      2. *-commutative 54.4%

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

      3. distribute-lft-out 54.4%

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

      4. associate-*r* 54.4%

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

      5. associate-*r* 57.4%

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

      6. distribute-lft-out 57.4%

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

      7. fma-define 57.4%

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

      8. associate-*r* 57.4%

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

      9. unpow2 57.4%

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

      10. cube-mult 57.4%

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

      11. *-commutative 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in l around inf 57.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 95000000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 115000000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -125.0)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 115000000000.0)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (+ U (* 0.3333333333333333 (* J (pow l 3.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -125.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 115000000000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-125.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 115000000000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -125.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 115000000000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -125.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 115000000000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -125.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 115000000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -125.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 115000000000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -125.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 115000000000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -125

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 76.1%

      \[\leadsto \color{blue}{J \cdot \left(27 - e^{-\ell}\right)} + U \]

   if -125 < l < 1.15e11

   1. Initial program 75.6%

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

   3. Taylor expanded in l around 0 95.9%

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

   if 1.15e11 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 76.8%

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

   4. Taylor expanded in K around 0 57.4%

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

      1. associate-*r* 54.4%

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

      2. *-commutative 54.4%

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

      3. distribute-lft-out 54.4%

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

      4. associate-*r* 54.4%

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

      5. associate-*r* 57.4%

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

      6. distribute-lft-out 57.4%

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

      7. fma-define 57.4%

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

      8. associate-*r* 57.4%

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

      9. unpow2 57.4%

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

      10. cube-mult 57.4%

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

      11. *-commutative 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in l around inf 57.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 115000000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -125 \lor \neg \left(\ell \leq 78000000000\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -125.0) (not (<= l 78000000000.0)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* l (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -125.0) || !(l <= 78000000000.0)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-125.0d0)) .or. (.not. (l <= 78000000000.0d0))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (l * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -125.0) || !(l <= 78000000000.0)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -125.0) or not (l <= 78000000000.0):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (l * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -125.0) || !(l <= 78000000000.0))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -125.0) || ~((l <= 78000000000.0)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (l * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -125.0], N[Not[LessEqual[l, 78000000000.0]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -125 or 7.8e10 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 78.5%

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

   4. Taylor expanded in K around 0 59.1%

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

      1. associate-*r* 56.9%

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

      2. *-commutative 56.9%

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

      3. distribute-lft-out 56.9%

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

      4. associate-*r* 56.9%

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

      5. associate-*r* 59.1%

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

      6. distribute-lft-out 59.1%

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

      7. fma-define 59.1%

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

      8. associate-*r* 59.1%

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

      9. unpow2 59.1%

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

      10. cube-mult 59.1%

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

      11. *-commutative 59.1%

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

   6. Simplified 59.1%

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

   7. Taylor expanded in l around inf 59.1%

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

   if -125 < l < 7.8e10

   1. Initial program 75.6%

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

   3. Taylor expanded in l around 0 95.9%

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

      1. log1p-expm1-u 98.1%

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

      2. *-commutative 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in K around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. *-commutative 85.3%

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

   8. Simplified 85.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125 \lor \neg \left(\ell \leq 78000000000\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 78000000000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -125.0)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 78000000000.0)
     (+ U (* l (* J 2.0)))
     (+ U (* 0.3333333333333333 (* J (pow l 3.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -125.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 78000000000.0) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-125.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 78000000000.0d0) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -125.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 78000000000.0) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -125.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 78000000000.0:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -125.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 78000000000.0)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -125.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 78000000000.0)
		tmp = U + (l * (J * 2.0));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -125.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 78000000000.0], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -125

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 76.1%

      \[\leadsto \color{blue}{J \cdot \left(27 - e^{-\ell}\right)} + U \]

   if -125 < l < 7.8e10

   1. Initial program 75.6%

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

   3. Taylor expanded in l around 0 95.9%

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

      1. log1p-expm1-u 98.1%

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

      2. *-commutative 98.1%

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

   5. Applied egg-rr 98.1%

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

   6. Taylor expanded in K around 0 85.3%

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

      1. associate-*r* 85.3%

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

      2. *-commutative 85.3%

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

   8. Simplified 85.3%

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

   if 7.8e10 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 76.8%

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

   4. Taylor expanded in K around 0 57.4%

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

      1. associate-*r* 54.4%

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

      2. *-commutative 54.4%

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

      3. distribute-lft-out 54.4%

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

      4. associate-*r* 54.4%

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

      5. associate-*r* 57.4%

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

      6. distribute-lft-out 57.4%

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

      7. fma-define 57.4%

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

      8. associate-*r* 57.4%

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

      9. unpow2 57.4%

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

      10. cube-mult 57.4%

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

      11. *-commutative 57.4%

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

   6. Simplified 57.4%

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

   7. Taylor expanded in l around inf 57.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 78000000000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 38.0% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{-6}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\frac{U}{J} - 4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l 4e-6) U (* J (- (/ U J) 4.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 4e-6) {
		tmp = U;
	} else {
		tmp = J * ((U / J) - 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 4d-6) then
        tmp = u
    else
        tmp = j * ((u / j) - 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 4e-6) {
		tmp = U;
	} else {
		tmp = J * ((U / J) - 4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 4e-6:
		tmp = U
	else:
		tmp = J * ((U / J) - 4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 4e-6)
		tmp = U;
	else
		tmp = Float64(J * Float64(Float64(U / J) - 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 4e-6)
		tmp = U;
	else
		tmp = J * ((U / J) - 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 4e-6], U, N[(J * N[(N[(U / J), $MachinePrecision] - 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.99999999999999982e-6

   1. Initial program 83.7%

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

   4. Applied egg-rr 32.0%

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

   5. Taylor expanded in J around 0 47.7%

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

   if 3.99999999999999982e-6 < l

   1. Initial program 99.6%

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

   4. Applied egg-rr 1.7%

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

   5. Taylor expanded in K around 0 2.2%

      \[\leadsto \color{blue}{-4 \cdot J} + U \]
   6. Step-by-step derivation

      1. *-commutative 2.2%

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

   7. Simplified 2.2%

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

   8. Taylor expanded in J around inf 9.2%

      \[\leadsto \color{blue}{J \cdot \left(\frac{U}{J} - 4\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 53.9% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

3. Taylor expanded in l around 0 60.3%

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

   1. log1p-expm1-u 99.1%

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

   2. *-commutative 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in K around 0 51.3%

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

   1. associate-*r* 51.3%

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

   2. *-commutative 51.3%

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

8. Simplified 51.3%

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

9. Final simplification 51.3%

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

Alternative 17: 36.7% 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 87.9%

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

4. Applied egg-rr 24.1%

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

5. Taylor expanded in J around 0 35.9%

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

Reproduce

herbie shell --seed 2024137 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))