Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 97.3%

Time: 9.6s

Alternatives: 13

Speedup: 2.4×

• Could not converge on a ground truth

  1. reg0 = (bf "-1.124959914631704985929278979783413744735e-224")
  2. reg1 = (bf "-2.320202130834822468796537176197458715599e290")
  3. reg2 = (bf #e16.50432421732004328873699705582112073898)
  4. reg3 = (bf "6.873628287164554505963600767664110019144e218")

• 50.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.4% 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: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot t\_1 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (- (exp l) (exp (- l))))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 5e+182)
     (+
      (*
       (*
        J
        (*
         l
         (+
          2.0
          (*
           (* l l)
           (+
            0.3333333333333333
            (*
             (* l l)
             (+ 0.016666666666666666 (* (* l l) 0.0003968253968253968))))))))
       t_1)
      U)
     (+ U (* t_0 t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (exp(l) - exp(-l));
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 5e+182) {
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))) * t_1) + U;
	} else {
		tmp = U + (t_0 * 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 * (exp(l) - exp(-l))
    t_1 = cos((k / 2.0d0))
    if (t_0 <= 5d+182) then
        tmp = ((j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))) * t_1) + u
    else
        tmp = u + (t_0 * t_1)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * (Math.exp(l) - Math.exp(-l));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 5e+182) {
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))) * t_1) + U;
	} else {
		tmp = U + (t_0 * t_1);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (math.exp(l) - math.exp(-l))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 5e+182:
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))) * t_1) + U
	else:
		tmp = U + (t_0 * t_1)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(exp(l) - exp(Float64(-l))))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 5e+182)
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968)))))))) * t_1) + U);
	else
		tmp = Float64(U + Float64(t_0 * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (exp(l) - exp(-l));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 5e+182)
		tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))) * t_1) + U;
	else
		tmp = U + (t_0 * t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 5e+182], N[(N[(N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 4.99999999999999973e182

   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 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.0%

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

   5. Simplified 99.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   6. Step-by-step derivation

      1. unpow2 99.0%

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

   7. Applied egg-rr 99.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   8. Step-by-step derivation

      1. unpow2 99.0%

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

   9. Applied egg-rr 99.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   10. Step-by-step derivation

       1. unpow2 99.0%

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

   11. Applied egg-rr 99.0%

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

   if 4.99999999999999973e182 < (*.f64 J (-.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
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \cdot \left(e^{\ell} - e^{-\ell}\right) \leq 5 \cdot 10^{+182}:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  (*
   (*
    J
    (*
     l
     (+
      2.0
      (*
       (* l l)
       (+
        0.3333333333333333
        (*
         (* l l)
         (+ 0.016666666666666666 (* (* l l) 0.0003968253968253968))))))))
   (cos (/ K 2.0)))
  U))

C

double code(double J, double l, double K, double U) {
	return ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))) * 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 * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * (0.016666666666666666d0 + ((l * l) * 0.0003968253968253968d0)))))))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * Float64(0.016666666666666666 + Float64(Float64(l * l) * 0.0003968253968253968)))))))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (l * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * (0.016666666666666666 + ((l * l) * 0.0003968253968253968)))))))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * N[(0.016666666666666666 + N[(N[(l * l), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in l around 0 97.3%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 97.3%

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

5. Simplified 97.3%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
6. Step-by-step derivation

   1. unpow2 97.3%

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

7. Applied egg-rr 97.3%

   \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
8. Step-by-step derivation

   1. unpow2 97.3%

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

9. Applied egg-rr 97.3%

   \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
10. Step-by-step derivation

    1. unpow2 97.3%

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

11. Applied egg-rr 97.3%

    \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot \left(0.016666666666666666 + \left(\ell \cdot \ell\right) \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
12. Add Preprocessing

Alternative 3: 64.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.12 \cdot 10^{-16}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 2.12e-16)
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.12e-16) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / 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 <= 2.12d-16) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.12e-16) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 2.12e-16:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 2.12e-16)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 2.12e-16)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 2.12e-16], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1199999999999999e-16

   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 79.0%

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

      1. *-commutative 79.0%

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

      2. associate-*l* 79.0%

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

   5. Simplified 79.0%

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

   if 2.1199999999999999e-16 < 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 30.0%

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

      1. *-commutative 30.0%

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

      2. *-commutative 30.0%

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

      3. associate-*l* 30.0%

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

      4. *-commutative 30.0%

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

      5. associate-*r* 30.0%

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

      6. *-commutative 30.0%

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

      7. associate-*l* 30.0%

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

   5. Simplified 30.0%

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

   6. Taylor expanded in U around inf 40.2%

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

      1. associate-/l* 45.5%

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

   8. Simplified 45.5%

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

   9. Taylor expanded in K around 0 35.9%

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

       1. associate-/l* 39.4%

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

   11. Simplified 39.4%

       \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \left(J \cdot \frac{\ell}{U}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.12 \cdot 10^{-16}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.12 \cdot 10^{-16}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 2.12e-16)
   (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.12e-16) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / 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 <= 2.12d-16) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.12e-16) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 2.12e-16:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 2.12e-16)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 2.12e-16)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 2.12e-16], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.1199999999999999e-16

   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 79.0%

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

      1. *-commutative 79.0%

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

      2. *-commutative 79.0%

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

      3. associate-*l* 79.0%

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

      4. *-commutative 79.0%

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

      5. associate-*r* 79.0%

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

      6. *-commutative 79.0%

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

      7. associate-*l* 79.0%

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

   5. Simplified 79.0%

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

   if 2.1199999999999999e-16 < 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 30.0%

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

      1. *-commutative 30.0%

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

      2. *-commutative 30.0%

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

      3. associate-*l* 30.0%

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

      4. *-commutative 30.0%

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

      5. associate-*r* 30.0%

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

      6. *-commutative 30.0%

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

      7. associate-*l* 30.0%

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

   5. Simplified 30.0%

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

   6. Taylor expanded in U around inf 40.2%

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

      1. associate-/l* 45.5%

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

   8. Simplified 45.5%

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

   9. Taylor expanded in K around 0 35.9%

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

       1. associate-/l* 39.4%

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

   11. Simplified 39.4%

       \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \left(J \cdot \frac{\ell}{U}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.12 \cdot 10^{-16}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + (2.0 * (J * (cos((K * 0.5)) * (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 * (cos((k * 0.5d0)) * (l / u)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.1%

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

3. Taylor expanded in l around 0 68.5%

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

   1. *-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. associate-*r* 68.4%

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

   6. *-commutative 68.4%

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

   7. associate-*l* 68.4%

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

5. Simplified 68.4%

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

6. Taylor expanded in U around inf 71.0%

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

   1. associate-/l* 73.6%

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

8. Simplified 73.6%

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

   1. *-commutative 73.6%

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

   2. *-un-lft-identity 73.6%

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

   3. times-frac 73.6%

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

10. Applied egg-rr 73.6%

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

11. Final simplification 73.6%

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

Alternative 6: 71.2% 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 87.1%

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

3. Taylor expanded in l around 0 68.5%

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

   1. *-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. associate-*r* 68.4%

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

   6. *-commutative 68.4%

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

   7. associate-*l* 68.4%

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

5. Simplified 68.4%

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

6. Taylor expanded in U around inf 71.0%

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

   1. associate-/l* 73.6%

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

8. Simplified 73.6%

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

9. Final simplification 73.6%

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

Alternative 7: 60.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -4.8 \cdot 10^{+160}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= J -4.8e+160)
   (* 2.0 (* J (* l (cos (* K 0.5)))))
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -4.8e+160) {
		tmp = 2.0 * (J * (l * cos((K * 0.5))));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (j <= (-4.8d+160)) then
        tmp = 2.0d0 * (j * (l * cos((k * 0.5d0))))
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (J <= -4.8e+160) {
		tmp = 2.0 * (J * (l * Math.cos((K * 0.5))));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if J <= -4.8e+160:
		tmp = 2.0 * (J * (l * math.cos((K * 0.5))))
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (J <= -4.8e+160)
		tmp = Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (J <= -4.8e+160)
		tmp = 2.0 * (J * (l * cos((K * 0.5))));
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[J, -4.8e+160], N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -4.8000000000000003e160

   1. Initial program 67.8%

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

   3. Taylor expanded in l around 0 93.7%

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

      1. *-commutative 93.7%

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

      2. *-commutative 93.7%

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

      3. associate-*l* 93.6%

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

      4. *-commutative 93.6%

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

      5. associate-*r* 93.6%

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

      6. *-commutative 93.6%

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

      7. associate-*l* 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in U around inf 84.2%

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

      1. associate-/l* 84.1%

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

   8. Simplified 84.1%

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

   9. Taylor expanded in U around 0 85.6%

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

   if -4.8000000000000003e160 < J

   1. Initial program 89.7%

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

   3. Taylor expanded in l around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. *-commutative 65.1%

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

      3. associate-*l* 65.1%

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

      4. *-commutative 65.1%

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

      5. associate-*r* 65.1%

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

      6. *-commutative 65.1%

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

      7. associate-*l* 65.1%

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

   5. Simplified 65.1%

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

   6. Taylor expanded in U around inf 69.2%

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

      1. associate-/l* 72.2%

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

   8. Simplified 72.2%

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

   9. Taylor expanded in K around 0 59.9%

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

       1. associate-/l* 61.1%

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

   11. Simplified 61.1%

       \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \left(J \cdot \frac{\ell}{U}\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -4.8 \cdot 10^{+160}:\\ \;\;\;\;2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 41.9% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -760 \lor \neg \left(\ell \leq 1000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -760.0) (not (<= l 1000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -760.0) || !(l <= 1000.0)) {
		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 <= (-760.0d0)) .or. (.not. (l <= 1000.0d0))) 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 <= -760.0) || !(l <= 1000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -760.0) or not (l <= 1000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -760.0) || !(l <= 1000.0))
		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 <= -760.0) || ~((l <= 1000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -760.0], N[Not[LessEqual[l, 1000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -760 or 1e3 < 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

   4. Applied egg-rr 17.8%

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

   if -760 < l < 1e3

   1. Initial program 76.3%

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

   3. Taylor expanded in J around 0 75.4%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -760 \lor \neg \left(\ell \leq 1000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.9% 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 87.1%

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

3. Taylor expanded in l around 0 68.5%

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

   1. *-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. associate-*r* 68.4%

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

   6. *-commutative 68.4%

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

   7. associate-*l* 68.4%

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

5. Simplified 68.4%

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

6. Taylor expanded in U around inf 71.0%

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

   1. associate-/l* 73.6%

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

8. Simplified 73.6%

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

9. Taylor expanded in K around 0 61.1%

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

    1. associate-/l* 62.2%

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

11. Simplified 62.2%

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

Alternative 10: 53.4% 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 87.1%

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

3. Taylor expanded in l around 0 68.5%

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

   1. *-commutative 68.5%

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

   2. *-commutative 68.5%

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

   3. associate-*l* 68.4%

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

   4. *-commutative 68.4%

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

   5. associate-*r* 68.4%

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

   6. *-commutative 68.4%

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

   7. associate-*l* 68.4%

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

5. Simplified 68.4%

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

6. Taylor expanded in K around 0 58.9%

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

   1. associate-*r* 58.9%

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

   2. *-commutative 58.9%

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

8. Simplified 58.9%

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

9. Final simplification 58.9%

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

Alternative 11: 36.1% 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 87.1%

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

3. Taylor expanded in J around 0 41.8%

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

Alternative 12: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 0.25)

C

double code(double J, double l, double K, double U) {
	return 0.25;
}

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 = 0.25d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 0.25;
}

Python

def code(J, l, K, U):
	return 0.25

Julia

function code(J, l, K, U)
	return 0.25
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 0.25;
end

Wolfram

code[J_, l_, K_, U_] := 0.25

Derivation

1. Initial program 87.1%

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

4. Applied egg-rr 26.2%

   \[\leadsto \color{blue}{0.25} + U \]

5. Taylor expanded in U around 0 2.7%

   \[\leadsto \color{blue}{0.25} \]
6. Add Preprocessing

Alternative 13: 2.7% 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 87.1%

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

4. Applied egg-rr 2.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-3, U, -4\right)} \]
5. Step-by-step derivation

   1. fma-undefine 2.5%

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

   2. *-commutative 2.5%

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

   3. fma-define 2.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(U, -3, -4\right)} \]

6. Simplified 2.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(U, -3, -4\right)} \]

7. Taylor expanded in U around 0 2.7%

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

Reproduce

herbie shell --seed 2024151 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))