Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.5%

Time: 7.5s

Alternatives: 11

Speedup: 1.4×

• 49.6% 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: 86.4% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $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 72.5%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{3} \cdot \left(J \cdot \left(t\_0 \cdot 0.3333333333333333\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -0.00033:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 3.0) (* J (* t_0 0.3333333333333333)))))
        (t_2 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -2.15e+74)
     t_1
     (if (<= l -0.00033)
       t_2
       (if (<= l 1.5e-21)
         (+ U (* 2.0 (* J (* l t_0))))
         (if (<= l 6.6e+45) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)));
	double t_2 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.15e+74) {
		tmp = t_1;
	} else if (l <= -0.00033) {
		tmp = t_2;
	} else if (l <= 1.5e-21) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 6.6e+45) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((l ** 3.0d0) * (j * (t_0 * 0.3333333333333333d0)))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-2.15d+74)) then
        tmp = t_1
    else if (l <= (-0.00033d0)) then
        tmp = t_2
    else if (l <= 1.5d-21) then
        tmp = u + (2.0d0 * (j * (l * t_0)))
    else if (l <= 6.6d+45) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (Math.pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)));
	double t_2 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.15e+74) {
		tmp = t_1;
	} else if (l <= -0.00033) {
		tmp = t_2;
	} else if (l <= 1.5e-21) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 6.6e+45) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (math.pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)))
	t_2 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -2.15e+74:
		tmp = t_1
	elif l <= -0.00033:
		tmp = t_2
	elif l <= 1.5e-21:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif l <= 6.6e+45:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64((l ^ 3.0) * Float64(J * Float64(t_0 * 0.3333333333333333))))
	t_2 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -2.15e+74)
		tmp = t_1;
	elseif (l <= -0.00033)
		tmp = t_2;
	elseif (l <= 1.5e-21)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif (l <= 6.6e+45)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((l ^ 3.0) * (J * (t_0 * 0.3333333333333333)));
	t_2 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -2.15e+74)
		tmp = t_1;
	elseif (l <= -0.00033)
		tmp = t_2;
	elseif (l <= 1.5e-21)
		tmp = U + (2.0 * (J * (l * t_0)));
	elseif (l <= 6.6e+45)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -2.15e+74], t$95$1, If[LessEqual[l, -0.00033], t$95$2, If[LessEqual[l, 1.5e-21], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.6e+45], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.15e74 or 6.6000000000000001e45 < 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 91.6%

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

   4. Taylor expanded in l around inf 91.6%

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

      1. *-commutative 91.6%

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

      2. associate-*r* 91.6%

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

      3. associate-*l* 91.6%

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

      4. *-commutative 91.6%

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

      5. associate-*r* 91.6%

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

      6. *-commutative 91.6%

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

      7. associate-*l* 91.6%

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

   6. Simplified 91.6%

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

   if -2.15e74 < l < -3.3e-4 or 1.49999999999999996e-21 < l < 6.6000000000000001e45

   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 79.4%

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

   if -3.3e-4 < l < 1.49999999999999996e-21

   1. Initial program 72.3%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+74}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -0.00033:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+45}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+45}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1.5e-21)
   (+
    U
    (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
   (if (<= l 6.6e+45)
     (+ (* (- (exp l) (exp (- l))) J) U)
     (+ U (* (pow l 3.0) (* J (* (cos (* K 0.5)) 0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.5e-21) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else if (l <= 6.6e+45) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (pow(l, 3.0) * (J * (cos((K * 0.5)) * 0.3333333333333333)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 1.5d-21) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    else if (l <= 6.6d+45) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + ((l ** 3.0d0) * (j * (cos((k * 0.5d0)) * 0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.5e-21) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else if (l <= 6.6e+45) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (Math.pow(l, 3.0) * (J * (Math.cos((K * 0.5)) * 0.3333333333333333)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1.5e-21:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	elif l <= 6.6e+45:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (math.pow(l, 3.0) * (J * (math.cos((K * 0.5)) * 0.3333333333333333)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1.5e-21)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	elseif (l <= 6.6e+45)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * Float64(cos(Float64(K * 0.5)) * 0.3333333333333333))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1.5e-21)
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	elseif (l <= 6.6e+45)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + ((l ^ 3.0) * (J * (cos((K * 0.5)) * 0.3333333333333333)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1.5e-21], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.6e+45], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 1.49999999999999996e-21

   1. Initial program 82.4%

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

   3. Taylor expanded in l around 0 92.8%

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

   if 1.49999999999999996e-21 < l < 6.6000000000000001e45

   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 88.2%

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

   if 6.6000000000000001e45 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 90.6%

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

   4. Taylor expanded in l around inf 90.6%

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

      1. *-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. associate-*l* 90.6%

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

      4. *-commutative 90.6%

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

      5. associate-*r* 90.6%

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

      6. *-commutative 90.6%

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

      7. associate-*l* 90.6%

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

   6. Simplified 90.6%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{+45}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (+ 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 + (2.0 * (J * (l * cos((K * 0.5)))));
	} 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 + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    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 + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} 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 + (2.0 * (J * (l * math.cos((K * 0.5)))))
	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(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	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 + (2.0 * (J * (l * cos((K * 0.5)))));
	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[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $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 84.7%

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

   3. Taylor expanded in l around 0 52.9%

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

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

   1. Initial program 87.9%

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

   3. Taylor expanded in l around 0 89.0%

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

   4. Taylor expanded in K around 0 85.6%

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

4. Final simplification 76.8%

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

Alternative 5: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.000118 \lor \neg \left(\ell \leq 1.5 \cdot 10^{-21}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.000118) (not (<= l 1.5e-21)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.000118) || !(l <= 1.5e-21)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.000118d0)) .or. (.not. (l <= 1.5d-21))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.000118) || !(l <= 1.5e-21)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.000118) or not (l <= 1.5e-21):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.000118) || !(l <= 1.5e-21))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.000118) || ~((l <= 1.5e-21)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.000118], N[Not[LessEqual[l, 1.5e-21]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.18e-4 or 1.49999999999999996e-21 < 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 69.1%

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

   if -1.18e-4 < l < 1.49999999999999996e-21

   1. Initial program 72.3%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.000118 \lor \neg \left(\ell \leq 1.5 \cdot 10^{-21}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -650 \lor \neg \left(\ell \leq 155000\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -650.0) (not (<= l 155000.0)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -650.0) || !(l <= 155000.0)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-650.0d0)) .or. (.not. (l <= 155000.0d0))) then
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -650.0) || !(l <= 155000.0)) {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -650.0) or not (l <= 155000.0):
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -650.0) || !(l <= 155000.0))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -650.0) || ~((l <= 155000.0)))
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -650.0], N[Not[LessEqual[l, 155000.0]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -650 or 155000 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 78.0%

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

   4. Taylor expanded in l around inf 78.0%

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

      1. *-commutative 78.0%

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

      2. associate-*r* 78.0%

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

      3. associate-*l* 78.0%

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

      4. *-commutative 78.0%

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

      5. associate-*r* 78.0%

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

      6. *-commutative 78.0%

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

      7. associate-*l* 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in K around 0 53.3%

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

   if -650 < l < 155000

   1. Initial program 73.6%

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

   3. Taylor expanded in l around 0 98.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -650 \lor \neg \left(\ell \leq 155000\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 7.5\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.5) (not (<= l 7.5)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (+ U (* l (* J 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5) || !(l <= 7.5)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.5d0)) .or. (.not. (l <= 7.5d0))) then
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    else
        tmp = u + (l * (j * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5) || !(l <= 7.5)) {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.5) or not (l <= 7.5):
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	else:
		tmp = U + (l * (J * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.5) || !(l <= 7.5))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.5) || ~((l <= 7.5)))
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	else
		tmp = U + (l * (J * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.5], N[Not[LessEqual[l, 7.5]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.5 or 7.5 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 77.4%

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

   4. Taylor expanded in l around inf 77.4%

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

      1. *-commutative 77.4%

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

      2. associate-*r* 77.4%

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

      3. associate-*l* 77.4%

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

      4. *-commutative 77.4%

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

      5. associate-*r* 77.4%

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

      6. *-commutative 77.4%

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

      7. associate-*l* 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in K around 0 52.9%

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

   if -2.5 < l < 7.5

   1. Initial program 73.4%

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

   3. Taylor expanded in l around 0 99.3%

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

      1. associate-*r* 99.3%

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

   5. Simplified 99.3%

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

   6. Taylor expanded in K around 0 85.6%

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

      1. associate-*r* 85.6%

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

      2. *-commutative 85.6%

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

      3. *-commutative 85.6%

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

   8. Simplified 85.6%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \lor \neg \left(\ell \leq 7.5\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.9% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -690 \lor \neg \left(\ell \leq 1350000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -690.0) (not (<= l 1350000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -690.0) || !(l <= 1350000.0)) {
		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 <= (-690.0d0)) .or. (.not. (l <= 1350000.0d0))) 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 <= -690.0) || !(l <= 1350000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -690.0) or not (l <= 1350000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -690.0) || !(l <= 1350000.0))
		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 <= -690.0) || ~((l <= 1350000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -690.0], N[Not[LessEqual[l, 1350000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -690 or 1.35e6 < 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 13.8%

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

   if -690 < l < 1.35e6

   1. Initial program 73.8%

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

   3. Taylor expanded in J around 0 71.4%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -690 \lor \neg \left(\ell \leq 1350000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.5% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.0%

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

3. Taylor expanded in l around 0 60.6%

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

   1. associate-*r* 60.6%

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

5. Simplified 60.6%

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

6. Taylor expanded in K around 0 50.4%

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

   1. associate-*r* 50.4%

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

   2. *-commutative 50.4%

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

   3. *-commutative 50.4%

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

8. Simplified 50.4%

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

9. Final simplification 50.4%

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

Alternative 10: 37.0% 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 87.0%

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

3. Taylor expanded in J around 0 36.8%

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

Alternative 11: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 1.0)

C

double code(double J, double l, double K, double U) {
	return 1.0;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = 1.0d0
end function

Java

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

Python

def code(J, l, K, U):
	return 1.0

Julia

function code(J, l, K, U)
	return 1.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 1.0;
end

Wolfram

code[J_, l_, K_, U_] := 1.0

Derivation

1. Initial program 87.0%

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

4. Applied egg-rr 2.5%

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

   1. *-inverses 2.5%

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

6. Simplified 2.5%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024111 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))