Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.7%

Time: 10.6s

Alternatives: 12

Speedup: 1.5×

• 50.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -0.05 \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;U + t_0 \cdot \left(J \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \end{array} \end{array} \]

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 -0.05) (not (<= t_1 0.0)))
     (+ U (* t_0 (* J t_1)))
     (+ U (* t_0 (* J (* 2.0 l)))))))

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 <= -0.05) || !(t_1 <= 0.0)) {
		tmp = U + (t_0 * (J * t_1));
	} else {
		tmp = U + (t_0 * (J * (2.0 * l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-0.05d0)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = u + (t_0 * (j * t_1))
    else
        tmp = u + (t_0 * (j * (2.0d0 * l)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -0.05) || !(t_1 <= 0.0)) {
		tmp = U + (t_0 * (J * t_1));
	} else {
		tmp = U + (t_0 * (J * (2.0 * l)));
	}
	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 <= -0.05) or not (t_1 <= 0.0):
		tmp = U + (t_0 * (J * t_1))
	else:
		tmp = U + (t_0 * (J * (2.0 * l)))
	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 <= -0.05) || !(t_1 <= 0.0))
		tmp = Float64(U + Float64(t_0 * Float64(J * t_1)));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(2.0 * l))));
	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 <= -0.05) || ~((t_1 <= 0.0)))
		tmp = U + (t_0 * (J * t_1));
	else
		tmp = U + (t_0 * (J * (2.0 * l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.05], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(U + N[(t$95$0 * N[(J * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.050000000000000003 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -0.050000000000000003 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 74.1%

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

   2. Taylor expanded in l around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.05 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \end{array} \]

Alternative 2: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.908:\\ \;\;\;\;t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.908)
     (+
      (*
       t_0
       (*
        J
        (+
         (* 0.3333333333333333 (pow l 3.0))
         (+
          (* 0.0003968253968253968 (pow l 7.0))
          (+ (* 0.016666666666666666 (pow l 5.0)) (* 2.0 l))))))
      U)
     (+ U (* J (* 2.0 (sinh l)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.908) {
		tmp = (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + ((0.0003968253968253968 * pow(l, 7.0)) + ((0.016666666666666666 * pow(l, 5.0)) + (2.0 * l)))))) + U;
	} else {
		tmp = U + (J * (2.0 * sinh(l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.908d0) then
        tmp = (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + ((0.0003968253968253968d0 * (l ** 7.0d0)) + ((0.016666666666666666d0 * (l ** 5.0d0)) + (2.0d0 * l)))))) + u
    else
        tmp = u + (j * (2.0d0 * sinh(l)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.908) {
		tmp = (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + ((0.0003968253968253968 * Math.pow(l, 7.0)) + ((0.016666666666666666 * Math.pow(l, 5.0)) + (2.0 * l)))))) + U;
	} else {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.908:
		tmp = (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + ((0.0003968253968253968 * math.pow(l, 7.0)) + ((0.016666666666666666 * math.pow(l, 5.0)) + (2.0 * l)))))) + U
	else:
		tmp = U + (J * (2.0 * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.908)
		tmp = Float64(Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(Float64(0.0003968253968253968 * (l ^ 7.0)) + Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(2.0 * l)))))) + U);
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	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 <= 0.908)
		tmp = (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + ((0.0003968253968253968 * (l ^ 7.0)) + ((0.016666666666666666 * (l ^ 5.0)) + (2.0 * l)))))) + U;
	else
		tmp = U + (J * (2.0 * sinh(l)));
	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, 0.908], N[(N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.0003968253968253968 * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.908000000000000029

   1. Initial program 88.8%

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

   2. Taylor expanded in l around 0 96.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 0.908000000000000029 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.2%

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

   2. Taylor expanded in K around 0 86.2%

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

      1. pow1 86.2%

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

      2. *-commutative 86.2%

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

      3. sinh-undef 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.908:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 3: 93.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.908:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.908)
     (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))
     (+ U (* J (* 2.0 (sinh l)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.908) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = U + (J * (2.0 * sinh(l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.908d0) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    else
        tmp = u + (j * (2.0d0 * sinh(l)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.908) {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.908:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	else:
		tmp = U + (J * (2.0 * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.908)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	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 <= 0.908)
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	else
		tmp = U + (J * (2.0 * sinh(l)));
	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, 0.908], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.908000000000000029

   1. Initial program 88.8%

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

   2. Taylor expanded in l around 0 96.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around 0 90.5%

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

      1. associate-*r* 90.5%

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

      2. associate-*r* 90.5%

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

      3. distribute-rgt-out 90.5%

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

      4. *-commutative 90.5%

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

      5. *-commutative 90.5%

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

      6. cube-mult 90.5%

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

      7. associate-*l* 90.5%

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

      8. distribute-lft-out 90.5%

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

   5. Simplified 90.5%

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

   if 0.908000000000000029 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.2%

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

   2. Taylor expanded in K around 0 86.2%

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

      1. pow1 86.2%

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

      2. *-commutative 86.2%

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

      3. sinh-undef 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. unpow1 100.0%

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

   6. Simplified 100.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.908:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 4: 87.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ U (* J (* (* 2.0 l) (cos (* K 0.5)))))
   (+ U (* J (* 2.0 (sinh l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = U + (J * ((2.0 * l) * cos((K * 0.5))));
	} else {
		tmp = U + (J * (2.0 * sinh(l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = u + (j * ((2.0d0 * l) * cos((k * 0.5d0))))
    else
        tmp = u + (j * (2.0d0 * sinh(l)))
    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)) <= -0.005) {
		tmp = U + (J * ((2.0 * l) * Math.cos((K * 0.5))));
	} else {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = U + (J * ((2.0 * l) * math.cos((K * 0.5))))
	else:
		tmp = U + (J * (2.0 * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(U + Float64(J * Float64(Float64(2.0 * l) * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = U + (J * ((2.0 * l) * cos((K * 0.5))));
	else
		tmp = U + (J * (2.0 * sinh(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(U + N[(J * N[(N[(2.0 * l), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.0050000000000000001

   1. Initial program 93.4%

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

   2. Taylor expanded in l around 0 98.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 98.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(2 \cdot \ell + 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} + U \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.0003968253968253968, {\ell}^{7}, \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \mathsf{fma}\left(\ell, 2, 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right)\right)} + U \]

   6. Taylor expanded in l around 0 56.4%

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

      1. *-commutative 56.4%

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

   8. Simplified 56.4%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.0%

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

   2. Taylor expanded in K around 0 85.0%

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

      1. pow1 85.0%

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

      2. *-commutative 85.0%

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

      3. sinh-undef 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. unpow1 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 5: 87.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.005:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.005)
     (+ U (* t_0 (* J (* 2.0 l))))
     (+ U (* J (* 2.0 (sinh l)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.005) {
		tmp = U + (t_0 * (J * (2.0 * l)));
	} else {
		tmp = U + (J * (2.0 * sinh(l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.005d0)) then
        tmp = u + (t_0 * (j * (2.0d0 * l)))
    else
        tmp = u + (j * (2.0d0 * sinh(l)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.005) {
		tmp = U + (t_0 * (J * (2.0 * l)));
	} else {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.005:
		tmp = U + (t_0 * (J * (2.0 * l)))
	else:
		tmp = U + (J * (2.0 * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.005)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(2.0 * l))));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	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 <= -0.005)
		tmp = U + (t_0 * (J * (2.0 * l)));
	else
		tmp = U + (J * (2.0 * sinh(l)));
	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, -0.005], N[(U + N[(t$95$0 * N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.0050000000000000001

   1. Initial program 93.4%

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

   2. Taylor expanded in l around 0 56.5%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.0%

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

   2. Taylor expanded in K around 0 85.0%

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

      1. pow1 85.0%

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

      2. *-commutative 85.0%

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

      3. sinh-undef 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. unpow1 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(2 \cdot \ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 6: 85.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ U (* J (* l (+ 2.0 (* (* K K) -0.25)))))
   (+ U (* J (* 2.0 (sinh l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	} else {
		tmp = U + (J * (2.0 * sinh(l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = u + (j * (l * (2.0d0 + ((k * k) * (-0.25d0)))))
    else
        tmp = u + (j * (2.0d0 * sinh(l)))
    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)) <= -0.005) {
		tmp = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	} else {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = U + (J * (l * (2.0 + ((K * K) * -0.25))))
	else:
		tmp = U + (J * (2.0 * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(K * K) * -0.25)))));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	else
		tmp = U + (J * (2.0 * sinh(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.0050000000000000001

   1. Initial program 93.4%

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

   2. Taylor expanded in l around 0 98.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 98.7%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(2 \cdot \ell + 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} + U \]

   5. Simplified 98.7%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.0003968253968253968, {\ell}^{7}, \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \mathsf{fma}\left(\ell, 2, 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right)\right)} + U \]

   6. Taylor expanded in l around 0 56.4%

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

      1. *-commutative 56.4%

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

   8. Simplified 56.4%

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

   9. Taylor expanded in K around 0 54.6%

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

       1. associate-*r* 54.6%

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

       2. distribute-rgt-out 54.6%

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

       3. *-commutative 54.6%

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

       4. unpow2 54.6%

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

   11. Simplified 54.6%

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

   if -0.0050000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.0%

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

   2. Taylor expanded in K around 0 85.0%

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

      1. pow1 85.0%

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

      2. *-commutative 85.0%

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

      3. sinh-undef 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. unpow1 96.6%

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

   6. Simplified 96.6%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 7: 59.6% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ t_1 := J \cdot \left(2 \cdot \ell\right)\\ \mathbf{if}\;\ell \leq -2 \cdot 10^{+187}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{t_1 \cdot t_1 - U \cdot U}{-U}\\ \mathbf{elif}\;\ell \leq 440:\\ \;\;\;\;U + t_1\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(J \cdot J\right)\right)}{t_1 - U}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* l (+ 2.0 (* (* K K) -0.25))))))
        (t_1 (* J (* 2.0 l))))
   (if (<= l -2e+187)
     t_0
     (if (<= l -9.5e+20)
       (/ (- (* t_1 t_1) (* U U)) (- U))
       (if (<= l 440.0)
         (+ U t_1)
         (if (<= l 1.2e+97) (/ (* 4.0 (* (* l l) (* J J))) (- t_1 U)) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	double t_1 = J * (2.0 * l);
	double tmp;
	if (l <= -2e+187) {
		tmp = t_0;
	} else if (l <= -9.5e+20) {
		tmp = ((t_1 * t_1) - (U * U)) / -U;
	} else if (l <= 440.0) {
		tmp = U + t_1;
	} else if (l <= 1.2e+97) {
		tmp = (4.0 * ((l * l) * (J * J))) / (t_1 - U);
	} 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 * (2.0d0 + ((k * k) * (-0.25d0)))))
    t_1 = j * (2.0d0 * l)
    if (l <= (-2d+187)) then
        tmp = t_0
    else if (l <= (-9.5d+20)) then
        tmp = ((t_1 * t_1) - (u * u)) / -u
    else if (l <= 440.0d0) then
        tmp = u + t_1
    else if (l <= 1.2d+97) then
        tmp = (4.0d0 * ((l * l) * (j * j))) / (t_1 - u)
    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 * (l * (2.0 + ((K * K) * -0.25))));
	double t_1 = J * (2.0 * l);
	double tmp;
	if (l <= -2e+187) {
		tmp = t_0;
	} else if (l <= -9.5e+20) {
		tmp = ((t_1 * t_1) - (U * U)) / -U;
	} else if (l <= 440.0) {
		tmp = U + t_1;
	} else if (l <= 1.2e+97) {
		tmp = (4.0 * ((l * l) * (J * J))) / (t_1 - U);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (l * (2.0 + ((K * K) * -0.25))))
	t_1 = J * (2.0 * l)
	tmp = 0
	if l <= -2e+187:
		tmp = t_0
	elif l <= -9.5e+20:
		tmp = ((t_1 * t_1) - (U * U)) / -U
	elif l <= 440.0:
		tmp = U + t_1
	elif l <= 1.2e+97:
		tmp = (4.0 * ((l * l) * (J * J))) / (t_1 - U)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(K * K) * -0.25)))))
	t_1 = Float64(J * Float64(2.0 * l))
	tmp = 0.0
	if (l <= -2e+187)
		tmp = t_0;
	elseif (l <= -9.5e+20)
		tmp = Float64(Float64(Float64(t_1 * t_1) - Float64(U * U)) / Float64(-U));
	elseif (l <= 440.0)
		tmp = Float64(U + t_1);
	elseif (l <= 1.2e+97)
		tmp = Float64(Float64(4.0 * Float64(Float64(l * l) * Float64(J * J))) / Float64(t_1 - U));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	t_1 = J * (2.0 * l);
	tmp = 0.0;
	if (l <= -2e+187)
		tmp = t_0;
	elseif (l <= -9.5e+20)
		tmp = ((t_1 * t_1) - (U * U)) / -U;
	elseif (l <= 440.0)
		tmp = U + t_1;
	elseif (l <= 1.2e+97)
		tmp = (4.0 * ((l * l) * (J * J))) / (t_1 - U);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(l * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2e+187], t$95$0, If[LessEqual[l, -9.5e+20], N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / (-U)), $MachinePrecision], If[LessEqual[l, 440.0], N[(U + t$95$1), $MachinePrecision], If[LessEqual[l, 1.2e+97], N[(N[(4.0 * N[(N[(l * l), $MachinePrecision] * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - U), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.99999999999999981e187 or 1.2e97 < 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%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(2 \cdot \ell + 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.0003968253968253968, {\ell}^{7}, \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \mathsf{fma}\left(\ell, 2, 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right)\right)} + U \]

   6. Taylor expanded in l around 0 33.9%

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

      1. *-commutative 33.9%

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

   8. Simplified 33.9%

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

   9. Taylor expanded in K around 0 55.9%

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

       1. associate-*r* 55.9%

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

       2. distribute-rgt-out 55.9%

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

       3. *-commutative 55.9%

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

       4. unpow2 55.9%

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

   11. Simplified 55.9%

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

   if -1.99999999999999981e187 < l < -9.5e20

   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 68.3%

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

   3. Taylor expanded in l around 0 8.3%

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

      1. flip-+ 28.2%

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

      2. *-commutative 28.2%

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

      3. *-commutative 28.2%

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

      4. *-commutative 28.2%

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

      5. *-commutative 28.2%

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

      6. *-commutative 28.2%

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

      7. *-commutative 28.2%

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

   5. Applied egg-rr 28.2%

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

   6. Taylor expanded in J around 0 37.3%

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

      1. neg-mul-1 37.3%

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

   8. Simplified 37.3%

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

   if -9.5e20 < l < 440

   1. Initial program 75.5%

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

   2. Taylor expanded in K around 0 75.5%

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

   3. Taylor expanded in l around 0 88.0%

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

   if 440 < l < 1.2e97

   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 85.7%

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

   3. Taylor expanded in l around 0 3.3%

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

      1. flip-+ 15.6%

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

      2. *-commutative 15.6%

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

      3. *-commutative 15.6%

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

      4. *-commutative 15.6%

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

      5. *-commutative 15.6%

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

      6. *-commutative 15.6%

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

      7. *-commutative 15.6%

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

   5. Applied egg-rr 15.6%

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

   6. Taylor expanded in J around inf 30.1%

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

      1. unpow2 30.1%

         \[\leadsto \frac{4 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot {J}^{2}\right)}{J \cdot \left(\ell \cdot 2\right) - U} \]

      2. unpow2 30.1%

         \[\leadsto \frac{4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(J \cdot J\right)}\right)}{J \cdot \left(\ell \cdot 2\right) - U} \]

   8. Simplified 30.1%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+187}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq -9.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \left(J \cdot \left(2 \cdot \ell\right)\right) - U \cdot U}{-U}\\ \mathbf{elif}\;\ell \leq 440:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(J \cdot J\right)\right)}{J \cdot \left(2 \cdot \ell\right) - U}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 8: 59.5% accurate, 13.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(2 \cdot \ell\right)\\ t_1 := U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+58}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 520:\\ \;\;\;\;U + t_0\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(J \cdot J\right)\right)}{t_0 - U}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 2.0 l)))
        (t_1 (+ U (* J (* l (+ 2.0 (* (* K K) -0.25)))))))
   (if (<= l -3.8e+58)
     t_1
     (if (<= l 520.0)
       (+ U t_0)
       (if (<= l 3.7e+96) (/ (* 4.0 (* (* l l) (* J J))) (- t_0 U)) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (2.0 * l);
	double t_1 = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	double tmp;
	if (l <= -3.8e+58) {
		tmp = t_1;
	} else if (l <= 520.0) {
		tmp = U + t_0;
	} else if (l <= 3.7e+96) {
		tmp = (4.0 * ((l * l) * (J * J))) / (t_0 - U);
	} 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 * (2.0d0 * l)
    t_1 = u + (j * (l * (2.0d0 + ((k * k) * (-0.25d0)))))
    if (l <= (-3.8d+58)) then
        tmp = t_1
    else if (l <= 520.0d0) then
        tmp = u + t_0
    else if (l <= 3.7d+96) then
        tmp = (4.0d0 * ((l * l) * (j * j))) / (t_0 - u)
    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 * (2.0 * l);
	double t_1 = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	double tmp;
	if (l <= -3.8e+58) {
		tmp = t_1;
	} else if (l <= 520.0) {
		tmp = U + t_0;
	} else if (l <= 3.7e+96) {
		tmp = (4.0 * ((l * l) * (J * J))) / (t_0 - U);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (2.0 * l)
	t_1 = U + (J * (l * (2.0 + ((K * K) * -0.25))))
	tmp = 0
	if l <= -3.8e+58:
		tmp = t_1
	elif l <= 520.0:
		tmp = U + t_0
	elif l <= 3.7e+96:
		tmp = (4.0 * ((l * l) * (J * J))) / (t_0 - U)
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(2.0 * l))
	t_1 = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(K * K) * -0.25)))))
	tmp = 0.0
	if (l <= -3.8e+58)
		tmp = t_1;
	elseif (l <= 520.0)
		tmp = Float64(U + t_0);
	elseif (l <= 3.7e+96)
		tmp = Float64(Float64(4.0 * Float64(Float64(l * l) * Float64(J * J))) / Float64(t_0 - U));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (2.0 * l);
	t_1 = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	tmp = 0.0;
	if (l <= -3.8e+58)
		tmp = t_1;
	elseif (l <= 520.0)
		tmp = U + t_0;
	elseif (l <= 3.7e+96)
		tmp = (4.0 * ((l * l) * (J * J))) / (t_0 - U);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(J * N[(l * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.8e+58], t$95$1, If[LessEqual[l, 520.0], N[(U + t$95$0), $MachinePrecision], If[LessEqual[l, 3.7e+96], N[(N[(4.0 * N[(N[(l * l), $MachinePrecision] * N[(J * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - U), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.7999999999999999e58 or 3.69999999999999991e96 < 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%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 100.0%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(2 \cdot \ell + 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.0003968253968253968, {\ell}^{7}, \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \mathsf{fma}\left(\ell, 2, 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right)\right)} + U \]

   6. Taylor expanded in l around 0 26.6%

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

      1. *-commutative 26.6%

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

   8. Simplified 26.6%

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

   9. Taylor expanded in K around 0 47.4%

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

       1. associate-*r* 47.4%

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

       2. distribute-rgt-out 47.4%

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

       3. *-commutative 47.4%

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

       4. unpow2 47.4%

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

   11. Simplified 47.4%

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

   if -3.7999999999999999e58 < l < 520

   1. Initial program 76.7%

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

   2. Taylor expanded in K around 0 76.7%

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

   3. Taylor expanded in l around 0 83.7%

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

   if 520 < l < 3.69999999999999991e96

   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 85.7%

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

   3. Taylor expanded in l around 0 3.3%

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

      1. flip-+ 15.6%

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

      2. *-commutative 15.6%

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

      3. *-commutative 15.6%

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

      4. *-commutative 15.6%

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

      5. *-commutative 15.6%

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

      6. *-commutative 15.6%

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

      7. *-commutative 15.6%

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

   5. Applied egg-rr 15.6%

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

   6. Taylor expanded in J around inf 30.1%

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

      1. unpow2 30.1%

         \[\leadsto \frac{4 \cdot \left(\color{blue}{\left(\ell \cdot \ell\right)} \cdot {J}^{2}\right)}{J \cdot \left(\ell \cdot 2\right) - U} \]

      2. unpow2 30.1%

         \[\leadsto \frac{4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \color{blue}{\left(J \cdot J\right)}\right)}{J \cdot \left(\ell \cdot 2\right) - U} \]

   8. Simplified 30.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+58}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq 520:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \mathbf{elif}\;\ell \leq 3.7 \cdot 10^{+96}:\\ \;\;\;\;\frac{4 \cdot \left(\left(\ell \cdot \ell\right) \cdot \left(J \cdot J\right)\right)}{J \cdot \left(2 \cdot \ell\right) - U}\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \end{array} \]

Alternative 9: 59.6% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+58} \lor \neg \left(\ell \leq 6.2 \cdot 10^{+34}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4.7e+58) (not (<= l 6.2e+34)))
   (+ U (* J (* l (+ 2.0 (* (* K K) -0.25)))))
   (+ U (* J (* 2.0 l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.7e+58) || !(l <= 6.2e+34)) {
		tmp = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	} else {
		tmp = U + (J * (2.0 * l));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4.7d+58)) .or. (.not. (l <= 6.2d+34))) then
        tmp = u + (j * (l * (2.0d0 + ((k * k) * (-0.25d0)))))
    else
        tmp = u + (j * (2.0d0 * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.7e+58) || !(l <= 6.2e+34)) {
		tmp = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	} else {
		tmp = U + (J * (2.0 * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4.7e+58) or not (l <= 6.2e+34):
		tmp = U + (J * (l * (2.0 + ((K * K) * -0.25))))
	else:
		tmp = U + (J * (2.0 * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4.7e+58) || !(l <= 6.2e+34))
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(K * K) * -0.25)))));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4.7e+58) || ~((l <= 6.2e+34)))
		tmp = U + (J * (l * (2.0 + ((K * K) * -0.25))));
	else
		tmp = U + (J * (2.0 * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4.7e+58], N[Not[LessEqual[l, 6.2e+34]], $MachinePrecision]], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.69999999999999972e58 or 6.19999999999999955e34 < 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 98.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in J around 0 98.2%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(2 \cdot \ell + 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right) \cdot \left(\cos \left(0.5 \cdot K\right) \cdot J\right)} + U \]

   5. Simplified 98.2%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.0003968253968253968, {\ell}^{7}, \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \mathsf{fma}\left(\ell, 2, 0.016666666666666666 \cdot {\ell}^{5}\right)\right)\right)\right)} + U \]

   6. Taylor expanded in l around 0 24.0%

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

      1. *-commutative 24.0%

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

   8. Simplified 24.0%

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

   9. Taylor expanded in K around 0 43.4%

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

       1. associate-*r* 43.4%

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

       2. distribute-rgt-out 43.4%

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

       3. *-commutative 43.4%

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

       4. unpow2 43.4%

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

   11. Simplified 43.4%

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

   if -4.69999999999999972e58 < l < 6.19999999999999955e34

   1. Initial program 78.1%

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

   2. Taylor expanded in K around 0 77.4%

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

   3. Taylor expanded in l around 0 78.9%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.7 \cdot 10^{+58} \lor \neg \left(\ell \leq 6.2 \cdot 10^{+34}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \ell\right)\\ \end{array} \]

Alternative 10: 46.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.049 \lor \neg \left(\ell \leq 420\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.049) (not (<= l 420.0))) (* l (* 2.0 J)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.049) || !(l <= 420.0)) {
		tmp = l * (2.0 * J);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.049d0)) .or. (.not. (l <= 420.0d0))) then
        tmp = l * (2.0d0 * j)
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.049) || !(l <= 420.0)) {
		tmp = l * (2.0 * J);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.049) or not (l <= 420.0):
		tmp = l * (2.0 * J)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.049) || !(l <= 420.0))
		tmp = Float64(l * Float64(2.0 * J));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.049) || ~((l <= 420.0)))
		tmp = l * (2.0 * J);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.049], N[Not[LessEqual[l, 420.0]], $MachinePrecision]], N[(l * N[(2.0 * J), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -0.049000000000000002 or 420 < 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 69.7%

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

   3. Taylor expanded in l around 0 15.5%

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

   4. Taylor expanded in l around inf 15.2%

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

      1. associate-*r* 15.2%

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

      2. *-commutative 15.2%

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

      3. associate-*l* 15.2%

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

   6. Simplified 15.2%

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

   if -0.049000000000000002 < l < 420

   1. Initial program 74.5%

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

   3. Applied egg-rr 51.4%

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

   4. Taylor expanded in J around 0 74.5%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.049 \lor \neg \left(\ell \leq 420\right):\\ \;\;\;\;\ell \cdot \left(2 \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 11: 53.8% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 K around 0 72.1%

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

3. Taylor expanded in l around 0 53.0%

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

4. Final simplification 53.0%

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

Alternative 12: 36.3% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.3%

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

3. Applied egg-rr 27.3%

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

4. Taylor expanded in J around 0 38.3%

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

5. Final simplification 38.3%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023214 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))