Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.8%

Time: 10.4s

Alternatives: 17

Speedup: 2.7×

• 49.5% 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: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{-\ell}\\ t_2 := e^{\ell} - t\_1\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(J \cdot \left(27 - t\_1\right)\right) \cdot t\_0 + U\\ \mathbf{elif}\;t\_2 \leq 0.0005:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(t\_2 \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (exp (- l))) (t_2 (- (exp l) t_1)))
   (if (<= t_2 (- INFINITY))
     (+ (* (* J (- 27.0 t_1)) t_0) U)
     (if (<= t_2 0.0005)
       (+ U (* t_0 (* l (+ (* 0.3333333333333333 (* J (* l l))) (* J 2.0)))))
       (+ U (* t_0 (* t_2 J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(-l);
	double t_2 = exp(l) - t_1;
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	} else if (t_2 <= 0.0005) {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	} else {
		tmp = U + (t_0 * (t_2 * J));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(-l);
	double t_2 = Math.exp(l) - t_1;
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	} else if (t_2 <= 0.0005) {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	} else {
		tmp = U + (t_0 * (t_2 * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(-l)
	t_2 = math.exp(l) - t_1
	tmp = 0
	if t_2 <= -math.inf:
		tmp = ((J * (27.0 - t_1)) * t_0) + U
	elif t_2 <= 0.0005:
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))))
	else:
		tmp = U + (t_0 * (t_2 * J))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = exp(Float64(-l))
	t_2 = Float64(exp(l) - t_1)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(J * Float64(27.0 - t_1)) * t_0) + U);
	elseif (t_2 <= 0.0005)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) + Float64(J * 2.0)))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(t_2 * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(-l);
	t_2 = exp(l) - t_1;
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	elseif (t_2 <= 0.0005)
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	else
		tmp = U + (t_0 * (t_2 * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[l], $MachinePrecision] - t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(J * N[(27.0 - t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$2, 0.0005], 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], N[(U + N[(t$95$0 * N[(t$95$2 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000001e-4

   1. Initial program 71.7%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.9%

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

   if 5.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\left(J \cdot \left(27 - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 0.0005:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -0.015:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 210:\\ \;\;\;\;U + t\_1 \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (- (exp l) (exp (- l))) J)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))
   (if (<= l -1.95e+77)
     t_2
     (if (<= l -0.015)
       t_0
       (if (<= l 210.0)
         (+ U (* t_1 (* l (+ (* 0.3333333333333333 (* J (* l l))) (* J 2.0)))))
         (if (<= l 2.4e+93) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((exp(l) - exp(-l)) * J);
	double t_1 = cos((K / 2.0));
	double t_2 = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double tmp;
	if (l <= -1.95e+77) {
		tmp = t_2;
	} else if (l <= -0.015) {
		tmp = t_0;
	} else if (l <= 210.0) {
		tmp = U + (t_1 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	} else if (l <= 2.4e+93) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = u + ((exp(l) - exp(-l)) * j)
    t_1 = cos((k / 2.0d0))
    t_2 = u + (t_1 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    if (l <= (-1.95d+77)) then
        tmp = t_2
    else if (l <= (-0.015d0)) then
        tmp = t_0
    else if (l <= 210.0d0) then
        tmp = u + (t_1 * (l * ((0.3333333333333333d0 * (j * (l * l))) + (j * 2.0d0))))
    else if (l <= 2.4d+93) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double tmp;
	if (l <= -1.95e+77) {
		tmp = t_2;
	} else if (l <= -0.015) {
		tmp = t_0;
	} else if (l <= 210.0) {
		tmp = U + (t_1 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	} else if (l <= 2.4e+93) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((math.exp(l) - math.exp(-l)) * J)
	t_1 = math.cos((K / 2.0))
	t_2 = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	tmp = 0
	if l <= -1.95e+77:
		tmp = t_2
	elif l <= -0.015:
		tmp = t_0
	elif l <= 210.0:
		tmp = U + (t_1 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))))
	elif l <= 2.4e+93:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))))
	tmp = 0.0
	if (l <= -1.95e+77)
		tmp = t_2;
	elseif (l <= -0.015)
		tmp = t_0;
	elseif (l <= 210.0)
		tmp = Float64(U + Float64(t_1 * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) + Float64(J * 2.0)))));
	elseif (l <= 2.4e+93)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((exp(l) - exp(-l)) * J);
	t_1 = cos((K / 2.0));
	t_2 = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	tmp = 0.0;
	if (l <= -1.95e+77)
		tmp = t_2;
	elseif (l <= -0.015)
		tmp = t_0;
	elseif (l <= 210.0)
		tmp = U + (t_1 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	elseif (l <= 2.4e+93)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.95e+77], t$95$2, If[LessEqual[l, -0.015], t$95$0, If[LessEqual[l, 210.0], N[(U + N[(t$95$1 * 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, 2.4e+93], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9499999999999999e77 or 2.4000000000000001e93 < 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 99.1%

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

      1. unpow2 90.2%

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

   5. Applied egg-rr 99.1%

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

   if -1.9499999999999999e77 < l < -0.014999999999999999 or 210 < l < 2.4000000000000001e93

   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%

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

   if -0.014999999999999999 < l < 210

   1. Initial program 71.9%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. unpow2 99.1%

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

   5. Applied egg-rr 99.1%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+77}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.015:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 210:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+93}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\left(J \cdot \left(27 - t\_1\right)\right) \cdot t\_0 + U\\ \mathbf{elif}\;\ell \leq 210:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+93}:\\ \;\;\;\;U + \left(e^{\ell} - t\_1\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (exp (- l))))
   (if (<= l -4.0)
     (+ (* (* J (- 27.0 t_1)) t_0) U)
     (if (<= l 210.0)
       (+ U (* t_0 (* l (+ (* 0.3333333333333333 (* J (* l l))) (* J 2.0)))))
       (if (<= l 3.2e+93)
         (+ U (* (- (exp l) t_1) J))
         (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(-l);
	double tmp;
	if (l <= -4.0) {
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	} else if (l <= 210.0) {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	} else if (l <= 3.2e+93) {
		tmp = U + ((exp(l) - t_1) * J);
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(-l)
    if (l <= (-4.0d0)) then
        tmp = ((j * (27.0d0 - t_1)) * t_0) + u
    else if (l <= 210.0d0) then
        tmp = u + (t_0 * (l * ((0.3333333333333333d0 * (j * (l * l))) + (j * 2.0d0))))
    else if (l <= 3.2d+93) then
        tmp = u + ((exp(l) - t_1) * j)
    else
        tmp = u + (t_0 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(-l);
	double tmp;
	if (l <= -4.0) {
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	} else if (l <= 210.0) {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	} else if (l <= 3.2e+93) {
		tmp = U + ((Math.exp(l) - t_1) * J);
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(-l)
	tmp = 0
	if l <= -4.0:
		tmp = ((J * (27.0 - t_1)) * t_0) + U
	elif l <= 210.0:
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))))
	elif l <= 3.2e+93:
		tmp = U + ((math.exp(l) - t_1) * J)
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -4.0)
		tmp = Float64(Float64(Float64(J * Float64(27.0 - t_1)) * t_0) + U);
	elseif (l <= 210.0)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * Float64(l * l))) + Float64(J * 2.0)))));
	elseif (l <= 3.2e+93)
		tmp = Float64(U + Float64(Float64(exp(l) - t_1) * J));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(-l);
	tmp = 0.0;
	if (l <= -4.0)
		tmp = ((J * (27.0 - t_1)) * t_0) + U;
	elseif (l <= 210.0)
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l * l))) + (J * 2.0))));
	elseif (l <= 3.2e+93)
		tmp = U + ((exp(l) - t_1) * J);
	else
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	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[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -4.0], N[(N[(N[(J * N[(27.0 - t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 210.0], 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], If[LessEqual[l, 3.2e+93], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - t$95$1), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

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

   if -4 < l < 210

   1. Initial program 71.9%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. unpow2 99.1%

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

   5. Applied egg-rr 99.1%

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

   if 210 < l < 3.2000000000000001e93

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 83.3%

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

   if 3.2000000000000001e93 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. unpow2 87.4%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4:\\ \;\;\;\;\left(J \cdot \left(27 - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;\ell \leq 210:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left(\ell \cdot \ell\right)\right) + J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+93}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.01:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.01)
   (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U)))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.01) {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.01d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.01) {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.01:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.01)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.01)
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.01], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l* 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in l around inf 51.6%

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

   7. Taylor expanded in U around inf 69.7%

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

      1. associate-/l* 74.7%

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

      2. *-commutative 74.7%

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

   9. Simplified 74.7%

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

   if 0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0 89.8%

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

   4. Taylor expanded in K around 0 85.9%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.01:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l* 64.9%

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

   5. Simplified 64.9%

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

   6. Taylor expanded in J around inf 68.0%

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

   if 0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0 89.8%

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

   4. Taylor expanded in K around 0 85.9%

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

4. Final simplification 81.9%

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

Alternative 6: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.01:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.01)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.01) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.01d0) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.01) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.01:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.01)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 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]}, If[LessEqual[t$95$0, 0.01], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.0100000000000000002

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. associate-*l* 64.9%

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

   5. Simplified 64.9%

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

   if 0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.1%

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

   3. Taylor expanded in l around 0 89.8%

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

   4. Taylor expanded in K around 0 85.9%

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

4. Final simplification 81.2%

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

Alternative 7: 67.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot 2\right)\\ t_1 := U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ t_2 := \ell \cdot \frac{U}{\ell}\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1.75 \cdot 10^{+19}:\\ \;\;\;\;\frac{t\_0 \cdot t\_0 - t\_2 \cdot t\_2}{t\_0 - t\_2}\\ \mathbf{elif}\;\ell \leq 100:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J 2.0)))
        (t_1 (* U (+ 1.0 (* 2.0 (* J (/ l U))))))
        (t_2 (* l (/ U l))))
   (if (<= l -6.5e+136)
     t_1
     (if (<= l -1.75e+19)
       (/ (- (* t_0 t_0) (* t_2 t_2)) (- t_0 t_2))
       (if (<= l 100.0) (+ U (* (cos (/ K 2.0)) (* J (* l 2.0)))) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	double t_2 = l * (U / l);
	double tmp;
	if (l <= -6.5e+136) {
		tmp = t_1;
	} else if (l <= -1.75e+19) {
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	} else if (l <= 100.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = l * (j * 2.0d0)
    t_1 = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    t_2 = l * (u / l)
    if (l <= (-6.5d+136)) then
        tmp = t_1
    else if (l <= (-1.75d+19)) then
        tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2)
    else if (l <= 100.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	double t_2 = l * (U / l);
	double tmp;
	if (l <= -6.5e+136) {
		tmp = t_1;
	} else if (l <= -1.75e+19) {
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	} else if (l <= 100.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * 2.0)
	t_1 = U * (1.0 + (2.0 * (J * (l / U))))
	t_2 = l * (U / l)
	tmp = 0
	if l <= -6.5e+136:
		tmp = t_1
	elif l <= -1.75e+19:
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2)
	elif l <= 100.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * 2.0))
	t_1 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))))
	t_2 = Float64(l * Float64(U / l))
	tmp = 0.0
	if (l <= -6.5e+136)
		tmp = t_1;
	elseif (l <= -1.75e+19)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(t_2 * t_2)) / Float64(t_0 - t_2));
	elseif (l <= 100.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (J * 2.0);
	t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	t_2 = l * (U / l);
	tmp = 0.0;
	if (l <= -6.5e+136)
		tmp = t_1;
	elseif (l <= -1.75e+19)
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	elseif (l <= 100.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(U / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.5e+136], t$95$1, If[LessEqual[l, -1.75e+19], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 100.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.4999999999999998e136 or 100 < 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 26.0%

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

      1. *-commutative 26.0%

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

      2. associate-*l* 26.9%

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

   5. Simplified 26.9%

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

   6. Taylor expanded in l around inf 26.0%

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

   7. Taylor expanded in K around 0 19.6%

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

   8. Taylor expanded in U around inf 33.4%

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

      1. associate-/l* 41.8%

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

   10. Simplified 41.8%

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

   if -6.4999999999999998e136 < l < -1.75e19

   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 4.7%

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

      1. *-commutative 4.7%

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

      2. associate-*l* 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in l around inf 4.7%

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

   7. Taylor expanded in K around 0 3.9%

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

      1. distribute-lft-in 3.9%

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

      2. flip-+ 34.6%

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

      3. *-commutative 34.6%

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

      4. *-commutative 34.6%

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

      5. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

   if -1.75e19 < l < 100

   1. Initial program 72.4%

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

   3. Taylor expanded in l around 0 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*l* 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+136}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.75 \cdot 10^{+19}:\\ \;\;\;\;\frac{\left(\ell \cdot \left(J \cdot 2\right)\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right) - \left(\ell \cdot \frac{U}{\ell}\right) \cdot \left(\ell \cdot \frac{U}{\ell}\right)}{\ell \cdot \left(J \cdot 2\right) - \ell \cdot \frac{U}{\ell}}\\ \mathbf{elif}\;\ell \leq 100:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot 2\right)\\ t_1 := U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ t_2 := \ell \cdot \frac{U}{\ell}\\ \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{t\_0 \cdot t\_0 - t\_2 \cdot t\_2}{t\_0 - t\_2}\\ \mathbf{elif}\;\ell \leq 30:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J 2.0)))
        (t_1 (* U (+ 1.0 (* 2.0 (* J (/ l U))))))
        (t_2 (* l (/ U l))))
   (if (<= l -3.05e+131)
     t_1
     (if (<= l -1.5e+19)
       (/ (- (* t_0 t_0) (* t_2 t_2)) (- t_0 t_2))
       (if (<= l 30.0) (+ U (* J (* 2.0 (* l (cos (* K 0.5)))))) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	double t_2 = l * (U / l);
	double tmp;
	if (l <= -3.05e+131) {
		tmp = t_1;
	} else if (l <= -1.5e+19) {
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	} else if (l <= 30.0) {
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = l * (j * 2.0d0)
    t_1 = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    t_2 = l * (u / l)
    if (l <= (-3.05d+131)) then
        tmp = t_1
    else if (l <= (-1.5d+19)) then
        tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2)
    else if (l <= 30.0d0) then
        tmp = u + (j * (2.0d0 * (l * cos((k * 0.5d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	double t_2 = l * (U / l);
	double tmp;
	if (l <= -3.05e+131) {
		tmp = t_1;
	} else if (l <= -1.5e+19) {
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	} else if (l <= 30.0) {
		tmp = U + (J * (2.0 * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * 2.0)
	t_1 = U * (1.0 + (2.0 * (J * (l / U))))
	t_2 = l * (U / l)
	tmp = 0
	if l <= -3.05e+131:
		tmp = t_1
	elif l <= -1.5e+19:
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2)
	elif l <= 30.0:
		tmp = U + (J * (2.0 * (l * math.cos((K * 0.5)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * 2.0))
	t_1 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))))
	t_2 = Float64(l * Float64(U / l))
	tmp = 0.0
	if (l <= -3.05e+131)
		tmp = t_1;
	elseif (l <= -1.5e+19)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(t_2 * t_2)) / Float64(t_0 - t_2));
	elseif (l <= 30.0)
		tmp = Float64(U + Float64(J * Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (J * 2.0);
	t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	t_2 = l * (U / l);
	tmp = 0.0;
	if (l <= -3.05e+131)
		tmp = t_1;
	elseif (l <= -1.5e+19)
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	elseif (l <= 30.0)
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(U / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.05e+131], t$95$1, If[LessEqual[l, -1.5e+19], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 30.0], N[(U + N[(J * N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.0499999999999999e131 or 30 < 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 26.0%

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

      1. *-commutative 26.0%

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

      2. associate-*l* 26.9%

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

   5. Simplified 26.9%

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

   6. Taylor expanded in l around inf 26.0%

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

   7. Taylor expanded in K around 0 19.6%

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

   8. Taylor expanded in U around inf 33.4%

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

      1. associate-/l* 41.8%

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

   10. Simplified 41.8%

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

   if -3.0499999999999999e131 < l < -1.5e19

   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 4.7%

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

      1. *-commutative 4.7%

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

      2. associate-*l* 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in l around inf 4.7%

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

   7. Taylor expanded in K around 0 3.9%

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

      1. distribute-lft-in 3.9%

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

      2. flip-+ 34.6%

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

      3. *-commutative 34.6%

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

      4. *-commutative 34.6%

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

      5. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

   if -1.5e19 < l < 30

   1. Initial program 72.4%

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

   3. Taylor expanded in l around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*l* 97.9%

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

      3. *-commutative 97.9%

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

      4. *-commutative 97.9%

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

   5. Simplified 97.9%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+131}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\left(\ell \cdot \left(J \cdot 2\right)\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right) - \left(\ell \cdot \frac{U}{\ell}\right) \cdot \left(\ell \cdot \frac{U}{\ell}\right)}{\ell \cdot \left(J \cdot 2\right) - \ell \cdot \frac{U}{\ell}}\\ \mathbf{elif}\;\ell \leq 30:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.0% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in l around 0 89.1%

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

   1. unpow2 85.3%

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

5. Applied egg-rr 89.1%

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

6. Final simplification 89.1%

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

Alternative 10: 61.3% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot 2\right)\\ t_1 := U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ t_2 := \ell \cdot \frac{U}{\ell}\\ \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{t\_0 \cdot t\_0 - t\_2 \cdot t\_2}{t\_0 - t\_2}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-39}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J 2.0)))
        (t_1 (* U (+ 1.0 (* 2.0 (* J (/ l U))))))
        (t_2 (* l (/ U l))))
   (if (<= l -2.25e+128)
     t_1
     (if (<= l -1.5e+19)
       (/ (- (* t_0 t_0) (* t_2 t_2)) (- t_0 t_2))
       (if (<= l 2e-39) (+ U (* 2.0 (* l J))) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	double t_2 = l * (U / l);
	double tmp;
	if (l <= -2.25e+128) {
		tmp = t_1;
	} else if (l <= -1.5e+19) {
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	} else if (l <= 2e-39) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = l * (j * 2.0d0)
    t_1 = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    t_2 = l * (u / l)
    if (l <= (-2.25d+128)) then
        tmp = t_1
    else if (l <= (-1.5d+19)) then
        tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2)
    else if (l <= 2d-39) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	double t_2 = l * (U / l);
	double tmp;
	if (l <= -2.25e+128) {
		tmp = t_1;
	} else if (l <= -1.5e+19) {
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	} else if (l <= 2e-39) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * 2.0)
	t_1 = U * (1.0 + (2.0 * (J * (l / U))))
	t_2 = l * (U / l)
	tmp = 0
	if l <= -2.25e+128:
		tmp = t_1
	elif l <= -1.5e+19:
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2)
	elif l <= 2e-39:
		tmp = U + (2.0 * (l * J))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * 2.0))
	t_1 = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))))
	t_2 = Float64(l * Float64(U / l))
	tmp = 0.0
	if (l <= -2.25e+128)
		tmp = t_1;
	elseif (l <= -1.5e+19)
		tmp = Float64(Float64(Float64(t_0 * t_0) - Float64(t_2 * t_2)) / Float64(t_0 - t_2));
	elseif (l <= 2e-39)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (J * 2.0);
	t_1 = U * (1.0 + (2.0 * (J * (l / U))));
	t_2 = l * (U / l);
	tmp = 0.0;
	if (l <= -2.25e+128)
		tmp = t_1;
	elseif (l <= -1.5e+19)
		tmp = ((t_0 * t_0) - (t_2 * t_2)) / (t_0 - t_2);
	elseif (l <= 2e-39)
		tmp = U + (2.0 * (l * J));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[(U / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.25e+128], t$95$1, If[LessEqual[l, -1.5e+19], N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e-39], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.2500000000000001e128 or 1.99999999999999986e-39 < l

   1. Initial program 97.4%

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

   3. Taylor expanded in l around 0 29.4%

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

      1. *-commutative 29.4%

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

      2. associate-*l* 30.1%

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

   5. Simplified 30.1%

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

   6. Taylor expanded in l around inf 29.4%

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

   7. Taylor expanded in K around 0 20.7%

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

   8. Taylor expanded in U around inf 33.8%

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

      1. associate-/l* 41.8%

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

   10. Simplified 41.8%

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

   if -2.2500000000000001e128 < l < -1.5e19

   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 4.7%

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

      1. *-commutative 4.7%

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

      2. associate-*l* 4.7%

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

   5. Simplified 4.7%

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

   6. Taylor expanded in l around inf 4.7%

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

   7. Taylor expanded in K around 0 3.9%

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

      1. distribute-lft-in 3.9%

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

      2. flip-+ 34.6%

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

      3. *-commutative 34.6%

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

      4. *-commutative 34.6%

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

      5. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

   if -1.5e19 < l < 1.99999999999999986e-39

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 98.2%

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

      1. *-commutative 98.2%

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

      2. associate-*l* 98.2%

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

   5. Simplified 98.2%

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

   6. Taylor expanded in K around 0 84.4%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.25 \cdot 10^{+128}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\frac{\left(\ell \cdot \left(J \cdot 2\right)\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right) - \left(\ell \cdot \frac{U}{\ell}\right) \cdot \left(\ell \cdot \frac{U}{\ell}\right)}{\ell \cdot \left(J \cdot 2\right) - \ell \cdot \frac{U}{\ell}}\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-39}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 46.7% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1150 \lor \neg \left(\ell \leq 0.00013\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1150.0) (not (<= l 0.00013))) (* J (* l 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1150.0) || !(l <= 0.00013)) {
		tmp = J * (l * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1150.0d0)) .or. (.not. (l <= 0.00013d0))) then
        tmp = j * (l * 2.0d0)
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1150.0) || !(l <= 0.00013)) {
		tmp = J * (l * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1150.0) or not (l <= 0.00013):
		tmp = J * (l * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1150.0) || !(l <= 0.00013))
		tmp = Float64(J * Float64(l * 2.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1150.0) || ~((l <= 0.00013)))
		tmp = J * (l * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1150.0], N[Not[LessEqual[l, 0.00013]], $MachinePrecision]], N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1150 or 1.29999999999999989e-4 < l

   1. Initial program 99.9%

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

   3. Taylor expanded in l around 0 23.3%

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

      1. *-commutative 23.3%

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

      2. associate-*l* 24.0%

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

   5. Simplified 24.0%

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

   6. Taylor expanded in l around inf 23.3%

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

   7. Taylor expanded in K around 0 17.6%

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

   8. Taylor expanded in l around inf 17.6%

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

      1. associate-*r* 17.6%

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

      2. *-commutative 17.6%

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

      3. associate-*r* 18.3%

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

   10. Simplified 18.3%

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

   if -1150 < l < 1.29999999999999989e-4

   1. Initial program 71.6%

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

   4. Applied egg-rr 50.0%

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

   5. Taylor expanded in U around inf 71.6%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1150 \lor \neg \left(\ell \leq 0.00013\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.0% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+19}:\\ \;\;\;\;J \cdot \left(\frac{U}{J} + \ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1e+19) (* J (+ (/ U J) (* l 2.0))) (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1e+19) {
		tmp = J * ((U / J) + (l * 2.0));
	} else {
		tmp = U + (2.0 * (l * 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 <= (-1d+19)) then
        tmp = j * ((u / j) + (l * 2.0d0))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1e+19) {
		tmp = J * ((U / J) + (l * 2.0));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1e+19:
		tmp = J * ((U / J) + (l * 2.0))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1e+19)
		tmp = Float64(J * Float64(Float64(U / J) + Float64(l * 2.0)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1e+19)
		tmp = J * ((U / J) + (l * 2.0));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1e+19], N[(J * N[(N[(U / J), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1e19

   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 24.8%

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

      1. *-commutative 24.8%

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

      2. associate-*l* 26.2%

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

   5. Simplified 26.2%

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

   6. Taylor expanded in l around inf 24.8%

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

   7. Taylor expanded in K around 0 18.6%

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

   8. Taylor expanded in J around inf 30.7%

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

   if -1e19 < l

   1. Initial program 82.2%

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

   3. Taylor expanded in l around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*l* 70.5%

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

   5. Simplified 70.5%

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

   6. Taylor expanded in K around 0 58.7%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+19}:\\ \;\;\;\;J \cdot \left(\frac{U}{J} + \ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.6% 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 87.1%

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

3. Taylor expanded in l around 0 58.0%

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

   1. *-commutative 58.0%

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

   2. associate-*l* 58.4%

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

5. Simplified 58.4%

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

6. Taylor expanded in l around inf 47.7%

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

7. Taylor expanded in K around 0 37.5%

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

8. Taylor expanded in U around inf 51.6%

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

   1. associate-/l* 55.3%

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

10. Simplified 55.3%

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

Alternative 14: 54.5% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in l around 0 58.0%

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

   1. *-commutative 58.0%

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

   2. associate-*l* 58.4%

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

5. Simplified 58.4%

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

6. Taylor expanded in K around 0 47.7%

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

7. Final simplification 47.7%

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

Alternative 15: 37.4% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

4. Applied egg-rr 24.0%

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

5. Taylor expanded in U around inf 33.7%

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

Alternative 16: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 8 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 8.0)

C

double code(double J, double l, double K, double U) {
	return 8.0;
}

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 = 8.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 8.0;
}

Python

def code(J, l, K, U):
	return 8.0

Julia

function code(J, l, K, U)
	return 8.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 8.0;
end

Wolfram

code[J_, l_, K_, U_] := 8.0

Derivation

1. Initial program 87.1%

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

4. Applied egg-rr 24.1%

   \[\leadsto \color{blue}{8} + U \]

5. Taylor expanded in U around 0 2.7%

   \[\leadsto \color{blue}{8} \]
6. Add Preprocessing

Alternative 17: 2.7% 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 87.1%

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

4. Applied egg-rr 24.0%

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

5. Taylor expanded in U around 0 2.7%

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

Reproduce

herbie shell --seed 2024145 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))