Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.4%

Time: 10.4s

Alternatives: 11

Speedup: 2.5×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot 2\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (log1p (expm1 (* l 2.0)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * log1p(expm1((l * 2.0)))) * cos((K / 2.0))) + U;
}

Java

public static double code(double J, double l, double K, double U) {
	return ((J * Math.log1p(Math.expm1((l * 2.0)))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * math.log1p(math.expm1((l * 2.0)))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * log1p(expm1(Float64(l * 2.0)))) * cos(Float64(K / 2.0))) + U)
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[Log[1 + N[(Exp[N[(l * 2.0), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Derivation

1. Initial program 81.6%

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

3. Taylor expanded in l around 0 68.3%

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

   1. *-commutative 68.3%

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

   2. associate-*l* 68.3%

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

5. Simplified 68.3%

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

   1. log1p-expm1-u 99.3%

      \[\leadsto \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

7. Applied egg-rr 99.3%

   \[\leadsto \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
8. Add Preprocessing

Alternative 2: 93.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + ((l * l) * (0.3333333333333333d0 + ((l * l) * 0.016666666666666666d0)))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 81.6%

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

3. Taylor expanded in l around 0 93.5%

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

   1. *-commutative 93.5%

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

5. Simplified 93.5%

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

   1. unpow2 93.5%

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

7. Applied egg-rr 93.5%

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

   1. unpow2 93.5%

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

9. Applied egg-rr 93.5%

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

10. Final simplification 93.5%

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

Alternative 3: 66.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+53}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 5.8e+53)
   (+ U (* J (* 2.0 (* l (cos (* K 0.5))))))
   (* U (+ 1.0 (* 10.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 5.8e+53) {
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	} else {
		tmp = U * (1.0 + (10.0 * (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 <= 5.8d+53) then
        tmp = u + (j * (2.0d0 * (l * cos((k * 0.5d0)))))
    else
        tmp = u * (1.0d0 + (10.0d0 * (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 <= 5.8e+53) {
		tmp = U + (J * (2.0 * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U * (1.0 + (10.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 5.8e+53:
		tmp = U + (J * (2.0 * (l * math.cos((K * 0.5)))))
	else:
		tmp = U * (1.0 + (10.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 5.8e+53)
		tmp = Float64(U + Float64(J * Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(10.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 5.8e+53)
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	else
		tmp = U * (1.0 + (10.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 5.8e+53], 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[(10.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.8000000000000004e53

   1. Initial program 77.3%

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

   3. Taylor expanded in l around 0 80.2%

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

      1. *-commutative 80.2%

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

      2. associate-*l* 80.2%

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

      3. *-commutative 80.2%

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

      4. *-commutative 80.2%

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

   5. Simplified 80.2%

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

   if 5.8000000000000004e53 < 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 17.1%

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

      1. *-commutative 17.1%

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

      2. associate-*l* 17.1%

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

   5. Simplified 17.1%

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

   7. Applied egg-rr 16.4%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 16.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 16.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 16.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 16.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 16.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 16.4%

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

   9. Simplified 16.4%

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

   10. Taylor expanded in K around 0 16.4%

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

       1. *-commutative 16.4%

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 10} + U \]

   12. Simplified 16.4%

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 10} + U \]

   13. Taylor expanded in U around inf 28.4%

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

       1. associate-/l* 34.4%

          \[\leadsto U \cdot \left(1 + 10 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   15. Simplified 34.4%

       \[\leadsto \color{blue}{U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.8 \cdot 10^{+53}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.6% 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 81.6%

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

3. Taylor expanded in l around 0 68.3%

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

   1. *-commutative 68.3%

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

   2. associate-*l* 68.3%

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

5. Simplified 68.3%

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

   1. log1p-expm1-u 99.3%

      \[\leadsto \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

7. Applied egg-rr 99.3%

   \[\leadsto \left(J \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot 2\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

8. Taylor expanded in U around inf 71.2%

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

   1. associate-/l* 73.5%

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

10. Simplified 73.5%

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

11. Final simplification 73.5%

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

Alternative 5: 58.0% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+17}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1.4e+17)
   (+ U (* l (* J 2.0)))
   (* U (+ 1.0 (* 10.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.4e+17) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U * (1.0 + (10.0 * (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 <= 1.4d+17) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u * (1.0d0 + (10.0d0 * (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 <= 1.4e+17) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U * (1.0 + (10.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1.4e+17:
		tmp = U + (l * (J * 2.0))
	else:
		tmp = U * (1.0 + (10.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1.4e+17)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(10.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1.4e+17)
		tmp = U + (l * (J * 2.0));
	else
		tmp = U * (1.0 + (10.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1.4e+17], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(10.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.4e17

   1. Initial program 76.4%

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

   3. Taylor expanded in l around 0 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-*l* 82.8%

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

   5. Simplified 82.8%

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

   6. Taylor expanded in K around 0 67.3%

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

      1. associate-*r* 67.3%

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

      2. *-commutative 67.3%

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

   8. Simplified 67.3%

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

   if 1.4e17 < 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 16.8%

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

      1. *-commutative 16.8%

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

      2. associate-*l* 16.8%

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

   5. Simplified 16.8%

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

   7. Applied egg-rr 14.4%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 14.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 14.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 14.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 14.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 14.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 14.4%

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

   9. Simplified 14.4%

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

   10. Taylor expanded in K around 0 14.4%

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

       1. *-commutative 14.4%

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 10} + U \]

   12. Simplified 14.4%

       \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot 10} + U \]

   13. Taylor expanded in U around inf 26.3%

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

       1. associate-/l* 31.4%

          \[\leadsto U \cdot \left(1 + 10 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   15. Simplified 31.4%

       \[\leadsto \color{blue}{U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.4 \cdot 10^{+17}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 42.1% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+14} \lor \neg \left(\ell \leq 5.2 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{U \cdot U}{U}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.2e+14) (not (<= l 5.2e-12))) (/ (* U U) U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.2e+14) || !(l <= 5.2e-12)) {
		tmp = (U * 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 <= (-1.2d+14)) .or. (.not. (l <= 5.2d-12))) then
        tmp = (u * 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 <= -1.2e+14) || !(l <= 5.2e-12)) {
		tmp = (U * U) / U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.2e+14) or not (l <= 5.2e-12):
		tmp = (U * U) / U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.2e+14) || !(l <= 5.2e-12))
		tmp = Float64(Float64(U * U) / U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.2e+14) || ~((l <= 5.2e-12)))
		tmp = (U * U) / U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.2e+14], N[Not[LessEqual[l, 5.2e-12]], $MachinePrecision]], N[(N[(U * U), $MachinePrecision] / U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.2e14 or 5.19999999999999965e-12 < l

   1. Initial program 99.2%

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

   4. Applied egg-rr 3.6%

      \[\leadsto \color{blue}{0} + U \]
   5. Step-by-step derivation

      1. flip-+ 16.9%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - U \cdot U}{0 - U}} \]

      2. metadata-eval 16.9%

         \[\leadsto \frac{\color{blue}{0} - U \cdot U}{0 - U} \]

      3. div-sub 16.9%

         \[\leadsto \color{blue}{\frac{0}{0 - U} - \frac{U \cdot U}{0 - U}} \]

      4. sub0-neg 16.9%

         \[\leadsto \frac{0}{\color{blue}{-U}} - \frac{U \cdot U}{0 - U} \]

      5. pow2 16.9%

         \[\leadsto \frac{0}{-U} - \frac{\color{blue}{{U}^{2}}}{0 - U} \]

      6. sub0-neg 16.9%

         \[\leadsto \frac{0}{-U} - \frac{{U}^{2}}{\color{blue}{-U}} \]

   6. Applied egg-rr 16.9%

      \[\leadsto \color{blue}{\frac{0}{-U} - \frac{{U}^{2}}{-U}} \]
   7. Step-by-step derivation

      1. div0 16.9%

         \[\leadsto \color{blue}{0} - \frac{{U}^{2}}{-U} \]

      2. neg-sub0 16.9%

         \[\leadsto \color{blue}{-\frac{{U}^{2}}{-U}} \]

      3. distribute-frac-neg2 16.9%

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

      4. remove-double-neg 16.9%

         \[\leadsto \frac{{U}^{2}}{\color{blue}{U}} \]

   8. Simplified 16.9%

      \[\leadsto \color{blue}{\frac{{U}^{2}}{U}} \]
   9. Step-by-step derivation

      1. unpow2 16.9%

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

   10. Applied egg-rr 16.9%

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

   if -1.2e14 < l < 5.19999999999999965e-12

   1. Initial program 69.3%

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

   4. Applied egg-rr 42.8%

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

   5. Taylor expanded in U around inf 67.5%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+14} \lor \neg \left(\ell \leq 5.2 \cdot 10^{-12}\right):\\ \;\;\;\;\frac{U \cdot U}{U}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.1% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;U + -6 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{U \cdot U}{U}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1.9e+16) (+ U (* -6.0 (* J l))) (/ (* U U) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.9e+16) {
		tmp = U + (-6.0 * (J * l));
	} else {
		tmp = (U * U) / 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 <= 1.9d+16) then
        tmp = u + ((-6.0d0) * (j * l))
    else
        tmp = (u * u) / u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.9e+16) {
		tmp = U + (-6.0 * (J * l));
	} else {
		tmp = (U * U) / U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1.9e+16:
		tmp = U + (-6.0 * (J * l))
	else:
		tmp = (U * U) / U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1.9e+16)
		tmp = Float64(U + Float64(-6.0 * Float64(J * l)));
	else
		tmp = Float64(Float64(U * U) / U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1.9e+16)
		tmp = U + (-6.0 * (J * l));
	else
		tmp = (U * U) / U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1.9e+16], N[(U + N[(-6.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(U * U), $MachinePrecision] / U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.9e16

   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 l around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. associate-*l* 83.2%

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

   5. Simplified 83.2%

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

   7. Applied egg-rr 55.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) \cdot -3\right)} + U \]

   8. Taylor expanded in K around 0 55.6%

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

      1. *-commutative 55.6%

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

   10. Simplified 55.6%

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

   if 1.9e16 < 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 2.8%

      \[\leadsto \color{blue}{0} + U \]
   5. Step-by-step derivation

      1. flip-+ 19.0%

         \[\leadsto \color{blue}{\frac{0 \cdot 0 - U \cdot U}{0 - U}} \]

      2. metadata-eval 19.0%

         \[\leadsto \frac{\color{blue}{0} - U \cdot U}{0 - U} \]

      3. div-sub 19.0%

         \[\leadsto \color{blue}{\frac{0}{0 - U} - \frac{U \cdot U}{0 - U}} \]

      4. sub0-neg 19.0%

         \[\leadsto \frac{0}{\color{blue}{-U}} - \frac{U \cdot U}{0 - U} \]

      5. pow2 19.0%

         \[\leadsto \frac{0}{-U} - \frac{\color{blue}{{U}^{2}}}{0 - U} \]

      6. sub0-neg 19.0%

         \[\leadsto \frac{0}{-U} - \frac{{U}^{2}}{\color{blue}{-U}} \]

   6. Applied egg-rr 19.0%

      \[\leadsto \color{blue}{\frac{0}{-U} - \frac{{U}^{2}}{-U}} \]
   7. Step-by-step derivation

      1. div0 19.0%

         \[\leadsto \color{blue}{0} - \frac{{U}^{2}}{-U} \]

      2. neg-sub0 19.0%

         \[\leadsto \color{blue}{-\frac{{U}^{2}}{-U}} \]

      3. distribute-frac-neg2 19.0%

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

      4. remove-double-neg 19.0%

         \[\leadsto \frac{{U}^{2}}{\color{blue}{U}} \]

   8. Simplified 19.0%

      \[\leadsto \color{blue}{\frac{{U}^{2}}{U}} \]
   9. Step-by-step derivation

      1. unpow2 19.0%

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

   10. Applied egg-rr 19.0%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;U + -6 \cdot \left(J \cdot \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{U \cdot U}{U}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.1% 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 81.6%

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

3. Taylor expanded in l around 0 68.3%

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

   1. *-commutative 68.3%

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

   2. associate-*l* 68.3%

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

5. Simplified 68.3%

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

6. Taylor expanded in K around 0 55.7%

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

   1. associate-*r* 55.7%

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

   2. *-commutative 55.7%

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

8. Simplified 55.7%

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

9. Final simplification 55.7%

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

Alternative 9: 36.9% 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 81.6%

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

4. Applied egg-rr 26.3%

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

5. Taylor expanded in U around inf 41.3%

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

Alternative 10: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 8 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 8.0)

C

double code(double J, double l, double K, double U) {
	return 8.0;
}

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 = 8.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 8.0;
}

Python

def code(J, l, K, U):
	return 8.0

Julia

function code(J, l, K, U)
	return 8.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 8.0;
end

Wolfram

code[J_, l_, K_, U_] := 8.0

Derivation

1. Initial program 81.6%

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

4. Applied egg-rr 26.4%

   \[\leadsto \color{blue}{8} + U \]

5. Taylor expanded in U around 0 3.0%

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

Alternative 11: 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 81.6%

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

4. Applied egg-rr 26.3%

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

5. Taylor expanded in U around 0 3.0%

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

Reproduce

herbie shell --seed 2024157 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))