Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.5% → 99.7%

Time: 10.3s

Alternatives: 13

Speedup: 1.4×

• 50.1% 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: 86.5% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 2e-12)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* 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 t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 2e-12)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e-12)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (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))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 2e-12):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (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))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 2e-12))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * 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));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 2e-12)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (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]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 2e-12]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $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 1.99999999999999996e-12 < (-.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))) < 1.99999999999999996e-12

   1. Initial program 63.9%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-12}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot t\_1\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)) (t_1 (cos (* K 0.5))))
   (if (<= l -5.4e+124)
     (* (* J (* 0.3333333333333333 (pow l 3.0))) t_1)
     (if (<= l -14500.0)
       t_0
       (if (<= l 0.39)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
         (if (<= l 3.5e+93)
           t_0
           (*
            U
            (+
             1.0
             (* 0.3333333333333333 (/ (* J (* (pow l 3.0) t_1)) U))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -5.4e+124) {
		tmp = (J * (0.3333333333333333 * pow(l, 3.0))) * t_1;
	} else if (l <= -14500.0) {
		tmp = t_0;
	} else if (l <= 0.39) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else if (l <= 3.5e+93) {
		tmp = t_0;
	} else {
		tmp = U * (1.0 + (0.3333333333333333 * ((J * (pow(l, 3.0) * t_1)) / U)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (exp(l) - exp(-l)) * j
    t_1 = cos((k * 0.5d0))
    if (l <= (-5.4d+124)) then
        tmp = (j * (0.3333333333333333d0 * (l ** 3.0d0))) * t_1
    else if (l <= (-14500.0d0)) then
        tmp = t_0
    else if (l <= 0.39d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    else if (l <= 3.5d+93) then
        tmp = t_0
    else
        tmp = u * (1.0d0 + (0.3333333333333333d0 * ((j * ((l ** 3.0d0) * t_1)) / u)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -5.4e+124) {
		tmp = (J * (0.3333333333333333 * Math.pow(l, 3.0))) * t_1;
	} else if (l <= -14500.0) {
		tmp = t_0;
	} else if (l <= 0.39) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else if (l <= 3.5e+93) {
		tmp = t_0;
	} else {
		tmp = U * (1.0 + (0.3333333333333333 * ((J * (Math.pow(l, 3.0) * t_1)) / U)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -5.4e+124:
		tmp = (J * (0.3333333333333333 * math.pow(l, 3.0))) * t_1
	elif l <= -14500.0:
		tmp = t_0
	elif l <= 0.39:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	elif l <= 3.5e+93:
		tmp = t_0
	else:
		tmp = U * (1.0 + (0.3333333333333333 * ((J * (math.pow(l, 3.0) * t_1)) / U)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -5.4e+124)
		tmp = Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) * t_1);
	elseif (l <= -14500.0)
		tmp = t_0;
	elseif (l <= 0.39)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	elseif (l <= 3.5e+93)
		tmp = t_0;
	else
		tmp = Float64(U * Float64(1.0 + Float64(0.3333333333333333 * Float64(Float64(J * Float64((l ^ 3.0) * t_1)) / U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -5.4e+124)
		tmp = (J * (0.3333333333333333 * (l ^ 3.0))) * t_1;
	elseif (l <= -14500.0)
		tmp = t_0;
	elseif (l <= 0.39)
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	elseif (l <= 3.5e+93)
		tmp = t_0;
	else
		tmp = U * (1.0 + (0.3333333333333333 * ((J * ((l ^ 3.0) * t_1)) / U)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.4e+124], N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, -14500.0], t$95$0, If[LessEqual[l, 0.39], 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, 3.5e+93], t$95$0, N[(U * N[(1.0 + N[(0.3333333333333333 * N[(N[(J * N[(N[Power[l, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.39999999999999956e124

   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. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in U around inf 100.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]

   8. Taylor expanded in U around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -5.39999999999999956e124 < l < -14500 or 0.39000000000000001 < l < 3.49999999999999998e93

   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 78.3%

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

   4. Taylor expanded in J around inf 78.3%

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

   if -14500 < l < 0.39000000000000001

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 99.0%

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

   if 3.49999999999999998e93 < 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 96.0%

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

   4. Taylor expanded in l around inf 96.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*l* 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in U around inf 98.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{+93}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot t\_1\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_1\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot t\_1\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)) (t_1 (cos (* K 0.5))))
   (if (<= l -5.4e+124)
     (* (* J (* 0.3333333333333333 (pow l 3.0))) t_1)
     (if (<= l -14500.0)
       t_0
       (if (<= l 0.39)
         (+ U (* 2.0 (* J (* l t_1))))
         (if (<= l 5e+93)
           t_0
           (*
            U
            (+
             1.0
             (* 0.3333333333333333 (/ (* J (* (pow l 3.0) t_1)) U))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -5.4e+124) {
		tmp = (J * (0.3333333333333333 * pow(l, 3.0))) * t_1;
	} else if (l <= -14500.0) {
		tmp = t_0;
	} else if (l <= 0.39) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 5e+93) {
		tmp = t_0;
	} else {
		tmp = U * (1.0 + (0.3333333333333333 * ((J * (pow(l, 3.0) * t_1)) / U)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (exp(l) - exp(-l)) * j
    t_1 = cos((k * 0.5d0))
    if (l <= (-5.4d+124)) then
        tmp = (j * (0.3333333333333333d0 * (l ** 3.0d0))) * t_1
    else if (l <= (-14500.0d0)) then
        tmp = t_0
    else if (l <= 0.39d0) then
        tmp = u + (2.0d0 * (j * (l * t_1)))
    else if (l <= 5d+93) then
        tmp = t_0
    else
        tmp = u * (1.0d0 + (0.3333333333333333d0 * ((j * ((l ** 3.0d0) * t_1)) / u)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -5.4e+124) {
		tmp = (J * (0.3333333333333333 * Math.pow(l, 3.0))) * t_1;
	} else if (l <= -14500.0) {
		tmp = t_0;
	} else if (l <= 0.39) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 5e+93) {
		tmp = t_0;
	} else {
		tmp = U * (1.0 + (0.3333333333333333 * ((J * (Math.pow(l, 3.0) * t_1)) / U)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -5.4e+124:
		tmp = (J * (0.3333333333333333 * math.pow(l, 3.0))) * t_1
	elif l <= -14500.0:
		tmp = t_0
	elif l <= 0.39:
		tmp = U + (2.0 * (J * (l * t_1)))
	elif l <= 5e+93:
		tmp = t_0
	else:
		tmp = U * (1.0 + (0.3333333333333333 * ((J * (math.pow(l, 3.0) * t_1)) / U)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -5.4e+124)
		tmp = Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) * t_1);
	elseif (l <= -14500.0)
		tmp = t_0;
	elseif (l <= 0.39)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_1))));
	elseif (l <= 5e+93)
		tmp = t_0;
	else
		tmp = Float64(U * Float64(1.0 + Float64(0.3333333333333333 * Float64(Float64(J * Float64((l ^ 3.0) * t_1)) / U))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -5.4e+124)
		tmp = (J * (0.3333333333333333 * (l ^ 3.0))) * t_1;
	elseif (l <= -14500.0)
		tmp = t_0;
	elseif (l <= 0.39)
		tmp = U + (2.0 * (J * (l * t_1)));
	elseif (l <= 5e+93)
		tmp = t_0;
	else
		tmp = U * (1.0 + (0.3333333333333333 * ((J * ((l ^ 3.0) * t_1)) / U)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -5.4e+124], N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[l, -14500.0], t$95$0, If[LessEqual[l, 0.39], N[(U + N[(2.0 * N[(J * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+93], t$95$0, N[(U * N[(1.0 + N[(0.3333333333333333 * N[(N[(J * N[(N[Power[l, 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.39999999999999956e124

   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. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in U around inf 100.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]

   8. Taylor expanded in U around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. associate-*l* 100.0%

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

      8. *-commutative 100.0%

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

   10. Simplified 100.0%

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

   if -5.39999999999999956e124 < l < -14500 or 0.39000000000000001 < l < 5.0000000000000001e93

   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 78.3%

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

   4. Taylor expanded in J around inf 78.3%

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

   if -14500 < l < 0.39000000000000001

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 98.8%

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

   if 5.0000000000000001e93 < 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 96.0%

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

   4. Taylor expanded in l around inf 96.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 96.0%

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

      2. *-commutative 96.0%

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

      3. associate-*l* 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in U around inf 98.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+93}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot t\_1\\ \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_1\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+93}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J))
        (t_1 (cos (* K 0.5)))
        (t_2 (* (* J (* 0.3333333333333333 (pow l 3.0))) t_1)))
   (if (<= l -5.4e+124)
     t_2
     (if (<= l -14500.0)
       t_0
       (if (<= l 0.39)
         (+ U (* 2.0 (* J (* l t_1))))
         (if (<= l 2.3e+93) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double t_1 = cos((K * 0.5));
	double t_2 = (J * (0.3333333333333333 * pow(l, 3.0))) * t_1;
	double tmp;
	if (l <= -5.4e+124) {
		tmp = t_2;
	} else if (l <= -14500.0) {
		tmp = t_0;
	} else if (l <= 0.39) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 2.3e+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 = (exp(l) - exp(-l)) * j
    t_1 = cos((k * 0.5d0))
    t_2 = (j * (0.3333333333333333d0 * (l ** 3.0d0))) * t_1
    if (l <= (-5.4d+124)) then
        tmp = t_2
    else if (l <= (-14500.0d0)) then
        tmp = t_0
    else if (l <= 0.39d0) then
        tmp = u + (2.0d0 * (j * (l * t_1)))
    else if (l <= 2.3d+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 = (Math.exp(l) - Math.exp(-l)) * J;
	double t_1 = Math.cos((K * 0.5));
	double t_2 = (J * (0.3333333333333333 * Math.pow(l, 3.0))) * t_1;
	double tmp;
	if (l <= -5.4e+124) {
		tmp = t_2;
	} else if (l <= -14500.0) {
		tmp = t_0;
	} else if (l <= 0.39) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 2.3e+93) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	t_1 = math.cos((K * 0.5))
	t_2 = (J * (0.3333333333333333 * math.pow(l, 3.0))) * t_1
	tmp = 0
	if l <= -5.4e+124:
		tmp = t_2
	elif l <= -14500.0:
		tmp = t_0
	elif l <= 0.39:
		tmp = U + (2.0 * (J * (l * t_1)))
	elif l <= 2.3e+93:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	t_1 = cos(Float64(K * 0.5))
	t_2 = Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) * t_1)
	tmp = 0.0
	if (l <= -5.4e+124)
		tmp = t_2;
	elseif (l <= -14500.0)
		tmp = t_0;
	elseif (l <= 0.39)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_1))));
	elseif (l <= 2.3e+93)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	t_1 = cos((K * 0.5));
	t_2 = (J * (0.3333333333333333 * (l ^ 3.0))) * t_1;
	tmp = 0.0;
	if (l <= -5.4e+124)
		tmp = t_2;
	elseif (l <= -14500.0)
		tmp = t_0;
	elseif (l <= 0.39)
		tmp = U + (2.0 * (J * (l * t_1)));
	elseif (l <= 2.3e+93)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[l, -5.4e+124], t$95$2, If[LessEqual[l, -14500.0], t$95$0, If[LessEqual[l, 0.39], N[(U + N[(2.0 * N[(J * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.3e+93], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.39999999999999956e124 or 2.3000000000000002e93 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 97.8%

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

   4. Taylor expanded in l around inf 97.8%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 97.8%

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

      2. *-commutative 97.8%

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

      3. associate-*l* 97.8%

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

   6. Simplified 97.8%

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

   7. Taylor expanded in U around inf 98.9%

      \[\leadsto \color{blue}{U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]

   8. Taylor expanded in U around 0 97.8%

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

      1. associate-*r* 97.8%

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

      2. associate-*l* 97.8%

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

      3. *-commutative 97.8%

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

      4. *-commutative 97.8%

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

      5. associate-*r* 97.8%

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

      6. *-commutative 97.8%

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

      7. associate-*l* 97.8%

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

      8. *-commutative 97.8%

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

   10. Simplified 97.8%

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

   if -5.39999999999999956e124 < l < -14500 or 0.39000000000000001 < l < 2.3000000000000002e93

   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 78.3%

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

   4. Taylor expanded in J around inf 78.3%

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

   if -14500 < l < 0.39000000000000001

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 98.8%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+124}:\\ \;\;\;\;\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \mathbf{elif}\;\ell \leq -14500:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.39:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+93}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right) \cdot \cos \left(K \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -14500 \lor \neg \left(\ell \leq 0.245\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -14500.0) (not (<= l 0.245)))
   (* (- (exp l) (exp (- l))) J)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -14500.0) || !(l <= 0.245)) {
		tmp = (exp(l) - exp(-l)) * J;
	} 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 <= (-14500.0d0)) .or. (.not. (l <= 0.245d0))) then
        tmp = (exp(l) - exp(-l)) * j
    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 <= -14500.0) || !(l <= 0.245)) {
		tmp = (Math.exp(l) - Math.exp(-l)) * J;
	} 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 <= -14500.0) or not (l <= 0.245):
		tmp = (math.exp(l) - math.exp(-l)) * J
	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 <= -14500.0) || !(l <= 0.245))
		tmp = Float64(Float64(exp(l) - exp(Float64(-l))) * J);
	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 <= -14500.0) || ~((l <= 0.245)))
		tmp = (exp(l) - exp(-l)) * J;
	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, -14500.0], N[Not[LessEqual[l, 0.245]], $MachinePrecision]], N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $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 < -14500 or 0.245 < 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 75.0%

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

   4. Taylor expanded in J around inf 75.0%

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

   if -14500 < l < 0.245

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 98.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -14500 \lor \neg \left(\ell \leq 0.245\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \mathbf{if}\;\ell \leq -230000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+39}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + {K}^{2} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* U (+ 1.0 (* 0.3333333333333333 (* J (/ (pow l 3.0) U)))))))
   (if (<= l -230000.0)
     t_0
     (if (<= l 8.4e-13)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 3.7e+39)
         (+ U (* (* l J) (+ 2.0 (* (pow K 2.0) -0.25))))
         t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (1.0 + (0.3333333333333333 * (J * (pow(l, 3.0) / U))));
	double tmp;
	if (l <= -230000.0) {
		tmp = t_0;
	} else if (l <= 8.4e-13) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 3.7e+39) {
		tmp = U + ((l * J) * (2.0 + (pow(K, 2.0) * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u * (1.0d0 + (0.3333333333333333d0 * (j * ((l ** 3.0d0) / u))))
    if (l <= (-230000.0d0)) then
        tmp = t_0
    else if (l <= 8.4d-13) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 3.7d+39) then
        tmp = u + ((l * j) * (2.0d0 + ((k ** 2.0d0) * (-0.25d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U * (1.0 + (0.3333333333333333 * (J * (Math.pow(l, 3.0) / U))));
	double tmp;
	if (l <= -230000.0) {
		tmp = t_0;
	} else if (l <= 8.4e-13) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 3.7e+39) {
		tmp = U + ((l * J) * (2.0 + (Math.pow(K, 2.0) * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (1.0 + (0.3333333333333333 * (J * (math.pow(l, 3.0) / U))))
	tmp = 0
	if l <= -230000.0:
		tmp = t_0
	elif l <= 8.4e-13:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 3.7e+39:
		tmp = U + ((l * J) * (2.0 + (math.pow(K, 2.0) * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(1.0 + Float64(0.3333333333333333 * Float64(J * Float64((l ^ 3.0) / U)))))
	tmp = 0.0
	if (l <= -230000.0)
		tmp = t_0;
	elseif (l <= 8.4e-13)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 3.7e+39)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64((K ^ 2.0) * -0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U * (1.0 + (0.3333333333333333 * (J * ((l ^ 3.0) / U))));
	tmp = 0.0;
	if (l <= -230000.0)
		tmp = t_0;
	elseif (l <= 8.4e-13)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 3.7e+39)
		tmp = U + ((l * J) * (2.0 + ((K ^ 2.0) * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U * N[(1.0 + N[(0.3333333333333333 * N[(J * N[(N[Power[l, 3.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -230000.0], t$95$0, If[LessEqual[l, 8.4e-13], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.7e+39], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[Power[K, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.3e5 or 3.70000000000000012e39 < 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 71.9%

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

   4. Taylor expanded in l around inf 71.9%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 71.9%

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

      2. *-commutative 71.9%

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

      3. associate-*l* 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in U around inf 78.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]

   8. Taylor expanded in K around 0 59.8%

      \[\leadsto U \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\frac{J \cdot {\ell}^{3}}{U}}\right) \]
   9. Step-by-step derivation

      1. associate-/l* 62.6%

         \[\leadsto U \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(J \cdot \frac{{\ell}^{3}}{U}\right)}\right) \]

   10. Simplified 62.6%

       \[\leadsto U \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(J \cdot \frac{{\ell}^{3}}{U}\right)}\right) \]

   if -2.3e5 < l < 8.39999999999999955e-13

   1. Initial program 64.5%

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

   3. Taylor expanded in l around 0 98.8%

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

   if 8.39999999999999955e-13 < l < 3.70000000000000012e39

   1. Initial program 95.0%

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

   3. Taylor expanded in l around 0 13.2%

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

   4. Taylor expanded in K around 0 61.2%

      \[\leadsto \color{blue}{\left(-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right) + 2 \cdot \left(J \cdot \ell\right)\right)} + U \]
   5. Step-by-step derivation

      1. +-commutative 61.2%

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right) + -0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)\right)} + U \]

      2. *-commutative 61.2%

         \[\leadsto \left(2 \cdot \left(J \cdot \ell\right) + -0.25 \cdot \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right)\right) + U \]

      3. associate-*r* 61.2%

         \[\leadsto \left(2 \cdot \left(J \cdot \ell\right) + -0.25 \cdot \color{blue}{\left(\left(J \cdot \ell\right) \cdot {K}^{2}\right)}\right) + U \]

      4. associate-*l* 61.2%

         \[\leadsto \left(2 \cdot \left(J \cdot \ell\right) + \color{blue}{\left(-0.25 \cdot \left(J \cdot \ell\right)\right) \cdot {K}^{2}}\right) + U \]

      5. *-commutative 61.2%

         \[\leadsto \left(2 \cdot \left(J \cdot \ell\right) + \color{blue}{{K}^{2} \cdot \left(-0.25 \cdot \left(J \cdot \ell\right)\right)}\right) + U \]

      6. associate-*r* 61.2%

         \[\leadsto \left(2 \cdot \left(J \cdot \ell\right) + \color{blue}{\left({K}^{2} \cdot -0.25\right) \cdot \left(J \cdot \ell\right)}\right) + U \]

      7. distribute-rgt-out 61.2%

         \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {K}^{2} \cdot -0.25\right)} + U \]

   6. Simplified 61.2%

      \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot \left(2 + {K}^{2} \cdot -0.25\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -230000:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 8.4 \cdot 10^{-13}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+39}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + {K}^{2} \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \mathbf{if}\;\ell \leq -65000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 800:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+36}:\\ \;\;\;\;U + -0.25 \cdot \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* U (+ 1.0 (* 0.3333333333333333 (* J (/ (pow l 3.0) U)))))))
   (if (<= l -65000.0)
     t_0
     (if (<= l 800.0)
       (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
       (if (<= l 6.8e+36) (+ U (* -0.25 (* J (* l (pow K 2.0))))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U * (1.0 + (0.3333333333333333 * (J * (pow(l, 3.0) / U))));
	double tmp;
	if (l <= -65000.0) {
		tmp = t_0;
	} else if (l <= 800.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 6.8e+36) {
		tmp = U + (-0.25 * (J * (l * pow(K, 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u * (1.0d0 + (0.3333333333333333d0 * (j * ((l ** 3.0d0) / u))))
    if (l <= (-65000.0d0)) then
        tmp = t_0
    else if (l <= 800.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 6.8d+36) then
        tmp = u + ((-0.25d0) * (j * (l * (k ** 2.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U * (1.0 + (0.3333333333333333 * (J * (Math.pow(l, 3.0) / U))));
	double tmp;
	if (l <= -65000.0) {
		tmp = t_0;
	} else if (l <= 800.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 6.8e+36) {
		tmp = U + (-0.25 * (J * (l * Math.pow(K, 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U * (1.0 + (0.3333333333333333 * (J * (math.pow(l, 3.0) / U))))
	tmp = 0
	if l <= -65000.0:
		tmp = t_0
	elif l <= 800.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 6.8e+36:
		tmp = U + (-0.25 * (J * (l * math.pow(K, 2.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U * Float64(1.0 + Float64(0.3333333333333333 * Float64(J * Float64((l ^ 3.0) / U)))))
	tmp = 0.0
	if (l <= -65000.0)
		tmp = t_0;
	elseif (l <= 800.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 6.8e+36)
		tmp = Float64(U + Float64(-0.25 * Float64(J * Float64(l * (K ^ 2.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U * (1.0 + (0.3333333333333333 * (J * ((l ^ 3.0) / U))));
	tmp = 0.0;
	if (l <= -65000.0)
		tmp = t_0;
	elseif (l <= 800.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 6.8e+36)
		tmp = U + (-0.25 * (J * (l * (K ^ 2.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U * N[(1.0 + N[(0.3333333333333333 * N[(J * N[(N[Power[l, 3.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -65000.0], t$95$0, If[LessEqual[l, 800.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.8e+36], N[(U + N[(-0.25 * N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -65000 or 6.7999999999999996e36 < 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 71.9%

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

   4. Taylor expanded in l around inf 71.9%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   5. Step-by-step derivation

      1. associate-*r* 71.9%

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

      2. *-commutative 71.9%

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

      3. associate-*l* 71.9%

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

   6. Simplified 71.9%

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

   7. Taylor expanded in U around inf 78.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 0.3333333333333333 \cdot \frac{J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]

   8. Taylor expanded in K around 0 59.8%

      \[\leadsto U \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\frac{J \cdot {\ell}^{3}}{U}}\right) \]
   9. Step-by-step derivation

      1. associate-/l* 62.6%

         \[\leadsto U \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(J \cdot \frac{{\ell}^{3}}{U}\right)}\right) \]

   10. Simplified 62.6%

       \[\leadsto U \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(J \cdot \frac{{\ell}^{3}}{U}\right)}\right) \]

   if -65000 < l < 800

   1. Initial program 64.6%

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

   3. Taylor expanded in l around 0 98.2%

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

   if 800 < l < 6.7999999999999996e36

   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 3.3%

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

   4. Taylor expanded in K around 0 60.8%

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

   5. Taylor expanded in K around inf 60.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot -0.25} + U \]

      2. *-commutative 60.1%

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \cdot -0.25 + U \]

   7. Simplified 60.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -65000:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 800:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{+36}:\\ \;\;\;\;U + -0.25 \cdot \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2400:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;U + -0.25 \cdot \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.55e-6)
   (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))
   (if (<= l 2400.0)
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (if (<= l 2.2e+41)
       (+ U (* -0.25 (* J (* l (pow K 2.0)))))
       (* 0.3333333333333333 (* J (pow l 3.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.55e-6) {
		tmp = J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))));
	} else if (l <= 2400.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 2.2e+41) {
		tmp = U + (-0.25 * (J * (l * pow(K, 2.0))));
	} else {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.55d-6)) then
        tmp = j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))
    else if (l <= 2400.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 2.2d+41) then
        tmp = u + ((-0.25d0) * (j * (l * (k ** 2.0d0))))
    else
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.55e-6) {
		tmp = J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))));
	} else if (l <= 2400.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 2.2e+41) {
		tmp = U + (-0.25 * (J * (l * Math.pow(K, 2.0))));
	} else {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.55e-6:
		tmp = J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))
	elif l <= 2400.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 2.2e+41:
		tmp = U + (-0.25 * (J * (l * math.pow(K, 2.0))))
	else:
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.55e-6)
		tmp = Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))));
	elseif (l <= 2400.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 2.2e+41)
		tmp = Float64(U + Float64(-0.25 * Float64(J * Float64(l * (K ^ 2.0)))));
	else
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.55e-6)
		tmp = J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))));
	elseif (l <= 2400.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 2.2e+41)
		tmp = U + (-0.25 * (J * (l * (K ^ 2.0))));
	else
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.55e-6], N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2400.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.2e+41], N[(U + N[(-0.25 * N[(J * N[(l * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.55e-6

   1. Initial program 99.6%

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

   3. Taylor expanded in l around 0 67.9%

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

   4. Taylor expanded in K around 0 51.8%

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

   5. Taylor expanded in J around inf 51.9%

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

   if -1.55e-6 < l < 2400

   1. Initial program 64.2%

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

   3. Taylor expanded in l around 0 99.2%

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

   if 2400 < l < 2.1999999999999999e41

   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 3.3%

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

   4. Taylor expanded in K around 0 60.8%

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

   5. Taylor expanded in K around inf 60.1%

      \[\leadsto \color{blue}{-0.25 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   6. Step-by-step derivation

      1. *-commutative 60.1%

         \[\leadsto \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot -0.25} + U \]

      2. *-commutative 60.1%

         \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot {K}^{2}\right)}\right) \cdot -0.25 + U \]

   7. Simplified 60.1%

      \[\leadsto \color{blue}{\left(J \cdot \left(\ell \cdot {K}^{2}\right)\right) \cdot -0.25} + U \]

   if 2.1999999999999999e41 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 76.0%

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

   4. Taylor expanded in K around 0 58.5%

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

   5. Taylor expanded in l around inf 58.5%

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

      1. associate-*r* 76.0%

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

      2. *-commutative 76.0%

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

      3. associate-*l* 76.0%

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

   7. Simplified 58.5%

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

   8. Taylor expanded in J around inf 58.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.55 \cdot 10^{-6}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 2400:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;U + -0.25 \cdot \left(J \cdot \left(\ell \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.6% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-11} \lor \neg \left(\ell \leq 4.8 \cdot 10^{-14}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5e-11) (not (<= l 4.8e-14))) (* J (* l 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5e-11) || !(l <= 4.8e-14)) {
		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 <= (-5d-11)) .or. (.not. (l <= 4.8d-14))) 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 <= -5e-11) || !(l <= 4.8e-14)) {
		tmp = J * (l * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5e-11) or not (l <= 4.8e-14):
		tmp = J * (l * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5e-11) || !(l <= 4.8e-14))
		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 <= -5e-11) || ~((l <= 4.8e-14)))
		tmp = J * (l * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5e-11], N[Not[LessEqual[l, 4.8e-14]], $MachinePrecision]], N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -5.00000000000000018e-11 or 4.8e-14 < l

   1. Initial program 99.4%

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

   3. Taylor expanded in l around 0 30.0%

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

   4. Taylor expanded in K around 0 33.5%

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

   5. Taylor expanded in K around 0 24.2%

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

      1. associate-*r* 24.2%

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

      2. *-commutative 24.2%

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

      3. associate-*l* 24.2%

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

      4. *-commutative 24.2%

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

   7. Simplified 24.2%

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

   8. Taylor expanded in J around inf 24.3%

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

      1. *-commutative 24.3%

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

      2. associate-*r* 24.3%

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

   10. Simplified 24.3%

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

   if -5.00000000000000018e-11 < l < 4.8e-14

   1. Initial program 64.0%

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

   3. Taylor expanded in K around 0 64.0%

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

   4. Taylor expanded in J around 0 64.0%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-11} \lor \neg \left(\ell \leq 4.8 \cdot 10^{-14}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.3% accurate, 20.8× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (l * ((l * (l * (J * 0.3333333333333333))) + (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 * ((l * (l * (j * 0.3333333333333333d0))) + (j * 2.0d0)))
end function

Java

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

Python

def code(J, l, K, U):
	return U + (l * ((l * (l * (J * 0.3333333333333333))) + (J * 2.0)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in K around 0 70.1%

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

4. Taylor expanded in l around 0 60.5%

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

   1. associate-*r* 60.5%

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

6. Applied egg-rr 60.5%

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

   1. unpow2 60.5%

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

   2. associate-*r* 60.5%

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

   3. *-commutative 60.5%

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

8. Applied egg-rr 60.5%

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

9. Final simplification 60.5%

   \[\leadsto U + \ell \cdot \left(\ell \cdot \left(\ell \cdot \left(J \cdot 0.3333333333333333\right)\right) + J \cdot 2\right) \]
10. Add Preprocessing

Alternative 11: 57.4% accurate, 28.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in l around 0 58.7%

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

4. Taylor expanded in K around 0 41.7%

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

5. Taylor expanded in K around 0 46.6%

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

   1. associate-*r* 46.6%

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

   2. *-commutative 46.6%

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

   3. associate-*l* 46.6%

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

   4. *-commutative 46.6%

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

7. Simplified 46.6%

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

8. Taylor expanded in U around inf 50.5%

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

9. Final simplification 50.5%

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

Alternative 12: 54.0% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.9%

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

3. Taylor expanded in K around 0 70.1%

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

4. Taylor expanded in l around 0 46.6%

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

5. Final simplification 46.6%

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

Alternative 13: 36.9% 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 84.9%

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

3. Taylor expanded in K around 0 70.1%

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

4. Taylor expanded in J around 0 27.5%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))