Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.7% → 96.0%

Time: 10.0s

Alternatives: 14

Speedup: 2.7×

• 48.7% 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: 86.7% 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: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;t\_0 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0)))))
   (if (<= t_0 0.0)
     (+
      (*
       J
       (*
        l
        (*
         (cos (* K 0.5))
         (+
          2.0
          (*
           (* l l)
           (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
      U)
     (+ t_0 U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (J * (l * (cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U;
	} else {
		tmp = t_0 + 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) :: tmp
    t_0 = (j * (exp(l) - exp(-l))) * cos((k / 2.0d0))
    if (t_0 <= 0.0d0) then
        tmp = (j * (l * (cos((k * 0.5d0)) * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))) + u
    else
        tmp = t_0 + u
    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))) * Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = (J * (l * (Math.cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U;
	} else {
		tmp = t_0 + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.0:
		tmp = (J * (l * (math.cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U
	else:
		tmp = t_0 + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0)))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))) + U);
	else
		tmp = Float64(t_0 + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (J * (exp(l) - exp(-l))) * cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (J * (l * (cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U;
	else
		tmp = t_0 + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(N[(J * N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * 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] + U), $MachinePrecision], N[(t$95$0 + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64)))) < -0.0

   1. Initial program 85.2%

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

   3. Taylor expanded in l around 0 98.3%

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

      1. *-commutative 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in J around 0 98.3%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. unpow2 98.3%

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

   8. Applied egg-rr 98.3%

      \[\leadsto J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right)\right) + U \]
   9. Step-by-step derivation

      1. unpow2 98.3%

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

   10. Applied egg-rr 98.3%

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

   if -0.0 < (*.f64 (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) (cos.f64 (/.f64 K #s(literal 2 binary64))))

   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 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) \leq 0:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ \mathbf{if}\;e^{\ell} - t\_0 \leq -\infty:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))))
   (if (<= (- (exp l) t_0) (- INFINITY))
     (+ U (* (cos (/ K 2.0)) (* J (- 27.0 t_0))))
     (+
      (*
       J
       (*
        l
        (*
         (cos (* K 0.5))
         (+
          2.0
          (*
           (* l l)
           (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
      U))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double tmp;
	if ((exp(l) - t_0) <= -((double) INFINITY)) {
		tmp = U + (cos((K / 2.0)) * (J * (27.0 - t_0)));
	} else {
		tmp = (J * (l * (cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double tmp;
	if ((Math.exp(l) - t_0) <= -Double.POSITIVE_INFINITY) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (27.0 - t_0)));
	} else {
		tmp = (J * (l * (Math.cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	tmp = 0
	if (math.exp(l) - t_0) <= -math.inf:
		tmp = U + (math.cos((K / 2.0)) * (J * (27.0 - t_0)))
	else:
		tmp = (J * (l * (math.cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	tmp = 0.0
	if (Float64(exp(l) - t_0) <= Float64(-Inf))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(27.0 - t_0))));
	else
		tmp = Float64(Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	tmp = 0.0;
	if ((exp(l) - t_0) <= -Inf)
		tmp = U + (cos((K / 2.0)) * (J * (27.0 - t_0)));
	else
		tmp = (J * (l * (cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision], (-Infinity)], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * 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] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0

   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 -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 85.8%

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

   3. Taylor expanded in l around 0 96.4%

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

      1. *-commutative 96.4%

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

   5. Simplified 96.4%

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

   6. Taylor expanded in J around 0 96.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. unpow2 96.5%

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

   8. Applied egg-rr 96.5%

      \[\leadsto J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right)\right) + U \]
   9. Step-by-step derivation

      1. unpow2 96.5%

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

   10. Applied egg-rr 96.5%

       \[\leadsto J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + \color{blue}{\left(\ell \cdot \ell\right)} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) + U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.9976:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right) \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.9976)
     (+ U (* t_0 (* J (* l (+ 2.0 (* (* l l) 0.3333333333333333))))))
     (+
      U
      (*
       J
       (*
        l
        (+
         2.0
         (*
          (+ 0.3333333333333333 (* (* l l) 0.016666666666666666))
          (pow l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.9976) {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else {
		tmp = U + (J * (l * (2.0 + ((0.3333333333333333 + ((l * l) * 0.016666666666666666)) * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.9976d0) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + ((l * l) * 0.3333333333333333d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + ((0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)) * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.9976) {
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	} else {
		tmp = U + (J * (l * (2.0 + ((0.3333333333333333 + ((l * l) * 0.016666666666666666)) * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.9976:
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))))
	else:
		tmp = U + (J * (l * (2.0 + ((0.3333333333333333 + ((l * l) * 0.016666666666666666)) * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.9976)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(Float64(l * l) * 0.3333333333333333))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666)) * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.9976)
		tmp = U + (t_0 * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
	else
		tmp = U + (J * (l * (2.0 + ((0.3333333333333333 + ((l * l) * 0.016666666666666666)) * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.9976], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(N[(0.3333333333333333 + N[(N[(l * l), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.99760000000000004

   1. Initial program 90.1%

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

   3. Taylor expanded in l around 0 87.8%

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

      1. unpow2 93.5%

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

   5. Applied egg-rr 87.8%

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

   if 0.99760000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 89.3%

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

   3. Taylor expanded in l around 0 96.5%

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

      1. *-commutative 96.5%

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

   5. Simplified 96.5%

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

   6. Taylor expanded in J around 0 96.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]
   7. Step-by-step derivation

      1. unpow2 96.5%

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

   8. Applied egg-rr 96.5%

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

   9. Taylor expanded in K around 0 96.5%

      \[\leadsto J \cdot \left(\ell \cdot \left(\color{blue}{1} \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.9976:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right) \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (* U (+ 1.0 (* 2.0 (* J (/ (* l (cos (* K 0.5))) U)))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.01d0)) then
        tmp = u * (1.0d0 + (2.0d0 * (j * ((l * cos((k * 0.5d0))) / u))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.01) {
		tmp = U * (1.0 + (2.0 * (J * ((l * Math.cos((K * 0.5))) / U))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.01:
		tmp = U * (1.0 + (2.0 * (J * ((l * math.cos((K * 0.5))) / U))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(Float64(l * cos(Float64(K * 0.5))) / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.01)
		tmp = U * (1.0 + (2.0 * (J * ((l * cos((K * 0.5))) / U))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], 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], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 88.6%

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

   3. Taylor expanded in l around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-*l* 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in U around inf 74.0%

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

      1. associate-/l* 73.9%

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

   8. Simplified 73.9%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

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

   3. Taylor expanded in l around 0 90.2%

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

   4. Taylor expanded in K around 0 88.3%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (+ U (* J (* 2.0 (* l (cos (* K 0.5))))))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.01) {
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.01d0)) then
        tmp = u + (j * (2.0d0 * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.01) {
		tmp = U + (J * (2.0 * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.01:
		tmp = U + (J * (2.0 * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.01)
		tmp = Float64(U + Float64(J * Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.01)
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(U + N[(J * N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 88.6%

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

   3. Taylor expanded in l around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-*l* 69.3%

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

      3. *-commutative 69.3%

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

      4. *-commutative 69.3%

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

   5. Simplified 69.3%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

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

   3. Taylor expanded in l around 0 90.2%

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

   4. Taylor expanded in K around 0 88.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.01)
   (+ U (* J (* 2.0 (* 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 (cos((K / 2.0)) <= -0.01) {
		tmp = U + (J * (2.0 * (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 (cos((k / 2.0d0)) <= (-0.01d0)) then
        tmp = u + (j * (2.0d0 * (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 (Math.cos((K / 2.0)) <= -0.01) {
		tmp = U + (J * (2.0 * (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 math.cos((K / 2.0)) <= -0.01:
		tmp = U + (J * (2.0 * (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 (cos(Float64(K / 2.0)) <= -0.01)
		tmp = Float64(U + Float64(J * Float64(2.0 * 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 (cos((K / 2.0)) <= -0.01)
		tmp = U + (J * (2.0 * (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[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.01], N[(U + N[(J * N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $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 (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0100000000000000002

   1. Initial program 88.6%

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

   3. Taylor expanded in l around 0 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-*l* 69.3%

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

      3. *-commutative 69.3%

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

      4. *-commutative 69.3%

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

   5. Simplified 69.3%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 90.0%

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

   3. Taylor expanded in l around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-*l* 62.5%

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

   5. Simplified 62.5%

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

   6. Taylor expanded in K around 0 60.5%

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

      1. associate-*r* 60.5%

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

      2. *-commutative 60.5%

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

   8. Simplified 60.5%

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

   9. Taylor expanded in U around inf 65.3%

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

       1. associate-/l* 68.3%

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

   11. Simplified 68.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  (*
   J
   (*
    l
    (*
     (cos (* K 0.5))
     (+
      2.0
      (* (* l l) (+ 0.3333333333333333 (* (* l l) 0.016666666666666666)))))))
  U))

C

double code(double J, double l, double K, double U) {
	return (J * (l * (cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + 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 * (cos((k * 0.5d0)) * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0))))))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return (J * (l * (Math.cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U;
}

Python

def code(J, l, K, U):
	return (J * (l * (math.cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(2.0 + Float64(Float64(l * l) * Float64(0.3333333333333333 + Float64(Float64(l * l) * 0.016666666666666666))))))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = (J * (l * (cos((K * 0.5)) * (2.0 + ((l * l) * (0.3333333333333333 + ((l * l) * 0.016666666666666666))))))) + U;
end

Wolfram

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

Derivation

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 95.1%

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

   1. *-commutative 95.1%

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

5. Simplified 95.1%

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

6. Taylor expanded in J around 0 95.2%

   \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]
7. Step-by-step derivation

   1. unpow2 95.2%

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

8. Applied egg-rr 95.2%

   \[\leadsto J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right)\right) + U \]
9. Step-by-step derivation

   1. unpow2 95.2%

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

10. Applied egg-rr 95.2%

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

11. Final simplification 95.2%

    \[\leadsto J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + \left(\ell \cdot \ell\right) \cdot \left(0.3333333333333333 + \left(\ell \cdot \ell\right) \cdot 0.016666666666666666\right)\right)\right)\right) + U \]
12. Add Preprocessing

Alternative 8: 87.5% accurate, 2.7× speedup

Math

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

FPCore

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

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)))));
}

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)))))
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)))));
}

Python

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

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) * 0.3333333333333333))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + ((l * l) * 0.3333333333333333)))));
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] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 89.2%

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

   1. unpow2 95.2%

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

5. Applied egg-rr 89.2%

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

6. Final simplification 89.2%

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

Alternative 9: 47.1% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9500000 \lor \neg \left(\ell \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -9500000.0) (not (<= l 8.5e-6))) (* J (* l 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -9500000.0) || !(l <= 8.5e-6)) {
		tmp = J * (l * 2.0);
	} 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 <= (-9500000.0d0)) .or. (.not. (l <= 8.5d-6))) then
        tmp = j * (l * 2.0d0)
    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 <= -9500000.0) || !(l <= 8.5e-6)) {
		tmp = J * (l * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -9500000.0) or not (l <= 8.5e-6):
		tmp = J * (l * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -9500000.0) || !(l <= 8.5e-6))
		tmp = Float64(J * Float64(l * 2.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -9500000.0) || ~((l <= 8.5e-6)))
		tmp = J * (l * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -9500000.0], N[Not[LessEqual[l, 8.5e-6]], $MachinePrecision]], N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -9.5e6 or 8.4999999999999999e-6 < l

   1. Initial program 99.8%

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

   3. Taylor expanded in l around 0 33.2%

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

      1. *-commutative 33.2%

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

      2. associate-*l* 33.2%

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

   5. Simplified 33.2%

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

   6. Taylor expanded in K around 0 26.6%

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

      1. associate-*r* 26.6%

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

      2. *-commutative 26.6%

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

   8. Simplified 26.6%

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

   9. Taylor expanded in l around inf 26.6%

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

   10. Taylor expanded in l around inf 26.5%

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

       1. associate-*r* 26.5%

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

       2. *-commutative 26.5%

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

       3. associate-*r* 26.5%

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

   12. Simplified 26.5%

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

   if -9.5e6 < l < 8.4999999999999999e-6

   1. Initial program 78.2%

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

   4. Applied egg-rr 50.9%

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

   5. Taylor expanded in U around inf 77.4%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9500000 \lor \neg \left(\ell \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 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 64.1%

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

   1. *-commutative 64.1%

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

   2. associate-*l* 64.1%

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

5. Simplified 64.1%

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

6. Taylor expanded in K around 0 56.3%

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

   1. associate-*r* 56.3%

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

   2. *-commutative 56.3%

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

8. Simplified 56.3%

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

9. Taylor expanded in U around inf 60.0%

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

    1. associate-/l* 62.3%

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

11. Simplified 62.3%

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

Alternative 11: 55.0% 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 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 64.1%

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

   1. *-commutative 64.1%

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

   2. associate-*l* 64.1%

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

5. Simplified 64.1%

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

6. Taylor expanded in K around 0 56.3%

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

   1. associate-*r* 56.3%

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

   2. *-commutative 56.3%

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

8. Simplified 56.3%

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

9. Final simplification 56.3%

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

Alternative 12: 38.1% 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 89.7%

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

4. Applied egg-rr 25.1%

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

5. Taylor expanded in U around inf 37.5%

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

Alternative 13: 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 89.7%

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

4. Applied egg-rr 25.5%

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

5. Taylor expanded in U around 0 2.9%

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

Alternative 14: 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 89.7%

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

4. Applied egg-rr 25.1%

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

5. Taylor expanded in U around 0 2.5%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))