Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.5%

Time: 11.7s

Alternatives: 20

Speedup: 2.6×

• 49.3% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.0)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* l (+ (* 0.3333333333333333 (* J (* l l))) (* J 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.0):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(l * N[(N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * 2.0), $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 0.0 < (-.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))) < 0.0

   1. Initial program 75.4%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ t_1 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - t\_1\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14} \lor \neg \left(\ell \leq 8.8 \cdot 10^{+136}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - t\_1\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (cos (/ K 2.0))
           (* l (+ (* 0.3333333333333333 (* J (* l l))) (* J 2.0))))))
        (t_1 (exp (- l))))
   (if (<= l -3.5e+145)
     t_0
     (if (<= l -63.0)
       (+ U (* J (- 27.0 t_1)))
       (if (or (<= l 2.25e+14) (not (<= l 8.8e+136)))
         t_0
         (+ (* (- (exp l) t_1) J) U))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	double t_1 = exp(-l);
	double tmp;
	if (l <= -3.5e+145) {
		tmp = t_0;
	} else if (l <= -63.0) {
		tmp = U + (J * (27.0 - t_1));
	} else if ((l <= 2.25e+14) || !(l <= 8.8e+136)) {
		tmp = t_0;
	} else {
		tmp = ((exp(l) - t_1) * J) + U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * (l * ((0.3333333333333333d0 * (j * (l * l))) + (j * 2.0d0))))
    t_1 = exp(-l)
    if (l <= (-3.5d+145)) then
        tmp = t_0
    else if (l <= (-63.0d0)) then
        tmp = u + (j * (27.0d0 - t_1))
    else if ((l <= 2.25d+14) .or. (.not. (l <= 8.8d+136))) then
        tmp = t_0
    else
        tmp = ((exp(l) - t_1) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	double t_1 = Math.exp(-l);
	double tmp;
	if (l <= -3.5e+145) {
		tmp = t_0;
	} else if (l <= -63.0) {
		tmp = U + (J * (27.0 - t_1));
	} else if ((l <= 2.25e+14) || !(l <= 8.8e+136)) {
		tmp = t_0;
	} else {
		tmp = ((Math.exp(l) - t_1) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))))
	t_1 = math.exp(-l)
	tmp = 0
	if l <= -3.5e+145:
		tmp = t_0
	elif l <= -63.0:
		tmp = U + (J * (27.0 - t_1))
	elif (l <= 2.25e+14) or not (l <= 8.8e+136):
		tmp = t_0
	else:
		tmp = ((math.exp(l) - t_1) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) + Float64(J * 2.0)))))
	t_1 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -3.5e+145)
		tmp = t_0;
	elseif (l <= -63.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - t_1)));
	elseif ((l <= 2.25e+14) || !(l <= 8.8e+136))
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(exp(l) - t_1) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	t_1 = exp(-l);
	tmp = 0.0;
	if (l <= -3.5e+145)
		tmp = t_0;
	elseif (l <= -63.0)
		tmp = U + (J * (27.0 - t_1));
	elseif ((l <= 2.25e+14) || ~((l <= 8.8e+136)))
		tmp = t_0;
	else
		tmp = ((exp(l) - t_1) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -3.5e+145], t$95$0, If[LessEqual[l, -63.0], N[(U + N[(J * N[(27.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 2.25e+14], N[Not[LessEqual[l, 8.8e+136]], $MachinePrecision]], t$95$0, N[(N[(N[(N[Exp[l], $MachinePrecision] - t$95$1), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.5000000000000001e145 or -63 < l < 2.25e14 or 8.7999999999999998e136 < l

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -3.5000000000000001e145 < l < -63

   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 93.5%

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

   5. Applied egg-rr 93.5%

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

   if 2.25e14 < l < 8.7999999999999998e136

   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 88.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5 \cdot 10^{+145}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14} \lor \neg \left(\ell \leq 8.8 \cdot 10^{+136}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 92.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+145}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14} \lor \neg \left(\ell \leq 4.8 \cdot 10^{+146}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (cos (/ K 2.0))
           (* l (+ (* 0.3333333333333333 (* J (* l l))) (* J 2.0)))))))
   (if (<= l -5.6e+145)
     t_0
     (if (<= l -63.0)
       (+ U (* J (- 27.0 (exp (- l)))))
       (if (or (<= l 2.25e+14) (not (<= l 4.8e+146)))
         t_0
         (+ U (* J (- (exp l) 27.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	double tmp;
	if (l <= -5.6e+145) {
		tmp = t_0;
	} else if (l <= -63.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if ((l <= 2.25e+14) || !(l <= 4.8e+146)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (exp(l) - 27.0));
	}
	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 + (cos((k / 2.0d0)) * (l * ((0.3333333333333333d0 * (j * (l * l))) + (j * 2.0d0))))
    if (l <= (-5.6d+145)) then
        tmp = t_0
    else if (l <= (-63.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if ((l <= 2.25d+14) .or. (.not. (l <= 4.8d+146))) then
        tmp = t_0
    else
        tmp = u + (j * (exp(l) - 27.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	double tmp;
	if (l <= -5.6e+145) {
		tmp = t_0;
	} else if (l <= -63.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if ((l <= 2.25e+14) || !(l <= 4.8e+146)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (Math.exp(l) - 27.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))))
	tmp = 0
	if l <= -5.6e+145:
		tmp = t_0
	elif l <= -63.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif (l <= 2.25e+14) or not (l <= 4.8e+146):
		tmp = t_0
	else:
		tmp = U + (J * (math.exp(l) - 27.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) + Float64(J * 2.0)))))
	tmp = 0.0
	if (l <= -5.6e+145)
		tmp = t_0;
	elseif (l <= -63.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif ((l <= 2.25e+14) || !(l <= 4.8e+146))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 27.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	tmp = 0.0;
	if (l <= -5.6e+145)
		tmp = t_0;
	elseif (l <= -63.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif ((l <= 2.25e+14) || ~((l <= 4.8e+146)))
		tmp = t_0;
	else
		tmp = U + (J * (exp(l) - 27.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(0.3333333333333333 * N[(J * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.6e+145], t$95$0, If[LessEqual[l, -63.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 2.25e+14], N[Not[LessEqual[l, 4.8e+146]], $MachinePrecision]], t$95$0, N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5999999999999997e145 or -63 < l < 2.25e14 or 4.8000000000000004e146 < l

   1. Initial program 84.5%

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

   3. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

   5. Applied egg-rr 99.5%

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

   if -5.5999999999999997e145 < l < -63

   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 93.5%

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

   5. Applied egg-rr 93.5%

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

   if 2.25e14 < l < 4.8000000000000004e146

   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 88.9%

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

   5. Applied egg-rr 88.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+145}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14} \lor \neg \left(\ell \leq 4.8 \cdot 10^{+146}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -63.0)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 2.25e+14)
     (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
     (+ U (* J (- (exp l) 27.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -63.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (exp(l) - 27.0));
	}
	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 <= (-63.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 2.25d+14) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else
        tmp = u + (j * (exp(l) - 27.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -63.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (Math.exp(l) - 27.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -63.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 2.25e+14:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = U + (J * (math.exp(l) - 27.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -63.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 2.25e+14)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 27.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -63.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 2.25e+14)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	else
		tmp = U + (J * (exp(l) - 27.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -63.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.25e+14], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -63

   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 80.0%

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

   5. Applied egg-rr 80.0%

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

   if -63 < l < 2.25e14

   1. Initial program 75.8%

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

   3. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.2%

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

   5. Simplified 99.2%

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

   if 2.25e14 < 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 85.9%

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

   5. Applied egg-rr 85.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -63.0)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 2.25e+14)
     (+ U (* l (* (* J 2.0) (cos (* K 0.5)))))
     (+ U (* J (- (exp l) 27.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -63.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = U + (l * ((J * 2.0) * cos((K * 0.5))));
	} else {
		tmp = U + (J * (exp(l) - 27.0));
	}
	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 <= (-63.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 2.25d+14) then
        tmp = u + (l * ((j * 2.0d0) * cos((k * 0.5d0))))
    else
        tmp = u + (j * (exp(l) - 27.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -63.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = U + (l * ((J * 2.0) * Math.cos((K * 0.5))));
	} else {
		tmp = U + (J * (Math.exp(l) - 27.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -63.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 2.25e+14:
		tmp = U + (l * ((J * 2.0) * math.cos((K * 0.5))))
	else:
		tmp = U + (J * (math.exp(l) - 27.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -63.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 2.25e+14)
		tmp = Float64(U + Float64(l * Float64(Float64(J * 2.0) * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 27.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -63.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 2.25e+14)
		tmp = U + (l * ((J * 2.0) * cos((K * 0.5))));
	else
		tmp = U + (J * (exp(l) - 27.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -63.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.25e+14], N[(U + N[(l * N[(N[(J * 2.0), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -63

   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 80.0%

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

   5. Applied egg-rr 80.0%

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

   if -63 < l < 2.25e14

   1. Initial program 75.8%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. *-commutative 99.2%

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

      7. associate-*l* 99.2%

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

   5. Simplified 99.2%

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

   if 2.25e14 < 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 85.9%

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

   5. Applied egg-rr 85.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;U + \ell \cdot \left(\left(J \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -63.0)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 2.25e+14)
     (+ U (* J (* l (* 2.0 (cos (* K 0.5))))))
     (+ U (* J (- (exp l) 27.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -63.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = U + (J * (l * (2.0 * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (exp(l) - 27.0));
	}
	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 <= (-63.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 2.25d+14) then
        tmp = u + (j * (l * (2.0d0 * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (exp(l) - 27.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -63.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = U + (J * (l * (2.0 * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (Math.exp(l) - 27.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -63.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 2.25e+14:
		tmp = U + (J * (l * (2.0 * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (math.exp(l) - 27.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -63.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 2.25e+14)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 27.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -63.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 2.25e+14)
		tmp = U + (J * (l * (2.0 * cos((K * 0.5)))));
	else
		tmp = U + (J * (exp(l) - 27.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -63.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.25e+14], N[(U + N[(J * N[(l * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -63

   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 80.0%

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

   5. Applied egg-rr 80.0%

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

   if -63 < l < 2.25e14

   1. Initial program 75.8%

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

   3. Taylor expanded in l around 0 99.2%

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

      1. unpow2 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r* 99.1%

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

      3. associate-*r* 99.1%

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

      4. *-commutative 99.1%

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

      5. *-commutative 99.1%

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

   8. Simplified 99.1%

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

   if 2.25e14 < 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 85.9%

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

   5. Applied egg-rr 85.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -63.0)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 2.25e+14)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (+ U (* J (- (exp l) 27.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -63.0) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (exp(l) - 27.0));
	}
	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 <= (-63.0d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 2.25d+14) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (exp(l) - 27.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -63.0) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (Math.exp(l) - 27.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -63.0:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 2.25e+14:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (math.exp(l) - 27.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -63.0)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 2.25e+14)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 27.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -63.0)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 2.25e+14)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + (J * (exp(l) - 27.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -63.0], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.25e+14], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -63

   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 80.0%

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

   5. Applied egg-rr 80.0%

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

   if -63 < l < 2.25e14

   1. Initial program 75.8%

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

   3. Taylor expanded in l around 0 99.1%

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

   if 2.25e14 < 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 85.9%

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

   5. Applied egg-rr 85.9%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -63:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-52}:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5e-52)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 2.25e+14) (fma J (* l 2.0) U) (+ U (* J (- (exp l) 27.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5e-52) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 2.25e+14) {
		tmp = fma(J, (l * 2.0), U);
	} else {
		tmp = U + (J * (exp(l) - 27.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5e-52)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 2.25e+14)
		tmp = fma(J, Float64(l * 2.0), U);
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 27.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5e-52], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.25e+14], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5e-52

   1. Initial program 94.4%

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

   3. Taylor expanded in K around 0 76.2%

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

   5. Applied egg-rr 76.1%

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

   if -5e-52 < l < 2.25e14

   1. Initial program 77.5%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. *-commutative 99.1%

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

      3. associate-*l* 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-*r* 99.1%

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

      6. *-commutative 99.1%

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

      7. associate-*l* 99.1%

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

   5. Simplified 99.1%

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

   6. Taylor expanded in K around 0 91.6%

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

      1. +-commutative 91.6%

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

      2. associate-*r* 91.6%

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

      3. *-commutative 91.6%

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

      4. associate-*r* 91.6%

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

      5. fma-define 91.6%

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

   8. Simplified 91.6%

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

   if 2.25e14 < 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 85.9%

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

   5. Applied egg-rr 85.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-52}:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.25 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 2.2e+16)
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))
   (+ U (* J (- (exp l) 27.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.2e+16) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (J * (exp(l) - 27.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 2.2d+16) then
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    else
        tmp = u + (j * (exp(l) - 27.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.2e+16) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (J * (Math.exp(l) - 27.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 2.2e+16:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	else:
		tmp = U + (J * (math.exp(l) - 27.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 2.2e+16)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(exp(l) - 27.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 2.2e+16)
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	else
		tmp = U + (J * (exp(l) - 27.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 2.2e+16], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - 27.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.2e16

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0 75.8%

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

      1. *-commutative 75.8%

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

      2. *-commutative 75.8%

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

      3. associate-*l* 75.8%

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

      4. *-commutative 75.8%

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

      5. associate-*r* 75.8%

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

      6. *-commutative 75.8%

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

      7. associate-*l* 75.8%

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

   5. Simplified 75.8%

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

   6. Taylor expanded in J around inf 69.3%

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

   7. Taylor expanded in K around 0 59.7%

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

      1. *-commutative 59.7%

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

   9. Simplified 59.7%

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

   10. Taylor expanded in U around inf 68.3%

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

       1. associate-/l* 69.8%

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

   12. Simplified 69.8%

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

   if 2.2e16 < 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 87.1%

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

   5. Applied egg-rr 87.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.2 \cdot 10^{+16}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - 27\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.8% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot 2\right)\\ \mathbf{if}\;\ell \leq -125000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+210}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J 2.0))))
   (if (<= l -125000000.0)
     t_0
     (if (<= l 3.6e+15) U (if (<= l 8.2e+210) (* U U) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double tmp;
	if (l <= -125000000.0) {
		tmp = t_0;
	} else if (l <= 3.6e+15) {
		tmp = U;
	} else if (l <= 8.2e+210) {
		tmp = U * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = l * (j * 2.0d0)
    if (l <= (-125000000.0d0)) then
        tmp = t_0
    else if (l <= 3.6d+15) then
        tmp = u
    else if (l <= 8.2d+210) then
        tmp = u * u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double tmp;
	if (l <= -125000000.0) {
		tmp = t_0;
	} else if (l <= 3.6e+15) {
		tmp = U;
	} else if (l <= 8.2e+210) {
		tmp = U * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * 2.0)
	tmp = 0
	if l <= -125000000.0:
		tmp = t_0
	elif l <= 3.6e+15:
		tmp = U
	elif l <= 8.2e+210:
		tmp = U * U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * 2.0))
	tmp = 0.0
	if (l <= -125000000.0)
		tmp = t_0;
	elseif (l <= 3.6e+15)
		tmp = U;
	elseif (l <= 8.2e+210)
		tmp = Float64(U * U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (J * 2.0);
	tmp = 0.0;
	if (l <= -125000000.0)
		tmp = t_0;
	elseif (l <= 3.6e+15)
		tmp = U;
	elseif (l <= 8.2e+210)
		tmp = U * U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -125000000.0], t$95$0, If[LessEqual[l, 3.6e+15], U, If[LessEqual[l, 8.2e+210], N[(U * U), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.25e8 or 8.20000000000000002e210 < 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 32.6%

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

      1. *-commutative 32.6%

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

      2. *-commutative 32.6%

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

      3. associate-*l* 32.6%

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

      4. *-commutative 32.6%

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

      5. associate-*r* 32.6%

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

      6. *-commutative 32.6%

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

      7. associate-*l* 32.6%

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

   5. Simplified 32.6%

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

   6. Taylor expanded in J around inf 34.7%

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

   7. Taylor expanded in K around 0 28.6%

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

      1. *-commutative 28.6%

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

   9. Simplified 28.6%

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

   10. Taylor expanded in J around inf 26.8%

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

       1. associate-*r* 26.8%

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

       2. *-commutative 26.8%

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

   12. Simplified 26.8%

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

   if -1.25e8 < l < 3.6e15

   1. Initial program 76.2%

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

   3. Taylor expanded in J around 0 73.5%

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

   if 3.6e15 < l < 8.20000000000000002e210

   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 31.7%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -125000000:\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 3.6 \cdot 10^{+15}:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 8.2 \cdot 10^{+210}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+15}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+210}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 4e+15)
   (+ U (* l (* J 2.0)))
   (if (<= l 6.2e+210) (* U U) (* J (+ (* l 2.0) (/ U J))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 4e+15) {
		tmp = U + (l * (J * 2.0));
	} else if (l <= 6.2e+210) {
		tmp = U * U;
	} else {
		tmp = J * ((l * 2.0) + (U / J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 4d+15) then
        tmp = u + (l * (j * 2.0d0))
    else if (l <= 6.2d+210) then
        tmp = u * u
    else
        tmp = j * ((l * 2.0d0) + (u / j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 4e+15) {
		tmp = U + (l * (J * 2.0));
	} else if (l <= 6.2e+210) {
		tmp = U * U;
	} else {
		tmp = J * ((l * 2.0) + (U / J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 4e+15:
		tmp = U + (l * (J * 2.0))
	elif l <= 6.2e+210:
		tmp = U * U
	else:
		tmp = J * ((l * 2.0) + (U / J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 4e+15)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	elseif (l <= 6.2e+210)
		tmp = Float64(U * U);
	else
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(U / J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 4e+15)
		tmp = U + (l * (J * 2.0));
	elseif (l <= 6.2e+210)
		tmp = U * U;
	else
		tmp = J * ((l * 2.0) + (U / J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 4e+15], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.2e+210], N[(U * U), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 4e15

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. associate-*l* 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. *-commutative 76.2%

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

      7. associate-*l* 76.2%

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

   5. Simplified 76.2%

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

   6. Taylor expanded in K around 0 66.4%

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

   if 4e15 < l < 6.1999999999999999e210

   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 31.7%

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

   if 6.1999999999999999e210 < 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 32.6%

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

      1. *-commutative 32.6%

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

      2. *-commutative 32.6%

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

      3. associate-*l* 32.6%

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

      4. *-commutative 32.6%

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

      5. associate-*r* 32.6%

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

      6. *-commutative 32.6%

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

      7. associate-*l* 32.6%

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

   5. Simplified 32.6%

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

   6. Taylor expanded in J around inf 36.1%

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

   7. Taylor expanded in K around 0 35.4%

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

      1. *-commutative 35.4%

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

   9. Simplified 35.4%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4 \cdot 10^{+15}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6.2 \cdot 10^{+210}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.1% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{+17}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + U \cdot \left(J \cdot -3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 2.35e+17)
   (+ U (* l (* J 2.0)))
   (* J (+ (* l 2.0) (* U (* J -3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.35e+17) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * ((l * 2.0) + (U * (J * -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 <= 2.35d+17) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = j * ((l * 2.0d0) + (u * (j * (-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 <= 2.35e+17) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * ((l * 2.0) + (U * (J * -3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 2.35e+17:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = J * ((l * 2.0) + (U * (J * -3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 2.35e+17)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(U * Float64(J * -3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 2.35e+17)
		tmp = U + (l * (J * 2.0));
	else
		tmp = J * ((l * 2.0) + (U * (J * -3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 2.35e+17], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(U * N[(J * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.35e17

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. *-commutative 75.4%

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

      3. associate-*l* 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*r* 75.4%

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

      6. *-commutative 75.4%

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

      7. associate-*l* 75.4%

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

   5. Simplified 75.4%

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

   6. Taylor expanded in K around 0 65.8%

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

   if 2.35e17 < 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 21.5%

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

      1. *-commutative 21.5%

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

      2. *-commutative 21.5%

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

      3. associate-*l* 21.5%

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

      4. *-commutative 21.5%

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

      5. associate-*r* 21.5%

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

      6. *-commutative 21.5%

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

      7. associate-*l* 21.5%

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

   5. Simplified 21.5%

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

   6. Taylor expanded in J around inf 26.2%

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

   7. Taylor expanded in K around 0 24.0%

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

      1. *-commutative 24.0%

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

   9. Simplified 24.0%

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

   11. Applied egg-rr 29.0%

       \[\leadsto J \cdot \left(\ell \cdot 2 + \color{blue}{\left(U \cdot J\right) \cdot -3}\right) \]
   12. Step-by-step derivation

       1. associate-*l* 29.0%

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

   13. Simplified 29.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.35 \cdot 10^{+17}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + U \cdot \left(J \cdot -3\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.6% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot 2\right)\\ \mathbf{if}\;\ell \leq 6 \cdot 10^{+15}:\\ \;\;\;\;U + t\_0\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+210}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J 2.0))))
   (if (<= l 6e+15) (+ U t_0) (if (<= l 6.5e+210) (* U U) t_0))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double tmp;
	if (l <= 6e+15) {
		tmp = U + t_0;
	} else if (l <= 6.5e+210) {
		tmp = U * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = l * (j * 2.0d0)
    if (l <= 6d+15) then
        tmp = u + t_0
    else if (l <= 6.5d+210) then
        tmp = u * u
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double tmp;
	if (l <= 6e+15) {
		tmp = U + t_0;
	} else if (l <= 6.5e+210) {
		tmp = U * U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * 2.0)
	tmp = 0
	if l <= 6e+15:
		tmp = U + t_0
	elif l <= 6.5e+210:
		tmp = U * U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * 2.0))
	tmp = 0.0
	if (l <= 6e+15)
		tmp = Float64(U + t_0);
	elseif (l <= 6.5e+210)
		tmp = Float64(U * U);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (J * 2.0);
	tmp = 0.0;
	if (l <= 6e+15)
		tmp = U + t_0;
	elseif (l <= 6.5e+210)
		tmp = U * U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 6e+15], N[(U + t$95$0), $MachinePrecision], If[LessEqual[l, 6.5e+210], N[(U * U), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if l < 6e15

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0 76.1%

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

      1. *-commutative 76.1%

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

      2. *-commutative 76.1%

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

      3. associate-*l* 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*r* 76.2%

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

      6. *-commutative 76.2%

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

      7. associate-*l* 76.2%

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

   5. Simplified 76.2%

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

   6. Taylor expanded in K around 0 66.4%

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

   if 6e15 < l < 6.4999999999999996e210

   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 31.7%

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

   if 6.4999999999999996e210 < 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 32.6%

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

      1. *-commutative 32.6%

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

      2. *-commutative 32.6%

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

      3. associate-*l* 32.6%

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

      4. *-commutative 32.6%

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

      5. associate-*r* 32.6%

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

      6. *-commutative 32.6%

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

      7. associate-*l* 32.6%

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

   5. Simplified 32.6%

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

   6. Taylor expanded in J around inf 36.1%

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

   7. Taylor expanded in K around 0 35.4%

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

      1. *-commutative 35.4%

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

   9. Simplified 35.4%

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

   10. Taylor expanded in J around inf 32.2%

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

       1. associate-*r* 32.2%

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

       2. *-commutative 32.2%

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

   12. Simplified 32.2%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 6 \cdot 10^{+15}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+210}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.1% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{+17}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 - J \cdot U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 3.4e+17) (+ U (* l (* J 2.0))) (* J (- (* l 2.0) (* J U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3.4e+17) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * ((l * 2.0) - (J * U));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 3.4d+17) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = j * ((l * 2.0d0) - (j * u))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 3.4e+17) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * ((l * 2.0) - (J * U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 3.4e+17:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = J * ((l * 2.0) - (J * U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 3.4e+17)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(J * Float64(Float64(l * 2.0) - Float64(J * U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 3.4e+17)
		tmp = U + (l * (J * 2.0));
	else
		tmp = J * ((l * 2.0) - (J * U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 3.4e+17], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] - N[(J * U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.4e17

   1. Initial program 84.2%

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

   3. Taylor expanded in l around 0 75.4%

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

      1. *-commutative 75.4%

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

      2. *-commutative 75.4%

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

      3. associate-*l* 75.4%

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

      4. *-commutative 75.4%

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

      5. associate-*r* 75.4%

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

      6. *-commutative 75.4%

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

      7. associate-*l* 75.4%

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

   5. Simplified 75.4%

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

   6. Taylor expanded in K around 0 65.8%

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

   if 3.4e17 < 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 21.5%

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

      1. *-commutative 21.5%

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

      2. *-commutative 21.5%

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

      3. associate-*l* 21.5%

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

      4. *-commutative 21.5%

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

      5. associate-*r* 21.5%

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

      6. *-commutative 21.5%

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

      7. associate-*l* 21.5%

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

   5. Simplified 21.5%

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

   6. Taylor expanded in J around inf 26.2%

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

   7. Taylor expanded in K around 0 24.0%

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

      1. *-commutative 24.0%

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

   9. Simplified 24.0%

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

   11. Applied egg-rr 20.8%

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

       1. fma-undefine 20.8%

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

       2. *-commutative 20.8%

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

       3. distribute-lft1-in 20.8%

          \[\leadsto J \cdot \left(\ell \cdot 2 + \left(U \cdot J + \color{blue}{\left(-3 + 1\right) \cdot \left(U \cdot J\right)}\right)\right) \]

       4. metadata-eval 20.8%

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

       5. distribute-rgt1-in 29.0%

          \[\leadsto J \cdot \left(\ell \cdot 2 + \color{blue}{\left(-2 + 1\right) \cdot \left(U \cdot J\right)}\right) \]

       6. metadata-eval 29.0%

          \[\leadsto J \cdot \left(\ell \cdot 2 + \color{blue}{-1} \cdot \left(U \cdot J\right)\right) \]

       7. neg-mul-1 29.0%

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

       8. distribute-rgt-neg-in 29.0%

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

   13. Simplified 29.0%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.4 \cdot 10^{+17}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 - J \cdot U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.3% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -35000000000 \lor \neg \left(\ell \leq 3.6 \cdot 10^{+15}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -35000000000.0) (not (<= l 3.6e+15))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -35000000000.0) || !(l <= 3.6e+15)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-35000000000.0d0)) .or. (.not. (l <= 3.6d+15))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -35000000000.0) || !(l <= 3.6e+15)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -35000000000.0) or not (l <= 3.6e+15):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -35000000000.0) || !(l <= 3.6e+15))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -35000000000.0) || ~((l <= 3.6e+15)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -35000000000.0], N[Not[LessEqual[l, 3.6e+15]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -3.5e10 or 3.6e15 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 19.0%

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

   if -3.5e10 < l < 3.6e15

   1. Initial program 76.5%

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

   3. Taylor expanded in J around 0 72.4%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -35000000000 \lor \neg \left(\ell \leq 3.6 \cdot 10^{+15}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 42.3% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+14}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+15}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -6.2e+14) (* U (- U -4.0)) (if (<= l 4e+15) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.2e+14) {
		tmp = U * (U - -4.0);
	} else if (l <= 4e+15) {
		tmp = U;
	} else {
		tmp = U * 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 <= (-6.2d+14)) then
        tmp = u * (u - (-4.0d0))
    else if (l <= 4d+15) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6.2e+14) {
		tmp = U * (U - -4.0);
	} else if (l <= 4e+15) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -6.2e+14:
		tmp = U * (U - -4.0)
	elif l <= 4e+15:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -6.2e+14)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 4e+15)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -6.2e+14)
		tmp = U * (U - -4.0);
	elseif (l <= 4e+15)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -6.2e+14], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4e+15], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.2e14

   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 15.8%

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

   if -6.2e14 < l < 4e15

   1. Initial program 76.5%

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

   3. Taylor expanded in J around 0 72.4%

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

   if 4e15 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 22.1%

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

Alternative 17: 60.2% accurate, 28.4× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* U (+ 1.0 (* 2.0 (* J (/ l U))))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * (l / 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 * (1.0d0 + (2.0d0 * (j * (l / u))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * (l / U))));
}

Python

def code(J, l, K, U):
	return U * (1.0 + (2.0 * (J * (l / U))))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + (2.0 * (J * (l / U))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in l around 0 62.6%

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

   1. *-commutative 62.6%

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

   2. *-commutative 62.6%

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

   3. associate-*l* 62.6%

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

   4. *-commutative 62.6%

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

   5. associate-*r* 62.6%

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

   6. *-commutative 62.6%

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

   7. associate-*l* 62.6%

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

5. Simplified 62.6%

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

6. Taylor expanded in J around inf 58.8%

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

7. Taylor expanded in K around 0 50.9%

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

   1. *-commutative 50.9%

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

9. Simplified 50.9%

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

10. Taylor expanded in U around inf 60.4%

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

    1. associate-/l* 63.4%

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

12. Simplified 63.4%

    \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)} \]
13. Add Preprocessing

Alternative 18: 36.8% 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 88.0%

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

3. Taylor expanded in J around 0 38.0%

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

Alternative 19: 3.0% accurate, 312.0× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return 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 = j
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in l around 0 62.6%

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

   1. *-commutative 62.6%

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

   2. *-commutative 62.6%

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

   3. associate-*l* 62.6%

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

   4. *-commutative 62.6%

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

   5. associate-*r* 62.6%

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

   6. *-commutative 62.6%

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

   7. associate-*l* 62.6%

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

5. Simplified 62.6%

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

6. Taylor expanded in J around inf 58.8%

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

7. Taylor expanded in K around 0 50.9%

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

   1. *-commutative 50.9%

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

9. Simplified 50.9%

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

11. Applied egg-rr 5.0%

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

    1. *-commutative 5.0%

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

    2. associate-/l* 3.1%

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

    3. *-inverses 3.1%

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

    4. *-rgt-identity 3.1%

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

13. Simplified 3.1%

    \[\leadsto \color{blue}{J} \]
14. Add Preprocessing

Alternative 20: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -4.0)

C

double code(double J, double l, double K, double U) {
	return -4.0;
}

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 = -4.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -4.0;
}

Python

def code(J, l, K, U):
	return -4.0

Julia

function code(J, l, K, U)
	return -4.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -4.0;
end

Wolfram

code[J_, l_, K_, U_] := -4.0

Derivation

1. Initial program 88.0%

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

4. Applied egg-rr 2.7%

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

5. Taylor expanded in U around 0 2.9%

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

Reproduce

herbie shell --seed 2024181 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))