Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.7%

Time: 11.2s

Alternatives: 17

Speedup: 2.5×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -0.2 \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -0.2) (not (<= t_0 0.0)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       l
       (* (cos (* K 0.5)) (* J (fma 0.3333333333333333 (pow l 2.0) 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -0.2) || !(t_0 <= 0.0)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * fma(0.3333333333333333, pow(l, 2.0), 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -0.2) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * fma(0.3333333333333333, (l ^ 2.0), 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.2], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision] + 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))) < -0.20000000000000001 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.9%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 69.8%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*r* 99.9%

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

      4. distribute-rgt-out 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*r* 99.9%

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

      7. distribute-rgt-out 99.9%

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

      8. fma-define 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.2 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -0.2 \lor \neg \left(t\_1 \leq 0.02\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -0.2) (not (<= t_1 0.02)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (*
        J
        (*
         l
         (+
          2.0
          (*
           (* l l)
           (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))))))

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 <= -0.2) || !(t_1 <= 0.02)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	}
	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 <= (-0.2d0)) .or. (.not. (t_1 <= 0.02d0))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))))
    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 <= -0.2) || !(t_1 <= 0.02)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	}
	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 <= -0.2) or not (t_1 <= 0.02):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))))
	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 <= -0.2) || !(t_1 <= 0.02))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))));
	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 <= -0.2) || ~((t_1 <= 0.02)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	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, -0.2], N[Not[LessEqual[t$95$1, 0.02]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.20000000000000001 or 0.0200000000000000004 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -0.20000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0200000000000000004

   1. Initial program 70.2%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.2 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.02\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -4.6:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 5200 \lor \neg \left(\ell \leq 7.2 \cdot 10^{+61}\right):\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - t\_0\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -4.6)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (or (<= l 5200.0) (not (<= l 7.2e+61)))
       (+
        U
        (*
         t_1
         (*
          J
          (*
           l
           (+
            2.0
            (*
             (* l l)
             (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))
       (+ (* (- (exp l) t_0) J) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -4.6) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if ((l <= 5200.0) || !(l <= 7.2e+61)) {
		tmp = U + (t_1 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	} else {
		tmp = ((exp(l) - t_0) * 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-4.6d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if ((l <= 5200.0d0) .or. (.not. (l <= 7.2d+61))) then
        tmp = u + (t_1 * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))))
    else
        tmp = ((exp(l) - t_0) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -4.6) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if ((l <= 5200.0) || !(l <= 7.2e+61)) {
		tmp = U + (t_1 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	} else {
		tmp = ((Math.exp(l) - t_0) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -4.6:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif (l <= 5200.0) or not (l <= 7.2e+61):
		tmp = U + (t_1 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))))
	else:
		tmp = ((math.exp(l) - t_0) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -4.6)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif ((l <= 5200.0) || !(l <= 7.2e+61))
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))));
	else
		tmp = Float64(Float64(Float64(exp(l) - t_0) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -4.6)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif ((l <= 5200.0) || ~((l <= 7.2e+61)))
		tmp = U + (t_1 * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	else
		tmp = ((exp(l) - t_0) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.6], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 5200.0], N[Not[LessEqual[l, 7.2e+61]], $MachinePrecision]], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.5999999999999996

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -4.5999999999999996 < l < 5200 or 7.20000000000000021e61 < l

   1. Initial program 78.9%

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

   3. Taylor expanded in l around 0 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 99.0%

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

   7. Applied egg-rr 99.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 99.0%

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

   9. Applied egg-rr 99.0%

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

   if 5200 < l < 7.20000000000000021e61

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 5200 \lor \neg \left(\ell \leq 7.2 \cdot 10^{+61}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5200 \lor \neg \left(\ell \leq 1.02 \cdot 10^{+62}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l 5200.0) (not (<= l 1.02e+62)))
   (+
    U
    (*
     (cos (/ K 2.0))
     (*
      J
      (*
       l
       (+
        2.0
        (*
         (* l l)
         (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))))))))
   (+ (* (- (exp l) (exp (- l))) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 5200.0) || !(l <= 1.02e+62)) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	} 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 ((l <= 5200.0d0) .or. (.not. (l <= 1.02d+62))) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))))
    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 ((l <= 5200.0) || !(l <= 1.02e+62)) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	} else {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= 5200.0) or not (l <= 1.02e+62):
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))))
	else:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= 5200.0) || !(l <= 1.02e+62))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))));
	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 ((l <= 5200.0) || ~((l <= 1.02e+62)))
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
	else
		tmp = ((exp(l) - exp(-l)) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, 5200.0], N[Not[LessEqual[l, 1.02e+62]], $MachinePrecision]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $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 l < 5200 or 1.02000000000000002e62 < l

   1. Initial program 83.6%

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

   3. Taylor expanded in l around 0 95.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 95.4%

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

   5. Simplified 95.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 95.4%

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

   7. Applied egg-rr 95.4%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 95.4%

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

   9. Applied egg-rr 95.4%

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

   if 5200 < l < 1.02000000000000002e62

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 100.0%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5200 \lor \neg \left(\ell \leq 1.02 \cdot 10^{+62}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   (cos (/ K 2.0))
   (*
    J
    (*
     l
     (+
      2.0
      (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
}

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 + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666)))))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in l around 0 91.5%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 91.5%

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

5. Simplified 91.5%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
6. Step-by-step derivation

   1. unpow2 91.5%

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

7. Applied egg-rr 91.5%

   \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.016666666666666666\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
8. Step-by-step derivation

   1. unpow2 91.5%

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

9. Applied egg-rr 91.5%

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

10. Final simplification 91.5%

    \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) \]
11. Add Preprocessing

Alternative 6: 64.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 100000000:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 100000000.0)
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))
   (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 100000000.0) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 100000000.0d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    else
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 100000000.0) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 100000000.0:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 100000000.0)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 100000000.0)
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 100000000.0], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K #s(literal 2 binary64)) < 1e8

   1. Initial program 86.2%

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

   3. Taylor expanded in l around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in U around inf 62.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 65.6%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 55.9%

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

       1. associate-/l* 59.4%

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

   11. Simplified 59.4%

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

   if 1e8 < (/.f64 K #s(literal 2 binary64))

   1. Initial program 79.0%

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

   3. Taylor expanded in l around 0 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*l* 75.6%

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

   5. Simplified 75.6%

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

   6. Taylor expanded in J around 0 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*r* 75.7%

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

      3. associate-*l* 75.7%

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

      4. *-commutative 75.7%

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

      5. *-commutative 75.7%

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

      6. *-commutative 75.7%

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

      7. associate-*l* 75.7%

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

      8. *-commutative 75.7%

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

   8. Simplified 75.7%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 100000000:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1450000000:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1450000000.0)
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))
   (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1450000000.0) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (J * (cos((K * 0.5)) * (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 (k <= 1450000000.0d0) then
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    else
        tmp = u + (j * (cos((k * 0.5d0)) * (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 (K <= 1450000000.0) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 1450000000.0:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	else:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 1450000000.0)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 1450000000.0)
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	else
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 1450000000.0], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.45e9

   1. Initial program 86.2%

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

   3. Taylor expanded in l around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in U around inf 62.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 65.6%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 55.9%

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

       1. associate-/l* 59.4%

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

   11. Simplified 59.4%

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

   if 1.45e9 < K

   1. Initial program 79.0%

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

   3. Taylor expanded in l around 0 75.6%

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

      1. *-commutative 75.6%

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

      2. associate-*l* 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-*l* 75.6%

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

   5. Simplified 75.6%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1450000000:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U))))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
}

Python

def code(J, l, K, U):
	return U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(2.0 * N[(J * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in l around 0 63.3%

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

   1. *-commutative 63.3%

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

   2. associate-*l* 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in U around inf 65.8%

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
7. Step-by-step derivation

   1. associate-/l* 68.9%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

8. Simplified 68.9%

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

9. Final simplification 68.9%

   \[\leadsto U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right) \]
10. Add Preprocessing

Alternative 9: 46.6% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot 2\right)\\ \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+237}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -780:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 44000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J 2.0))))
   (if (<= l -3.05e+237)
     t_0
     (if (<= l -780.0) (* U (- U -4.0)) (if (<= l 44000000.0) U t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double tmp;
	if (l <= -3.05e+237) {
		tmp = t_0;
	} else if (l <= -780.0) {
		tmp = U * (U - -4.0);
	} else if (l <= 44000000.0) {
		tmp = 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) :: tmp
    t_0 = l * (j * 2.0d0)
    if (l <= (-3.05d+237)) then
        tmp = t_0
    else if (l <= (-780.0d0)) then
        tmp = u * (u - (-4.0d0))
    else if (l <= 44000000.0d0) then
        tmp = 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 = l * (J * 2.0);
	double tmp;
	if (l <= -3.05e+237) {
		tmp = t_0;
	} else if (l <= -780.0) {
		tmp = U * (U - -4.0);
	} else if (l <= 44000000.0) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * 2.0)
	tmp = 0
	if l <= -3.05e+237:
		tmp = t_0
	elif l <= -780.0:
		tmp = U * (U - -4.0)
	elif l <= 44000000.0:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * 2.0))
	tmp = 0.0
	if (l <= -3.05e+237)
		tmp = t_0;
	elseif (l <= -780.0)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 44000000.0)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (J * 2.0);
	tmp = 0.0;
	if (l <= -3.05e+237)
		tmp = t_0;
	elseif (l <= -780.0)
		tmp = U * (U - -4.0);
	elseif (l <= 44000000.0)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.05e+237], t$95$0, If[LessEqual[l, -780.0], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 44000000.0], U, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.0500000000000001e237 or 4.4e7 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 32.9%

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

      1. *-commutative 32.9%

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

      2. associate-*l* 32.9%

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

   5. Simplified 32.9%

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

   6. Taylor expanded in U around inf 41.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 50.1%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 50.1%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 28.2%

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

   10. Taylor expanded in U around 0 23.4%

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

       1. associate-*r* 23.4%

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

       2. *-commutative 23.4%

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

   12. Simplified 23.4%

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

   if -3.0500000000000001e237 < l < -780

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 20.5%

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

   if -780 < l < 4.4e7

   1. Initial program 71.2%

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

      1. associate-*l* 71.2%

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

      2. fma-define 71.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   3. Simplified 71.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 67.0%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+237}:\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -780:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 44000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.1% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1350 \lor \neg \left(\ell \leq 18000\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1350.0) (not (<= l 18000.0))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1350.0) || !(l <= 18000.0)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1350.0d0)) .or. (.not. (l <= 18000.0d0))) then
        tmp = u * (u - (-4.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1350.0) || !(l <= 18000.0)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1350.0) or not (l <= 18000.0):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1350.0) || !(l <= 18000.0))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1350.0) || ~((l <= 18000.0)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1350.0], N[Not[LessEqual[l, 18000.0]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1350 or 18000 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 14.1%

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

   if -1350 < l < 18000

   1. Initial program 71.0%

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

      1. associate-*l* 71.0%

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

      2. fma-define 71.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 67.4%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1350 \lor \neg \left(\ell \leq 18000\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 56.0% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 90000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 90000.0) (+ U (* l (* J 2.0))) (* J (+ (* l 2.0) (/ U J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 90000.0) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * ((l * 2.0) + (U / J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 90000.0d0) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = j * ((l * 2.0d0) + (u / j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 90000.0) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = J * ((l * 2.0) + (U / J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 90000.0:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = J * ((l * 2.0) + (U / J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 90000.0)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(U / J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 90000.0)
		tmp = U + (l * (J * 2.0));
	else
		tmp = J * ((l * 2.0) + (U / J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 90000.0], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 9e4

   1. Initial program 79.2%

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

   3. Taylor expanded in l around 0 74.9%

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

      1. *-commutative 74.9%

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

      2. associate-*l* 74.9%

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

   5. Simplified 74.9%

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

   6. Taylor expanded in K around 0 62.8%

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

      1. associate-*r* 62.8%

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

      2. *-commutative 62.8%

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

   8. Simplified 62.8%

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

   if 9e4 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 29.0%

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

      1. *-commutative 29.0%

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

      2. associate-*l* 29.0%

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

   5. Simplified 29.0%

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

   6. Taylor expanded in U around inf 34.8%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-/l* 42.3%

         \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

   8. Simplified 42.3%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

   9. Taylor expanded in K around 0 23.9%

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

   10. Taylor expanded in J around inf 26.8%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 90000:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.1% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1000 \lor \neg \left(\ell \leq 11800\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1000.0) (not (<= l 11800.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1000.0) || !(l <= 11800.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1000.0d0)) .or. (.not. (l <= 11800.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1000.0) || !(l <= 11800.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1000.0) or not (l <= 11800.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1000.0) || !(l <= 11800.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1000.0) || ~((l <= 11800.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1000.0], N[Not[LessEqual[l, 11800.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1e3 or 11800 < 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. Step-by-step derivation

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 14.0%

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

   if -1e3 < l < 11800

   1. Initial program 71.0%

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

      1. associate-*l* 71.0%

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

      2. fma-define 71.0%

         \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in J around 0 67.4%

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

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1000 \lor \neg \left(\ell \leq 11800\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 28.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u * (1.0d0 + (2.0d0 * (j * (l / u))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in l around 0 63.3%

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

   1. *-commutative 63.3%

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

   2. associate-*l* 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in U around inf 65.8%

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
7. Step-by-step derivation

   1. associate-/l* 68.9%

      \[\leadsto U \cdot \left(1 + 2 \cdot \color{blue}{\left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)}\right) \]

8. Simplified 68.9%

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)\right)} \]

9. Taylor expanded in K around 0 54.9%

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

    1. associate-/l* 57.5%

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

11. Simplified 57.5%

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

Alternative 14: 54.7% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Taylor expanded in l around 0 63.3%

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

   1. *-commutative 63.3%

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

   2. associate-*l* 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in K around 0 51.8%

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

   1. associate-*r* 51.8%

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

   2. *-commutative 51.8%

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

8. Simplified 51.8%

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

9. Final simplification 51.8%

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

Alternative 15: 37.3% 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 84.5%

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

   1. associate-*l* 84.5%

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

   2. fma-define 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

3. Simplified 84.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
4. Add Preprocessing

5. Taylor expanded in J around 0 37.1%

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

Alternative 16: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -0.3333333333333333)

C

double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

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 = -0.3333333333333333d0
end function

Java

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

Python

def code(J, l, K, U):
	return -0.3333333333333333

Julia

function code(J, l, K, U)
	return -0.3333333333333333
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -0.3333333333333333;
end

Wolfram

code[J_, l_, K_, U_] := -0.3333333333333333

Derivation

1. Initial program 84.5%

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

   1. associate-*l* 84.5%

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

   2. fma-define 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

3. Simplified 84.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
4. Add Preprocessing

6. Applied egg-rr 2.4%

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

   1. associate-+r+ 2.4%

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

   2. distribute-rgt1-in 2.4%

      \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

   3. metadata-eval 2.4%

      \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

   4. *-commutative 2.4%

      \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

   5. distribute-lft-out 2.4%

      \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

   6. associate-/r* 2.3%

      \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

   7. *-inverses 2.3%

      \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

   8. *-commutative 2.3%

      \[\leadsto \frac{1}{-3 + \color{blue}{U \cdot -4}} \]

8. Simplified 2.3%

   \[\leadsto \color{blue}{\frac{1}{-3 + U \cdot -4}} \]

9. Taylor expanded in U around 0 2.7%

   \[\leadsto \color{blue}{-0.3333333333333333} \]
10. Add Preprocessing

Alternative 17: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -4.0)

C

double code(double J, double l, double K, double U) {
	return -4.0;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = -4.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -4.0;
}

Python

def code(J, l, K, U):
	return -4.0

Julia

function code(J, l, K, U)
	return -4.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -4.0;
end

Wolfram

code[J_, l_, K_, U_] := -4.0

Derivation

1. Initial program 84.5%

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

   1. associate-*l* 84.5%

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

   2. fma-define 84.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]

3. Simplified 84.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
4. Add Preprocessing

6. Applied egg-rr 2.4%

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

   1. sub-neg 2.4%

      \[\leadsto \color{blue}{-4 - U} \]

8. Simplified 2.4%

   \[\leadsto \color{blue}{-4 - U} \]

9. Taylor expanded in U around 0 2.7%

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

Reproduce

herbie shell --seed 2024149 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))