Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.8%

Time: 12.1s

Alternatives: 12

Speedup: 1.4×

• 49.7% 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: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{-5}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e-5)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (* t_0 (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e-5)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (l * J))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 5e-5)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e-5):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e-5))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e-5)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e-5]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 5.00000000000000024e-5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.00000000000000024e-5

   1. Initial program 78.5%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

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

Alternative 2: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + 0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot t_0\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -60000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 0.00022:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* 0.3333333333333333 (* (* J (pow l 3.0)) t_0))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -2.25e+92)
     t_1
     (if (<= l -60000000000.0)
       t_2
       (if (<= l 0.00022)
         (+ U (* (* l 2.0) (* J t_0)))
         (if (<= l 1.08e+102) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (0.3333333333333333 * ((J * pow(l, 3.0)) * t_0));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.25e+92) {
		tmp = t_1;
	} else if (l <= -60000000000.0) {
		tmp = t_2;
	} else if (l <= 0.00022) {
		tmp = U + ((l * 2.0) * (J * t_0));
	} else if (l <= 1.08e+102) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + (0.3333333333333333d0 * ((j * (l ** 3.0d0)) * t_0))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-2.25d+92)) then
        tmp = t_1
    else if (l <= (-60000000000.0d0)) then
        tmp = t_2
    else if (l <= 0.00022d0) then
        tmp = u + ((l * 2.0d0) * (j * t_0))
    else if (l <= 1.08d+102) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (0.3333333333333333 * ((J * Math.pow(l, 3.0)) * t_0));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.25e+92) {
		tmp = t_1;
	} else if (l <= -60000000000.0) {
		tmp = t_2;
	} else if (l <= 0.00022) {
		tmp = U + ((l * 2.0) * (J * t_0));
	} else if (l <= 1.08e+102) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (0.3333333333333333 * ((J * math.pow(l, 3.0)) * t_0))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -2.25e+92:
		tmp = t_1
	elif l <= -60000000000.0:
		tmp = t_2
	elif l <= 0.00022:
		tmp = U + ((l * 2.0) * (J * t_0))
	elif l <= 1.08e+102:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(0.3333333333333333 * Float64(Float64(J * (l ^ 3.0)) * t_0)))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -2.25e+92)
		tmp = t_1;
	elseif (l <= -60000000000.0)
		tmp = t_2;
	elseif (l <= 0.00022)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * t_0)));
	elseif (l <= 1.08e+102)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + (0.3333333333333333 * ((J * (l ^ 3.0)) * t_0));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -2.25e+92)
		tmp = t_1;
	elseif (l <= -60000000000.0)
		tmp = t_2;
	elseif (l <= 0.00022)
		tmp = U + ((l * 2.0) * (J * t_0));
	elseif (l <= 1.08e+102)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(0.3333333333333333 * N[(N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -2.25e+92], t$95$1, If[LessEqual[l, -60000000000.0], t$95$2, If[LessEqual[l, 0.00022], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.08e+102], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.25e92 or 1.08000000000000002e102 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 97.6%

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

   4. Taylor expanded in l around inf 97.6%

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

      1. associate-*r* 97.6%

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

   6. Simplified 97.6%

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

   if -2.25e92 < l < -6e10 or 2.20000000000000008e-4 < l < 1.08000000000000002e102

   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 83.9%

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

   if -6e10 < l < 2.20000000000000008e-4

   1. Initial program 78.7%

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

   3. Taylor expanded in l around 0 99.0%

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

      1. associate-*r* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l* 99.1%

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

      4. associate-*r* 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+92}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -60000000000:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 0.00022:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.08 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.01], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 84.3%

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

   3. Taylor expanded in l around 0 75.3%

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

      1. associate-*r* 75.3%

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

      2. *-commutative 75.3%

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

      3. associate-*l* 75.3%

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

      4. associate-*r* 75.3%

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

   5. Simplified 75.3%

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

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

   1. Initial program 89.5%

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

   3. Taylor expanded in l around 0 91.3%

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

   4. Taylor expanded in K around 0 88.8%

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

4. Final simplification 84.5%

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

Alternative 4: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -60000000000 \lor \neg \left(\ell \leq 0.007\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -60000000000.0) (not (<= l 0.007)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -60000000000.0) || !(l <= 0.007)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-60000000000.0d0)) .or. (.not. (l <= 0.007d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -60000000000.0) || !(l <= 0.007)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -60000000000.0) or not (l <= 0.007):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -60000000000.0) || !(l <= 0.007))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -60000000000.0) || ~((l <= 0.007)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -60000000000.0], N[Not[LessEqual[l, 0.007]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6e10 or 0.00700000000000000015 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 70.9%

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

   if -6e10 < l < 0.00700000000000000015

   1. Initial program 78.7%

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

   3. Taylor expanded in l around 0 99.0%

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

      1. associate-*r* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l* 99.1%

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

      4. associate-*r* 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -60000000000 \lor \neg \left(\ell \leq 0.007\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   (cos (/ K 2.0))
   (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (l * J))));
}

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 + (cos((k / 2.0d0)) * ((0.3333333333333333d0 * (j * (l ** 3.0d0))) + (2.0d0 * (l * j))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in l around 0 91.4%

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

4. Final simplification 91.4%

   \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right) \]
5. Add Preprocessing

Alternative 6: 87.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (* (cos (/ K 2.0)) (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in l around 0 91.4%

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

4. Final simplification 91.4%

   \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -470:\\ \;\;\;\;U + \left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332\\ \mathbf{elif}\;\ell \leq 12500000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -7e+212)
     t_0
     (if (<= l -470.0)
       (+ U (* (* J (pow K 4.0)) 0.020833333333333332))
       (if (<= l 12500000.0) (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -7e+212) {
		tmp = t_0;
	} else if (l <= -470.0) {
		tmp = U + ((J * pow(K, 4.0)) * 0.020833333333333332);
	} else if (l <= 12500000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-7d+212)) then
        tmp = t_0
    else if (l <= (-470.0d0)) then
        tmp = u + ((j * (k ** 4.0d0)) * 0.020833333333333332d0)
    else if (l <= 12500000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -7e+212) {
		tmp = t_0;
	} else if (l <= -470.0) {
		tmp = U + ((J * Math.pow(K, 4.0)) * 0.020833333333333332);
	} else if (l <= 12500000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -7e+212:
		tmp = t_0
	elif l <= -470.0:
		tmp = U + ((J * math.pow(K, 4.0)) * 0.020833333333333332)
	elif l <= 12500000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -7e+212)
		tmp = t_0;
	elseif (l <= -470.0)
		tmp = Float64(U + Float64(Float64(J * (K ^ 4.0)) * 0.020833333333333332));
	elseif (l <= 12500000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -7e+212)
		tmp = t_0;
	elseif (l <= -470.0)
		tmp = U + ((J * (K ^ 4.0)) * 0.020833333333333332);
	elseif (l <= 12500000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7e+212], t$95$0, If[LessEqual[l, -470.0], N[(U + N[(N[(J * N[Power[K, 4.0], $MachinePrecision]), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 12500000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.99999999999999974e212 or 1.25e7 < 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 90.4%

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

   4. Taylor expanded in l around inf 90.4%

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

   5. Taylor expanded in K around 0 70.1%

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

   if -6.99999999999999974e212 < l < -470

   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 2.0%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 12.9%

      \[\leadsto \left(J \cdot 8\right) \cdot \color{blue}{\left(1 + \left(-0.125 \cdot {K}^{2} + 0.0026041666666666665 \cdot {K}^{4}\right)\right)} + U \]

   6. Taylor expanded in K around inf 44.3%

      \[\leadsto \color{blue}{0.020833333333333332 \cdot \left(J \cdot {K}^{4}\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 44.3%

         \[\leadsto \color{blue}{\left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332} + U \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332} + U \]

   if -470 < l < 1.25e7

   1. Initial program 78.5%

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

   3. Taylor expanded in l around 0 99.7%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+212}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -470:\\ \;\;\;\;U + \left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332\\ \mathbf{elif}\;\ell \leq 12500000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -7 \cdot 10^{+212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -600:\\ \;\;\;\;U + \left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332\\ \mathbf{elif}\;\ell \leq 27000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -7e+212)
     t_0
     (if (<= l -600.0)
       (+ U (* (* J (pow K 4.0)) 0.020833333333333332))
       (if (<= l 27000000.0) (+ U (* (* l 2.0) (* J (cos (* K 0.5))))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -7e+212) {
		tmp = t_0;
	} else if (l <= -600.0) {
		tmp = U + ((J * pow(K, 4.0)) * 0.020833333333333332);
	} else if (l <= 27000000.0) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-7d+212)) then
        tmp = t_0
    else if (l <= (-600.0d0)) then
        tmp = u + ((j * (k ** 4.0d0)) * 0.020833333333333332d0)
    else if (l <= 27000000.0d0) then
        tmp = u + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -7e+212) {
		tmp = t_0;
	} else if (l <= -600.0) {
		tmp = U + ((J * Math.pow(K, 4.0)) * 0.020833333333333332);
	} else if (l <= 27000000.0) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -7e+212:
		tmp = t_0
	elif l <= -600.0:
		tmp = U + ((J * math.pow(K, 4.0)) * 0.020833333333333332)
	elif l <= 27000000.0:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -7e+212)
		tmp = t_0;
	elseif (l <= -600.0)
		tmp = Float64(U + Float64(Float64(J * (K ^ 4.0)) * 0.020833333333333332));
	elseif (l <= 27000000.0)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -7e+212)
		tmp = t_0;
	elseif (l <= -600.0)
		tmp = U + ((J * (K ^ 4.0)) * 0.020833333333333332);
	elseif (l <= 27000000.0)
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7e+212], t$95$0, If[LessEqual[l, -600.0], N[(U + N[(N[(J * N[Power[K, 4.0], $MachinePrecision]), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 27000000.0], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.99999999999999974e212 or 2.7e7 < 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 90.4%

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

   4. Taylor expanded in l around inf 90.4%

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

   5. Taylor expanded in K around 0 70.1%

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

   if -6.99999999999999974e212 < l < -600

   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 2.0%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 12.9%

      \[\leadsto \left(J \cdot 8\right) \cdot \color{blue}{\left(1 + \left(-0.125 \cdot {K}^{2} + 0.0026041666666666665 \cdot {K}^{4}\right)\right)} + U \]

   6. Taylor expanded in K around inf 44.3%

      \[\leadsto \color{blue}{0.020833333333333332 \cdot \left(J \cdot {K}^{4}\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 44.3%

         \[\leadsto \color{blue}{\left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332} + U \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332} + U \]

   if -600 < l < 2.7e7

   1. Initial program 78.5%

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

   3. Taylor expanded in l around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+212}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -600:\\ \;\;\;\;U + \left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332\\ \mathbf{elif}\;\ell \leq 27000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -860:\\ \;\;\;\;U + \left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332\\ \mathbf{elif}\;\ell \leq 2.5:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -1e+213)
     t_0
     (if (<= l -860.0)
       (+ U (* (* J (pow K 4.0)) 0.020833333333333332))
       (if (<= l 2.5) (+ U (* J (* l 2.0))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -1e+213) {
		tmp = t_0;
	} else if (l <= -860.0) {
		tmp = U + ((J * pow(K, 4.0)) * 0.020833333333333332);
	} else if (l <= 2.5) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-1d+213)) then
        tmp = t_0
    else if (l <= (-860.0d0)) then
        tmp = u + ((j * (k ** 4.0d0)) * 0.020833333333333332d0)
    else if (l <= 2.5d0) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -1e+213) {
		tmp = t_0;
	} else if (l <= -860.0) {
		tmp = U + ((J * Math.pow(K, 4.0)) * 0.020833333333333332);
	} else if (l <= 2.5) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -1e+213:
		tmp = t_0
	elif l <= -860.0:
		tmp = U + ((J * math.pow(K, 4.0)) * 0.020833333333333332)
	elif l <= 2.5:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -1e+213)
		tmp = t_0;
	elseif (l <= -860.0)
		tmp = Float64(U + Float64(Float64(J * (K ^ 4.0)) * 0.020833333333333332));
	elseif (l <= 2.5)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -1e+213)
		tmp = t_0;
	elseif (l <= -860.0)
		tmp = U + ((J * (K ^ 4.0)) * 0.020833333333333332);
	elseif (l <= 2.5)
		tmp = U + (J * (l * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1e+213], t$95$0, If[LessEqual[l, -860.0], N[(U + N[(N[(J * N[Power[K, 4.0], $MachinePrecision]), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.5], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -9.99999999999999984e212 or 2.5 < 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 90.4%

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

   4. Taylor expanded in l around inf 90.4%

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

   5. Taylor expanded in K around 0 70.1%

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

   if -9.99999999999999984e212 < l < -860

   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 2.0%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 12.9%

      \[\leadsto \left(J \cdot 8\right) \cdot \color{blue}{\left(1 + \left(-0.125 \cdot {K}^{2} + 0.0026041666666666665 \cdot {K}^{4}\right)\right)} + U \]

   6. Taylor expanded in K around inf 44.3%

      \[\leadsto \color{blue}{0.020833333333333332 \cdot \left(J \cdot {K}^{4}\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 44.3%

         \[\leadsto \color{blue}{\left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332} + U \]

   8. Simplified 44.3%

      \[\leadsto \color{blue}{\left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332} + U \]

   if -860 < l < 2.5

   1. Initial program 78.5%

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

   3. Taylor expanded in l around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*l* 99.7%

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

      4. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in K around 0 87.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+213}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -860:\\ \;\;\;\;U + \left(J \cdot {K}^{4}\right) \cdot 0.020833333333333332\\ \mathbf{elif}\;\ell \leq 2.5:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -60000000000 \lor \neg \left(\ell \leq 2.5\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -60000000000.0) (not (<= l 2.5)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -60000000000.0) || !(l <= 2.5)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-60000000000.0d0)) .or. (.not. (l <= 2.5d0))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -60000000000.0) || !(l <= 2.5)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -60000000000.0) or not (l <= 2.5):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -60000000000.0) || !(l <= 2.5))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -60000000000.0) || ~((l <= 2.5)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -60000000000.0], N[Not[LessEqual[l, 2.5]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6e10 or 2.5 < 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 81.0%

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

   4. Taylor expanded in l around inf 81.0%

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

   5. Taylor expanded in K around 0 57.3%

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

   if -6e10 < l < 2.5

   1. Initial program 78.7%

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

   3. Taylor expanded in l around 0 99.0%

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

      1. associate-*r* 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l* 99.1%

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

      4. associate-*r* 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in K around 0 87.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -60000000000 \lor \neg \left(\ell \leq 2.5\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.4% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.8%

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

3. Taylor expanded in l around 0 71.2%

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

   1. associate-*r* 71.2%

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

   2. *-commutative 71.2%

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

   3. associate-*l* 71.2%

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

   4. associate-*r* 71.2%

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

5. Simplified 71.2%

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

6. Taylor expanded in K around 0 59.5%

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

7. Final simplification 59.5%

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

Alternative 12: 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 87.8%

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

4. Applied egg-rr 33.3%

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

5. Taylor expanded in J around 0 45.0%

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

6. Final simplification 45.0%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))