Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.8%

Time: 14.0s

Alternatives: 11

Speedup: 1.4×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+24} \lor \neg \left(t_1 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot t_0\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot t_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -2e+24) (not (<= t_1 2e-6)))
     (+ (* (* t_1 J) (cos (/ K 2.0))) U)
     (+
      U
      (+
       (* 0.3333333333333333 (* J (* (pow l 3.0) t_0)))
       (* 2.0 (* J (* l t_0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -2e+24) || !(t_1 <= 2e-6)) {
		tmp = ((t_1 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + ((0.3333333333333333 * (J * (pow(l, 3.0) * t_0))) + (2.0 * (J * (l * 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 = cos((k * 0.5d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-2d+24)) .or. (.not. (t_1 <= 2d-6))) then
        tmp = ((t_1 * j) * cos((k / 2.0d0))) + u
    else
        tmp = u + ((0.3333333333333333d0 * (j * ((l ** 3.0d0) * t_0))) + (2.0d0 * (j * (l * t_0))))
    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 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -2e+24) || !(t_1 <= 2e-6)) {
		tmp = ((t_1 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + ((0.3333333333333333 * (J * (Math.pow(l, 3.0) * t_0))) + (2.0 * (J * (l * t_0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -2e+24) or not (t_1 <= 2e-6):
		tmp = ((t_1 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + ((0.3333333333333333 * (J * (math.pow(l, 3.0) * t_0))) + (2.0 * (J * (l * t_0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -2e+24) || !(t_1 <= 2e-6))
		tmp = Float64(Float64(Float64(t_1 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(Float64(0.3333333333333333 * Float64(J * Float64((l ^ 3.0) * t_0))) + Float64(2.0 * Float64(J * Float64(l * t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -2e+24) || ~((t_1 <= 2e-6)))
		tmp = ((t_1 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + ((0.3333333333333333 * (J * ((l ^ 3.0) * t_0))) + (2.0 * (J * (l * t_0))));
	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[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+24], N[Not[LessEqual[t$95$1, 2e-6]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(0.3333333333333333 * N[(J * N[(N[Power[l, 3.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -2e24 or 1.99999999999999991e-6 < (-.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 -2e24 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999991e-6

   1. Initial program 69.9%

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

   2. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -2 \cdot 10^{+24} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+24} \lor \neg \left(t_1 \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 -2e+24) (not (<= t_1 2e-6)))
     (+ (* (* t_1 J) t_0) 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 <= -2e+24) || !(t_1 <= 2e-6)) {
		tmp = ((t_1 * J) * t_0) + 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 <= (-2d+24)) .or. (.not. (t_1 <= 2d-6))) then
        tmp = ((t_1 * j) * t_0) + 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 <= -2e+24) || !(t_1 <= 2e-6)) {
		tmp = ((t_1 * J) * t_0) + 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 <= -2e+24) or not (t_1 <= 2e-6):
		tmp = ((t_1 * J) * t_0) + 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 <= -2e+24) || !(t_1 <= 2e-6))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + 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 <= -2e+24) || ~((t_1 <= 2e-6)))
		tmp = ((t_1 * J) * t_0) + 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, -2e+24], N[Not[LessEqual[t$95$1, 2e-6]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $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))) < -2e24 or 1.99999999999999991e-6 < (-.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 -2e24 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.99999999999999991e-6

   1. Initial program 69.9%

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

   2. Taylor expanded in l around 0 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 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -2 \cdot 10^{+24} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-6}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 95.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot {\ell}^{3}\\ t_1 := e^{\ell} - e^{-\ell}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := U + t_2 \cdot \left(J \cdot t_0\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+125}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;t_1 \cdot J + U\\ \mathbf{elif}\;\ell \leq 13.6:\\ \;\;\;\;U + t_2 \cdot \left(J \cdot \left(t_0 + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+100}:\\ \;\;\;\;U + t_1 \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (pow l 3.0)))
        (t_1 (- (exp l) (exp (- l))))
        (t_2 (cos (/ K 2.0)))
        (t_3 (+ U (* t_2 (* J t_0)))))
   (if (<= l -5e+125)
     t_3
     (if (<= l -57.0)
       (+ (* t_1 J) U)
       (if (<= l 13.6)
         (+ U (* t_2 (* J (+ t_0 (* l 2.0)))))
         (if (<= l 6.5e+100)
           (+ U (* t_1 (+ J (* J (* -0.125 (* K K))))))
           t_3))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * pow(l, 3.0);
	double t_1 = exp(l) - exp(-l);
	double t_2 = cos((K / 2.0));
	double t_3 = U + (t_2 * (J * t_0));
	double tmp;
	if (l <= -5e+125) {
		tmp = t_3;
	} else if (l <= -57.0) {
		tmp = (t_1 * J) + U;
	} else if (l <= 13.6) {
		tmp = U + (t_2 * (J * (t_0 + (l * 2.0))));
	} else if (l <= 6.5e+100) {
		tmp = U + (t_1 * (J + (J * (-0.125 * (K * K)))));
	} else {
		tmp = t_3;
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (l ** 3.0d0)
    t_1 = exp(l) - exp(-l)
    t_2 = cos((k / 2.0d0))
    t_3 = u + (t_2 * (j * t_0))
    if (l <= (-5d+125)) then
        tmp = t_3
    else if (l <= (-57.0d0)) then
        tmp = (t_1 * j) + u
    else if (l <= 13.6d0) then
        tmp = u + (t_2 * (j * (t_0 + (l * 2.0d0))))
    else if (l <= 6.5d+100) then
        tmp = u + (t_1 * (j + (j * ((-0.125d0) * (k * k)))))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * Math.pow(l, 3.0);
	double t_1 = Math.exp(l) - Math.exp(-l);
	double t_2 = Math.cos((K / 2.0));
	double t_3 = U + (t_2 * (J * t_0));
	double tmp;
	if (l <= -5e+125) {
		tmp = t_3;
	} else if (l <= -57.0) {
		tmp = (t_1 * J) + U;
	} else if (l <= 13.6) {
		tmp = U + (t_2 * (J * (t_0 + (l * 2.0))));
	} else if (l <= 6.5e+100) {
		tmp = U + (t_1 * (J + (J * (-0.125 * (K * K)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * math.pow(l, 3.0)
	t_1 = math.exp(l) - math.exp(-l)
	t_2 = math.cos((K / 2.0))
	t_3 = U + (t_2 * (J * t_0))
	tmp = 0
	if l <= -5e+125:
		tmp = t_3
	elif l <= -57.0:
		tmp = (t_1 * J) + U
	elif l <= 13.6:
		tmp = U + (t_2 * (J * (t_0 + (l * 2.0))))
	elif l <= 6.5e+100:
		tmp = U + (t_1 * (J + (J * (-0.125 * (K * K)))))
	else:
		tmp = t_3
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * (l ^ 3.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(U + Float64(t_2 * Float64(J * t_0)))
	tmp = 0.0
	if (l <= -5e+125)
		tmp = t_3;
	elseif (l <= -57.0)
		tmp = Float64(Float64(t_1 * J) + U);
	elseif (l <= 13.6)
		tmp = Float64(U + Float64(t_2 * Float64(J * Float64(t_0 + Float64(l * 2.0)))));
	elseif (l <= 6.5e+100)
		tmp = Float64(U + Float64(t_1 * Float64(J + Float64(J * Float64(-0.125 * Float64(K * K))))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 * (l ^ 3.0);
	t_1 = exp(l) - exp(-l);
	t_2 = cos((K / 2.0));
	t_3 = U + (t_2 * (J * t_0));
	tmp = 0.0;
	if (l <= -5e+125)
		tmp = t_3;
	elseif (l <= -57.0)
		tmp = (t_1 * J) + U;
	elseif (l <= 13.6)
		tmp = U + (t_2 * (J * (t_0 + (l * 2.0))));
	elseif (l <= 6.5e+100)
		tmp = U + (t_1 * (J + (J * (-0.125 * (K * K)))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(U + N[(t$95$2 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e+125], t$95$3, If[LessEqual[l, -57.0], N[(N[(t$95$1 * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 13.6], N[(U + N[(t$95$2 * N[(J * N[(t$95$0 + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e+100], N[(U + N[(t$95$1 * N[(J + N[(J * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.99999999999999962e125 or 6.50000000000000001e100 < 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%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -4.99999999999999962e125 < l < -57

   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 76.9%

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

   if -57 < l < 13.5999999999999996

   1. Initial program 70.2%

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

   2. Taylor expanded in l around 0 99.1%

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

   if 13.5999999999999996 < l < 6.50000000000000001e100

   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%

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

      1. +-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. associate-*r* 0.0%

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

      4. distribute-rgt-out 91.7%

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

      5. associate-*r* 91.7%

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

      6. *-commutative 91.7%

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

      7. associate-*l* 91.7%

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

      8. unpow2 91.7%

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

   4. Simplified 91.7%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+125}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 13.6:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+100}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 4: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot {\ell}^{3}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := U + t_1 \cdot \left(J \cdot t_0\right)\\ t_3 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -1.96 \cdot 10^{+105}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 16500000000:\\ \;\;\;\;U + t_1 \cdot \left(J \cdot \left(t_0 + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+100}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (pow l 3.0)))
        (t_1 (cos (/ K 2.0)))
        (t_2 (+ U (* t_1 (* J t_0))))
        (t_3 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -1.96e+105)
     t_2
     (if (<= l -57.0)
       t_3
       (if (<= l 16500000000.0)
         (+ U (* t_1 (* J (+ t_0 (* l 2.0)))))
         (if (<= l 6.3e+100) t_3 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = 0.3333333333333333 * pow(l, 3.0);
	double t_1 = cos((K / 2.0));
	double t_2 = U + (t_1 * (J * t_0));
	double t_3 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -1.96e+105) {
		tmp = t_2;
	} else if (l <= -57.0) {
		tmp = t_3;
	} else if (l <= 16500000000.0) {
		tmp = U + (t_1 * (J * (t_0 + (l * 2.0))));
	} else if (l <= 6.3e+100) {
		tmp = t_3;
	} 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) :: t_3
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (l ** 3.0d0)
    t_1 = cos((k / 2.0d0))
    t_2 = u + (t_1 * (j * t_0))
    t_3 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-1.96d+105)) then
        tmp = t_2
    else if (l <= (-57.0d0)) then
        tmp = t_3
    else if (l <= 16500000000.0d0) then
        tmp = u + (t_1 * (j * (t_0 + (l * 2.0d0))))
    else if (l <= 6.3d+100) then
        tmp = t_3
    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 = 0.3333333333333333 * Math.pow(l, 3.0);
	double t_1 = Math.cos((K / 2.0));
	double t_2 = U + (t_1 * (J * t_0));
	double t_3 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -1.96e+105) {
		tmp = t_2;
	} else if (l <= -57.0) {
		tmp = t_3;
	} else if (l <= 16500000000.0) {
		tmp = U + (t_1 * (J * (t_0 + (l * 2.0))));
	} else if (l <= 6.3e+100) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = 0.3333333333333333 * math.pow(l, 3.0)
	t_1 = math.cos((K / 2.0))
	t_2 = U + (t_1 * (J * t_0))
	t_3 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -1.96e+105:
		tmp = t_2
	elif l <= -57.0:
		tmp = t_3
	elif l <= 16500000000.0:
		tmp = U + (t_1 * (J * (t_0 + (l * 2.0))))
	elif l <= 6.3e+100:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(0.3333333333333333 * (l ^ 3.0))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(U + Float64(t_1 * Float64(J * t_0)))
	t_3 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -1.96e+105)
		tmp = t_2;
	elseif (l <= -57.0)
		tmp = t_3;
	elseif (l <= 16500000000.0)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(t_0 + Float64(l * 2.0)))));
	elseif (l <= 6.3e+100)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = 0.3333333333333333 * (l ^ 3.0);
	t_1 = cos((K / 2.0));
	t_2 = U + (t_1 * (J * t_0));
	t_3 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -1.96e+105)
		tmp = t_2;
	elseif (l <= -57.0)
		tmp = t_3;
	elseif (l <= 16500000000.0)
		tmp = U + (t_1 * (J * (t_0 + (l * 2.0))));
	elseif (l <= 6.3e+100)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(t$95$1 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1.96e+105], t$95$2, If[LessEqual[l, -57.0], t$95$3, If[LessEqual[l, 16500000000.0], N[(U + N[(t$95$1 * N[(J * N[(t$95$0 + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.3e+100], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.95999999999999994e105 or 6.3000000000000004e100 < 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%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -1.95999999999999994e105 < l < -57 or 1.65e10 < l < 6.3000000000000004e100

   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 79.6%

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

   if -57 < l < 1.65e10

   1. Initial program 70.5%

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

   2. Taylor expanded in l around 0 98.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.96 \cdot 10^{+105}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 16500000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.3 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 5: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -1.96 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 16500000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (cos (/ K 2.0)) (* J (* 0.3333333333333333 (pow l 3.0))))))
        (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -1.96e+105)
     t_0
     (if (<= l -57.0)
       t_1
       (if (<= l 16500000000.0)
         (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
         (if (<= l 4.3e+100) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * (0.3333333333333333 * pow(l, 3.0))));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -1.96e+105) {
		tmp = t_0;
	} else if (l <= -57.0) {
		tmp = t_1;
	} else if (l <= 16500000000.0) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else if (l <= 4.3e+100) {
		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 + (cos((k / 2.0d0)) * (j * (0.3333333333333333d0 * (l ** 3.0d0))))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-1.96d+105)) then
        tmp = t_0
    else if (l <= (-57.0d0)) then
        tmp = t_1
    else if (l <= 16500000000.0d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else if (l <= 4.3d+100) 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 + (Math.cos((K / 2.0)) * (J * (0.3333333333333333 * Math.pow(l, 3.0))));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -1.96e+105) {
		tmp = t_0;
	} else if (l <= -57.0) {
		tmp = t_1;
	} else if (l <= 16500000000.0) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else if (l <= 4.3e+100) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * (0.3333333333333333 * math.pow(l, 3.0))))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -1.96e+105:
		tmp = t_0
	elif l <= -57.0:
		tmp = t_1
	elif l <= 16500000000.0:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	elif l <= 4.3e+100:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -1.96e+105)
		tmp = t_0;
	elseif (l <= -57.0)
		tmp = t_1;
	elseif (l <= 16500000000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	elseif (l <= 4.3e+100)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * (0.3333333333333333 * (l ^ 3.0))));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -1.96e+105)
		tmp = t_0;
	elseif (l <= -57.0)
		tmp = t_1;
	elseif (l <= 16500000000.0)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	elseif (l <= 4.3e+100)
		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[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -1.96e+105], t$95$0, If[LessEqual[l, -57.0], t$95$1, If[LessEqual[l, 16500000000.0], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.3e+100], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.95999999999999994e105 or 4.29999999999999993e100 < 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%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -1.95999999999999994e105 < l < -57 or 1.65e10 < l < 4.29999999999999993e100

   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 79.6%

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

   if -57 < l < 1.65e10

   1. Initial program 70.5%

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

   2. Taylor expanded in l around 0 98.2%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.96 \cdot 10^{+105}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -57:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 16500000000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 4.3 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]

Alternative 6: 59.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.04:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \mathbf{elif}\;t_0 \leq 0.832:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot \left(\ell \cdot K\right)\right) \cdot 192\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.04)
     (+ U (* J (* l -6.0)))
     (if (<= t_0 0.832)
       (+ U (* J (* (* K (* l K)) 192.0)))
       (+ U (* 2.0 (* l J)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.04) {
		tmp = U + (J * (l * -6.0));
	} else if (t_0 <= 0.832) {
		tmp = U + (J * ((K * (l * K)) * 192.0));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.04d0)) then
        tmp = u + (j * (l * (-6.0d0)))
    else if (t_0 <= 0.832d0) then
        tmp = u + (j * ((k * (l * k)) * 192.0d0))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.04) {
		tmp = U + (J * (l * -6.0));
	} else if (t_0 <= 0.832) {
		tmp = U + (J * ((K * (l * K)) * 192.0));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.04:
		tmp = U + (J * (l * -6.0))
	elif t_0 <= 0.832:
		tmp = U + (J * ((K * (l * K)) * 192.0))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.04)
		tmp = Float64(U + Float64(J * Float64(l * -6.0)));
	elseif (t_0 <= 0.832)
		tmp = Float64(U + Float64(J * Float64(Float64(K * Float64(l * K)) * 192.0)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.04)
		tmp = U + (J * (l * -6.0));
	elseif (t_0 <= 0.832)
		tmp = U + (J * ((K * (l * K)) * 192.0));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.04], N[(U + N[(J * N[(l * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.832], N[(U + N[(J * N[(N[(K * N[(l * K), $MachinePrecision]), $MachinePrecision] * 192.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.0400000000000000008

   1. Initial program 82.3%

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

   2. Taylor expanded in l around 0 72.8%

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

      1. associate-*r* 72.8%

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

      2. *-commutative 72.8%

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

   4. Simplified 72.8%

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

   6. Applied egg-rr 46.8%

      \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \color{blue}{\left(\cos \left(-8 \cdot K\right) \cdot -3\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 46.8%

         \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -8\right)} \cdot -3\right) + U \]

   8. Simplified 46.8%

      \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \color{blue}{\left(\cos \left(K \cdot -8\right) \cdot -3\right)} + U \]

   9. Taylor expanded in K around 0 57.5%

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

       1. *-commutative 57.5%

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

       2. associate-*l* 58.8%

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   11. Simplified 58.8%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   if -0.0400000000000000008 < (cos.f64 (/.f64 K 2)) < 0.831999999999999962

   1. Initial program 83.7%

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

   2. Taylor expanded in l around 0 61.5%

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

      1. associate-*r* 61.5%

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

      2. *-commutative 61.5%

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

   4. Simplified 61.5%

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

   6. Applied egg-rr 28.1%

      \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \color{blue}{\left(\cos \left(-8 \cdot K\right) \cdot -3\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 28.1%

         \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -8\right)} \cdot -3\right) + U \]

   8. Simplified 28.1%

      \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \color{blue}{\left(\cos \left(K \cdot -8\right) \cdot -3\right)} + U \]

   9. Taylor expanded in K around 0 58.4%

      \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \left(\color{blue}{\left(1 + -32 \cdot {K}^{2}\right)} \cdot -3\right) + U \]
   10. Step-by-step derivation

       1. *-commutative 58.4%

          \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \left(\left(1 + \color{blue}{{K}^{2} \cdot -32}\right) \cdot -3\right) + U \]

       2. unpow2 58.4%

          \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \left(\left(1 + \color{blue}{\left(K \cdot K\right)} \cdot -32\right) \cdot -3\right) + U \]

   11. Simplified 58.4%

       \[\leadsto \left(\ell \cdot \left(2 \cdot J\right)\right) \cdot \left(\color{blue}{\left(1 + \left(K \cdot K\right) \cdot -32\right)} \cdot -3\right) + U \]

   12. Taylor expanded in K around inf 58.4%

       \[\leadsto \color{blue}{192 \cdot \left(J \cdot \left({K}^{2} \cdot \ell\right)\right)} + U \]
   13. Step-by-step derivation

       1. *-commutative 58.4%

          \[\leadsto \color{blue}{\left(J \cdot \left({K}^{2} \cdot \ell\right)\right) \cdot 192} + U \]

       2. associate-*l* 58.4%

          \[\leadsto \color{blue}{J \cdot \left(\left({K}^{2} \cdot \ell\right) \cdot 192\right)} + U \]

       3. *-commutative 58.4%

          \[\leadsto J \cdot \left(\color{blue}{\left(\ell \cdot {K}^{2}\right)} \cdot 192\right) + U \]

       4. unpow2 58.4%

          \[\leadsto J \cdot \left(\left(\ell \cdot \color{blue}{\left(K \cdot K\right)}\right) \cdot 192\right) + U \]

       5. associate-*r* 58.5%

          \[\leadsto J \cdot \left(\color{blue}{\left(\left(\ell \cdot K\right) \cdot K\right)} \cdot 192\right) + U \]

   14. Simplified 58.5%

       \[\leadsto \color{blue}{J \cdot \left(\left(\left(\ell \cdot K\right) \cdot K\right) \cdot 192\right)} + U \]

   if 0.831999999999999962 < (cos.f64 (/.f64 K 2))

   1. Initial program 90.2%

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

   2. Taylor expanded in l around 0 58.0%

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

      1. associate-*r* 58.0%

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

      2. *-commutative 58.0%

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

   4. Simplified 58.0%

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

   5. Taylor expanded in K around 0 56.8%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.04:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.832:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot \left(\ell \cdot K\right)\right) \cdot 192\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 7: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.55:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.55)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.55) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (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) :: tmp
    if (cos((k / 2.0d0)) <= 0.55d0) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (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 tmp;
	if (Math.cos((K / 2.0)) <= 0.55) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.55:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.55)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + 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)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.55)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.55], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.55000000000000004

   1. Initial program 80.4%

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

   2. Taylor expanded in l around 0 75.1%

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

   if 0.55000000000000004 < (cos.f64 (/.f64 K 2))

   1. Initial program 90.8%

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

   2. Taylor expanded in l around 0 83.2%

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

   3. Taylor expanded in K around 0 81.7%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.55:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 8: 81.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -2.2 \cdot 10^{+141} \lor \neg \left(J \leq 2.6 \cdot 10^{+72}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J -2.2e+141) (not (<= J 2.6e+72)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (+ (* (- (exp l) (exp (- l))) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -2.2e+141) || !(J <= 2.6e+72)) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((j <= (-2.2d+141)) .or. (.not. (j <= 2.6d+72))) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = ((exp(l) - exp(-l)) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -2.2e+141) || !(J <= 2.6e+72)) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= -2.2e+141) or not (J <= 2.6e+72):
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= -2.2e+141) || !(J <= 2.6e+72))
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= -2.2e+141) || ~((J <= 2.6e+72)))
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = ((exp(l) - exp(-l)) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, -2.2e+141], N[Not[LessEqual[J, 2.6e+72]], $MachinePrecision]], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -2.2e141 or 2.59999999999999981e72 < J

   1. Initial program 70.5%

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

   2. Taylor expanded in l around 0 88.0%

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

   if -2.2e141 < J < 2.59999999999999981e72

   1. Initial program 97.5%

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

   2. Taylor expanded in K around 0 86.8%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -2.2 \cdot 10^{+141} \lor \neg \left(J \leq 2.6 \cdot 10^{+72}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]

Alternative 9: 64.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* 2.0 (* J (* l (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	return U + (2.0 * (J * (l * cos((K * 0.5)))));
}

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 * (j * (l * cos((k * 0.5d0)))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
}

Python

def code(J, l, K, U):
	return U + (2.0 * (J * (l * math.cos((K * 0.5)))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
end

Wolfram

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

Derivation

1. Initial program 86.8%

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

2. Taylor expanded in l around 0 62.9%

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

3. Final simplification 62.9%

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

Alternative 10: 54.1% 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 86.8%

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

2. Taylor expanded in l around 0 62.9%

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

   1. associate-*r* 62.9%

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

   2. *-commutative 62.9%

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

4. Simplified 62.9%

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

5. Taylor expanded in K around 0 48.3%

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

6. Final simplification 48.3%

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

Alternative 11: 36.5% 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 86.8%

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

3. Applied egg-rr 23.1%

   \[\leadsto \left(J \cdot \color{blue}{512}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

4. Taylor expanded in J around 0 31.6%

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

5. Final simplification 31.6%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023275 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))