Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.8% → 99.9%

Time: 12.5s

Alternatives: 12

Speedup: 1.4×

• 50.2% 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: 86.8% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -500000 \lor \neg \left(t_1 \leq 0.001\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

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 -500000.0) (not (<= t_1 0.001)))
     (+ (* t_0 (* t_1 J)) U)
     (+
      U
      (*
       t_0
       (*
        J
        (+
         (* 0.3333333333333333 (pow l 3.0))
         (+ (* 0.016666666666666666 (pow l 5.0)) (* 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 <= -500000.0) || !(t_1 <= 0.001)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + ((0.016666666666666666 * pow(l, 5.0)) + (l * 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-500000.0d0)) .or. (.not. (t_1 <= 0.001d0))) then
        tmp = (t_0 * (t_1 * j)) + u
    else
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + ((0.016666666666666666d0 * (l ** 5.0d0)) + (l * 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -500000.0) || !(t_1 <= 0.001)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + ((0.016666666666666666 * Math.pow(l, 5.0)) + (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 <= -500000.0) or not (t_1 <= 0.001):
		tmp = (t_0 * (t_1 * J)) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + ((0.016666666666666666 * math.pow(l, 5.0)) + (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 <= -500000.0) || !(t_1 <= 0.001))
		tmp = Float64(Float64(t_0 * Float64(t_1 * J)) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(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 <= -500000.0) || ~((t_1 <= 0.001)))
		tmp = (t_0 * (t_1 * J)) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + ((0.016666666666666666 * (l ^ 5.0)) + (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, -500000.0], N[Not[LessEqual[t$95$1, 0.001]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -5e5 or 1e-3 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -5e5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1e-3

   1. Initial program 72.6%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -500000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.001\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -500000 \lor \neg \left(t_1 \leq 0.001\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -500000.0) (not (<= t_1 0.001)))
     (+ (* t_0 (* t_1 J)) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* 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 <= -500000.0) || !(t_1 <= 0.001)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-500000.0d0)) .or. (.not. (t_1 <= 0.001d0))) then
        tmp = (t_0 * (t_1 * j)) + u
    else
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -500000.0) || !(t_1 <= 0.001)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (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 <= -500000.0) or not (t_1 <= 0.001):
		tmp = (t_0 * (t_1 * J)) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (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 <= -500000.0) || !(t_1 <= 0.001))
		tmp = Float64(Float64(t_0 * Float64(t_1 * J)) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -500000.0) || ~((t_1 <= 0.001)))
		tmp = (t_0 * (t_1 * J)) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (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, -500000.0], N[Not[LessEqual[t$95$1, 0.001]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -5e5 or 1e-3 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -5e5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1e-3

   1. Initial program 72.6%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -500000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.001\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot \cos \left(K \cdot 0.5\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.23:\\ \;\;\;\;U + t_1\\ \mathbf{elif}\;\ell \leq 85:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+61}:\\ \;\;\;\;U + t_1 \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (* J (pow l 5.0)) (* 0.016666666666666666 (cos (* K 0.5))))))
        (t_1 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -4.5e+65)
     t_0
     (if (<= l -0.23)
       (+ U t_1)
       (if (<= l 85.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 3.2e+61) (+ U (* t_1 (+ (* -0.125 (* K K)) 1.0))) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * pow(l, 5.0)) * (0.016666666666666666 * cos((K * 0.5))));
	double t_1 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -4.5e+65) {
		tmp = t_0;
	} else if (l <= -0.23) {
		tmp = U + t_1;
	} else if (l <= 85.0) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 3.2e+61) {
		tmp = U + (t_1 * ((-0.125 * (K * K)) + 1.0));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = u + ((j * (l ** 5.0d0)) * (0.016666666666666666d0 * cos((k * 0.5d0))))
    t_1 = (exp(l) - exp(-l)) * j
    if (l <= (-4.5d+65)) then
        tmp = t_0
    else if (l <= (-0.23d0)) then
        tmp = u + t_1
    else if (l <= 85.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 3.2d+61) then
        tmp = u + (t_1 * (((-0.125d0) * (k * k)) + 1.0d0))
    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 + ((J * Math.pow(l, 5.0)) * (0.016666666666666666 * Math.cos((K * 0.5))));
	double t_1 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -4.5e+65) {
		tmp = t_0;
	} else if (l <= -0.23) {
		tmp = U + t_1;
	} else if (l <= 85.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 3.2e+61) {
		tmp = U + (t_1 * ((-0.125 * (K * K)) + 1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * math.pow(l, 5.0)) * (0.016666666666666666 * math.cos((K * 0.5))))
	t_1 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -4.5e+65:
		tmp = t_0
	elif l <= -0.23:
		tmp = U + t_1
	elif l <= 85.0:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 3.2e+61:
		tmp = U + (t_1 * ((-0.125 * (K * K)) + 1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * (l ^ 5.0)) * Float64(0.016666666666666666 * cos(Float64(K * 0.5)))))
	t_1 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -4.5e+65)
		tmp = t_0;
	elseif (l <= -0.23)
		tmp = Float64(U + t_1);
	elseif (l <= 85.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 3.2e+61)
		tmp = Float64(U + Float64(t_1 * Float64(Float64(-0.125 * Float64(K * K)) + 1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (l ^ 5.0)) * (0.016666666666666666 * cos((K * 0.5))));
	t_1 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -4.5e+65)
		tmp = t_0;
	elseif (l <= -0.23)
		tmp = U + t_1;
	elseif (l <= 85.0)
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 3.2e+61)
		tmp = U + (t_1 * ((-0.125 * (K * K)) + 1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * N[(0.016666666666666666 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -4.5e+65], t$95$0, If[LessEqual[l, -0.23], N[(U + t$95$1), $MachinePrecision], If[LessEqual[l, 85.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.2e+61], N[(U + N[(t$95$1 * N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.5e65 or 3.1999999999999998e61 < 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. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left({\ell}^{5} \cdot J\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -4.5e65 < l < -0.23000000000000001

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 93.9%

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

   if -0.23000000000000001 < l < 85

   1. Initial program 72.6%

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

   2. Taylor expanded in l around 0 99.9%

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

   if 85 < l < 3.1999999999999998e61

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 0.0%

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J + -0.125 \cdot \left({K}^{2} \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 0.0%

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

      2. distribute-rgt1-in 73.7%

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

      3. unpow2 73.7%

         \[\leadsto \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]

   4. Simplified 73.7%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.23:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 85:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+61}:\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 4: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot \cos \left(K \cdot 0.5\right)\right)\\ t_1 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.053:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (* J (pow l 5.0)) (* 0.016666666666666666 (cos (* K 0.5))))))
        (t_1 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -4.5e+65)
     t_0
     (if (<= l -0.053)
       t_1
       (if (<= l 200.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 4.5e+61) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((J * pow(l, 5.0)) * (0.016666666666666666 * cos((K * 0.5))));
	double t_1 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -4.5e+65) {
		tmp = t_0;
	} else if (l <= -0.053) {
		tmp = t_1;
	} else if (l <= 200.0) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 4.5e+61) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = u + ((j * (l ** 5.0d0)) * (0.016666666666666666d0 * cos((k * 0.5d0))))
    t_1 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-4.5d+65)) then
        tmp = t_0
    else if (l <= (-0.053d0)) then
        tmp = t_1
    else if (l <= 200.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 4.5d+61) then
        tmp = t_1
    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 + ((J * Math.pow(l, 5.0)) * (0.016666666666666666 * Math.cos((K * 0.5))));
	double t_1 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -4.5e+65) {
		tmp = t_0;
	} else if (l <= -0.053) {
		tmp = t_1;
	} else if (l <= 200.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 4.5e+61) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((J * math.pow(l, 5.0)) * (0.016666666666666666 * math.cos((K * 0.5))))
	t_1 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -4.5e+65:
		tmp = t_0
	elif l <= -0.053:
		tmp = t_1
	elif l <= 200.0:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 4.5e+61:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(J * (l ^ 5.0)) * Float64(0.016666666666666666 * cos(Float64(K * 0.5)))))
	t_1 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -4.5e+65)
		tmp = t_0;
	elseif (l <= -0.053)
		tmp = t_1;
	elseif (l <= 200.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 4.5e+61)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((J * (l ^ 5.0)) * (0.016666666666666666 * cos((K * 0.5))));
	t_1 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -4.5e+65)
		tmp = t_0;
	elseif (l <= -0.053)
		tmp = t_1;
	elseif (l <= 200.0)
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 4.5e+61)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * N[(0.016666666666666666 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+65], t$95$0, If[LessEqual[l, -0.053], t$95$1, If[LessEqual[l, 200.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.5e+61], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.5e65 or 4.5e61 < 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. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left({\ell}^{5} \cdot J\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -4.5e65 < l < -0.0529999999999999985 or 200 < l < 4.5e61

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 81.3%

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

   if -0.0529999999999999985 < l < 200

   1. Initial program 72.8%

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

   2. Taylor expanded in l around 0 99.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.053:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot t_0\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.0026:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot t_0\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (* J (pow l 5.0)) (* 0.016666666666666666 t_0))))
        (t_2 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -4.5e+65)
     t_1
     (if (<= l -0.0026)
       t_2
       (if (<= l 200.0)
         (+ U (* 2.0 (* l (* J t_0))))
         (if (<= l 1.95e+61) 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 + ((J * pow(l, 5.0)) * (0.016666666666666666 * t_0));
	double t_2 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -4.5e+65) {
		tmp = t_1;
	} else if (l <= -0.0026) {
		tmp = t_2;
	} else if (l <= 200.0) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else if (l <= 1.95e+61) {
		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 + ((j * (l ** 5.0d0)) * (0.016666666666666666d0 * t_0))
    t_2 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-4.5d+65)) then
        tmp = t_1
    else if (l <= (-0.0026d0)) then
        tmp = t_2
    else if (l <= 200.0d0) then
        tmp = u + (2.0d0 * (l * (j * t_0)))
    else if (l <= 1.95d+61) 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 + ((J * Math.pow(l, 5.0)) * (0.016666666666666666 * t_0));
	double t_2 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -4.5e+65) {
		tmp = t_1;
	} else if (l <= -0.0026) {
		tmp = t_2;
	} else if (l <= 200.0) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else if (l <= 1.95e+61) {
		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 + ((J * math.pow(l, 5.0)) * (0.016666666666666666 * t_0))
	t_2 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -4.5e+65:
		tmp = t_1
	elif l <= -0.0026:
		tmp = t_2
	elif l <= 200.0:
		tmp = U + (2.0 * (l * (J * t_0)))
	elif l <= 1.95e+61:
		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(Float64(J * (l ^ 5.0)) * Float64(0.016666666666666666 * t_0)))
	t_2 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -4.5e+65)
		tmp = t_1;
	elseif (l <= -0.0026)
		tmp = t_2;
	elseif (l <= 200.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * t_0))));
	elseif (l <= 1.95e+61)
		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 + ((J * (l ^ 5.0)) * (0.016666666666666666 * t_0));
	t_2 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -4.5e+65)
		tmp = t_1;
	elseif (l <= -0.0026)
		tmp = t_2;
	elseif (l <= 200.0)
		tmp = U + (2.0 * (l * (J * t_0)));
	elseif (l <= 1.95e+61)
		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[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * N[(0.016666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+65], t$95$1, If[LessEqual[l, -0.0026], t$95$2, If[LessEqual[l, 200.0], N[(U + N[(2.0 * N[(l * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.95e+61], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.5e65 or 1.94999999999999994e61 < 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. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left({\ell}^{5} \cdot J\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   if -4.5e65 < l < -0.0025999999999999999 or 200 < l < 1.94999999999999994e61

   1. Initial program 99.9%

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

   2. Taylor expanded in K around 0 81.3%

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

   if -0.0025999999999999999 < l < 200

   1. Initial program 72.8%

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

   2. Taylor expanded in l around 0 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l* 98.7%

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

      3. *-commutative 98.7%

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

   4. Simplified 98.7%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.0026:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 200:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.95 \cdot 10^{+61}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 86.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00325 \lor \neg \left(\ell \leq 200\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.00325) (not (<= l 200.0)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00325) || !(l <= 200.0)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} else {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.00325d0)) .or. (.not. (l <= 200.0d0))) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    else
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00325) || !(l <= 200.0)) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} else {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.00325) or not (l <= 200.0):
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	else:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.00325) || !(l <= 200.0))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.00325) || ~((l <= 200.0)))
		tmp = U + ((exp(l) - exp(-l)) * J);
	else
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.00325], N[Not[LessEqual[l, 200.0]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.00324999999999999985 or 200 < 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. Taylor expanded in K around 0 76.1%

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

   if -0.00324999999999999985 < l < 200

   1. Initial program 72.8%

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

   2. Taylor expanded in l around 0 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l* 98.7%

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

      3. *-commutative 98.7%

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

   4. Simplified 98.7%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00325 \lor \neg \left(\ell \leq 200\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 7: 71.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.06:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.06)
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* 0.016666666666666666 (* J (pow l 5.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.06) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (0.016666666666666666 * (J * pow(l, 5.0)));
	}
	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)) <= (-0.06d0)) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (0.016666666666666666d0 * (j * (l ** 5.0d0)))
    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)) <= -0.06) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (0.016666666666666666 * (J * Math.pow(l, 5.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.06:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (0.016666666666666666 * (J * math.pow(l, 5.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.06)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(0.016666666666666666 * Float64(J * (l ^ 5.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.06)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (0.016666666666666666 * (J * (l ^ 5.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.06], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.016666666666666666 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.059999999999999998

   1. Initial program 85.5%

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

   2. Taylor expanded in l around 0 56.5%

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

   3. Taylor expanded in K around 0 43.0%

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

      1. associate-*r* 43.0%

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

      2. distribute-rgt-out 55.5%

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

      3. *-commutative 55.5%

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

      4. unpow2 55.5%

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

   5. Simplified 55.5%

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

   if -0.059999999999999998 < (cos.f64 (/.f64 K 2))

   1. Initial program 87.3%

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

   2. Taylor expanded in l around 0 92.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 79.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 79.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left({\ell}^{5} \cdot J\right)} + U \]

      2. *-commutative 79.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 79.0%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in K around 0 79.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left({\ell}^{5} \cdot J\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.06:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \end{array} \]

Alternative 8: 82.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot {\ell}^{5}\\ t_1 := U + 0.016666666666666666 \cdot t_0\\ \mathbf{if}\;\ell \leq -6.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 196:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+47}:\\ \;\;\;\;U + t_0 \cdot \left(0.016666666666666666 + \left(K \cdot K\right) \cdot -0.0020833333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (pow l 5.0))) (t_1 (+ U (* 0.016666666666666666 t_0))))
   (if (<= l -6.5)
     t_1
     (if (<= l 196.0)
       (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
       (if (<= l 1.65e+47)
         (+
          U
          (* t_0 (+ 0.016666666666666666 (* (* K K) -0.0020833333333333333))))
         t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * pow(l, 5.0);
	double t_1 = U + (0.016666666666666666 * t_0);
	double tmp;
	if (l <= -6.5) {
		tmp = t_1;
	} else if (l <= 196.0) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 1.65e+47) {
		tmp = U + (t_0 * (0.016666666666666666 + ((K * K) * -0.0020833333333333333)));
	} 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) :: tmp
    t_0 = j * (l ** 5.0d0)
    t_1 = u + (0.016666666666666666d0 * t_0)
    if (l <= (-6.5d0)) then
        tmp = t_1
    else if (l <= 196.0d0) then
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    else if (l <= 1.65d+47) then
        tmp = u + (t_0 * (0.016666666666666666d0 + ((k * k) * (-0.0020833333333333333d0))))
    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 = J * Math.pow(l, 5.0);
	double t_1 = U + (0.016666666666666666 * t_0);
	double tmp;
	if (l <= -6.5) {
		tmp = t_1;
	} else if (l <= 196.0) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 1.65e+47) {
		tmp = U + (t_0 * (0.016666666666666666 + ((K * K) * -0.0020833333333333333)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * math.pow(l, 5.0)
	t_1 = U + (0.016666666666666666 * t_0)
	tmp = 0
	if l <= -6.5:
		tmp = t_1
	elif l <= 196.0:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 1.65e+47:
		tmp = U + (t_0 * (0.016666666666666666 + ((K * K) * -0.0020833333333333333)))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * (l ^ 5.0))
	t_1 = Float64(U + Float64(0.016666666666666666 * t_0))
	tmp = 0.0
	if (l <= -6.5)
		tmp = t_1;
	elseif (l <= 196.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 1.65e+47)
		tmp = Float64(U + Float64(t_0 * Float64(0.016666666666666666 + Float64(Float64(K * K) * -0.0020833333333333333))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l ^ 5.0);
	t_1 = U + (0.016666666666666666 * t_0);
	tmp = 0.0;
	if (l <= -6.5)
		tmp = t_1;
	elseif (l <= 196.0)
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	elseif (l <= 1.65e+47)
		tmp = U + (t_0 * (0.016666666666666666 + ((K * K) * -0.0020833333333333333)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(0.016666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.5], t$95$1, If[LessEqual[l, 196.0], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.65e+47], N[(U + N[(t$95$0 * N[(0.016666666666666666 + N[(N[(K * K), $MachinePrecision] * -0.0020833333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.5 or 1.65e47 < 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. Taylor expanded in l around 0 92.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 92.3%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 92.3%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left({\ell}^{5} \cdot J\right)} + U \]

      2. *-commutative 92.3%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 92.3%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in K around 0 70.8%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left({\ell}^{5} \cdot J\right)} + U \]

   if -6.5 < l < 196

   1. Initial program 72.6%

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

   2. Taylor expanded in l around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. *-commutative 99.4%

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

   4. Simplified 99.4%

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

   if 196 < l < 1.65e47

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 13.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 13.0%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 13.0%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left({\ell}^{5} \cdot J\right)} + U \]

      2. *-commutative 13.0%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 13.0%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in K around 0 32.8%

      \[\leadsto \color{blue}{\left(-0.0020833333333333333 \cdot \left({K}^{2} \cdot \left({\ell}^{5} \cdot J\right)\right) + 0.016666666666666666 \cdot \left({\ell}^{5} \cdot J\right)\right)} + U \]
   7. Step-by-step derivation

      1. +-commutative 32.8%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \left({\ell}^{5} \cdot J\right) + -0.0020833333333333333 \cdot \left({K}^{2} \cdot \left({\ell}^{5} \cdot J\right)\right)\right)} + U \]

      2. associate-*r* 32.8%

         \[\leadsto \left(0.016666666666666666 \cdot \left({\ell}^{5} \cdot J\right) + \color{blue}{\left(-0.0020833333333333333 \cdot {K}^{2}\right) \cdot \left({\ell}^{5} \cdot J\right)}\right) + U \]

      3. distribute-rgt-out 40.5%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 + -0.0020833333333333333 \cdot {K}^{2}\right)} + U \]

      4. unpow2 40.5%

         \[\leadsto \left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 + -0.0020833333333333333 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   8. Simplified 40.5%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 + -0.0020833333333333333 \cdot \left(K \cdot K\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \mathbf{elif}\;\ell \leq 196:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+47}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{5}\right) \cdot \left(0.016666666666666666 + \left(K \cdot K\right) \cdot -0.0020833333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \end{array} \]

Alternative 9: 81.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \lor \neg \left(\ell \leq 1520000\right):\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.6) (not (<= l 1520000.0)))
   (+ U (* 0.016666666666666666 (* J (pow l 5.0))))
   (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.6) || !(l <= 1520000.0)) {
		tmp = U + (0.016666666666666666 * (J * pow(l, 5.0)));
	} else {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-6.6d0)) .or. (.not. (l <= 1520000.0d0))) then
        tmp = u + (0.016666666666666666d0 * (j * (l ** 5.0d0)))
    else
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.6) || !(l <= 1520000.0)) {
		tmp = U + (0.016666666666666666 * (J * Math.pow(l, 5.0)));
	} else {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.6) or not (l <= 1520000.0):
		tmp = U + (0.016666666666666666 * (J * math.pow(l, 5.0)))
	else:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.6) || !(l <= 1520000.0))
		tmp = Float64(U + Float64(0.016666666666666666 * Float64(J * (l ^ 5.0))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -6.6) || ~((l <= 1520000.0)))
		tmp = U + (0.016666666666666666 * (J * (l ^ 5.0)));
	else
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.6], N[Not[LessEqual[l, 1520000.0]], $MachinePrecision]], N[(U + N[(0.016666666666666666 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -6.5999999999999996 or 1.52e6 < 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. Taylor expanded in l around 0 85.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 85.1%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{5} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 85.1%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left({\ell}^{5} \cdot J\right)} + U \]

      2. *-commutative 85.1%

         \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   5. Simplified 85.1%

      \[\leadsto \color{blue}{\left({\ell}^{5} \cdot J\right) \cdot \left(0.016666666666666666 \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   6. Taylor expanded in K around 0 64.8%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left({\ell}^{5} \cdot J\right)} + U \]

   if -6.5999999999999996 < l < 1.52e6

   1. Initial program 72.8%

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

   2. Taylor expanded in l around 0 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-*l* 98.7%

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

      3. *-commutative 98.7%

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

   4. Simplified 98.7%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.6 \lor \neg \left(\ell \leq 1520000\right):\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 10: 58.2% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+78} \lor \neg \left(\ell \leq 1.45 \cdot 10^{-43}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.45e+78) (not (<= l 1.45e-43)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.45e+78) || !(l <= 1.45e-43)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.45d+78)) .or. (.not. (l <= 1.45d-43))) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.45e+78) || !(l <= 1.45e-43)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.45e+78) or not (l <= 1.45e-43):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.45e+78) || !(l <= 1.45e-43))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.45e+78) || ~((l <= 1.45e-43)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.45e+78], N[Not[LessEqual[l, 1.45e-43]], $MachinePrecision]], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.45000000000000008e78 or 1.4500000000000001e-43 < l

   1. Initial program 97.4%

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

   2. Taylor expanded in l around 0 36.9%

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

   3. Taylor expanded in K around 0 20.1%

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

      1. associate-*r* 20.1%

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

      2. distribute-rgt-out 39.7%

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

      3. *-commutative 39.7%

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

      4. unpow2 39.7%

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

   5. Simplified 39.7%

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

   if -1.45000000000000008e78 < l < 1.4500000000000001e-43

   1. Initial program 76.4%

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

   2. Taylor expanded in l around 0 87.2%

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

   3. Taylor expanded in K around 0 75.9%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.45 \cdot 10^{+78} \lor \neg \left(\ell \leq 1.45 \cdot 10^{-43}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 11: 53.5% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

2. Taylor expanded in l around 0 62.1%

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

3. Taylor expanded in K around 0 52.4%

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

4. Final simplification 52.4%

   \[\leadsto U + 2 \cdot \left(\ell \cdot J\right) \]

Alternative 12: 36.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

3. Applied egg-rr 24.5%

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

4. Taylor expanded in J around 0 35.2%

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

5. Final simplification 35.2%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023268 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))