Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.0% → 99.4%

Time: 10.7s

Alternatives: 15

Speedup: 2.7×

• 49.7% 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: 86.0% 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 87.6%

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

3. Taylor expanded in l around 0 62.4%

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

   1. *-commutative 62.4%

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

   2. associate-*l* 62.4%

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

5. Simplified 62.4%

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

   1. log1p-expm1-u 99.6%

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

7. Applied egg-rr 99.6%

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

Alternative 2: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ t_1 := U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -0.000108:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))
        (t_1 (+ U (* J (- (exp l) (exp (- l)))))))
   (if (<= l -2.5e+124)
     t_0
     (if (<= l -0.000108)
       t_1
       (if (<= l 2.8e-11)
         (+ U (* (* J (cos (* K 0.5))) (* l 2.0)))
         (if (<= l 5.3e+88) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double t_1 = U + (J * (exp(l) - exp(-l)));
	double tmp;
	if (l <= -2.5e+124) {
		tmp = t_0;
	} else if (l <= -0.000108) {
		tmp = t_1;
	} else if (l <= 2.8e-11) {
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	} else if (l <= 5.3e+88) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    t_1 = u + (j * (exp(l) - exp(-l)))
    if (l <= (-2.5d+124)) then
        tmp = t_0
    else if (l <= (-0.000108d0)) then
        tmp = t_1
    else if (l <= 2.8d-11) then
        tmp = u + ((j * cos((k * 0.5d0))) * (l * 2.0d0))
    else if (l <= 5.3d+88) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	double t_1 = U + (J * (Math.exp(l) - Math.exp(-l)));
	double tmp;
	if (l <= -2.5e+124) {
		tmp = t_0;
	} else if (l <= -0.000108) {
		tmp = t_1;
	} else if (l <= 2.8e-11) {
		tmp = U + ((J * Math.cos((K * 0.5))) * (l * 2.0));
	} else if (l <= 5.3e+88) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	t_1 = U + (J * (math.exp(l) - math.exp(-l)))
	tmp = 0
	if l <= -2.5e+124:
		tmp = t_0
	elif l <= -0.000108:
		tmp = t_1
	elif l <= 2.8e-11:
		tmp = U + ((J * math.cos((K * 0.5))) * (l * 2.0))
	elif l <= 5.3e+88:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))))
	t_1 = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))))
	tmp = 0.0
	if (l <= -2.5e+124)
		tmp = t_0;
	elseif (l <= -0.000108)
		tmp = t_1;
	elseif (l <= 2.8e-11)
		tmp = Float64(U + Float64(Float64(J * cos(Float64(K * 0.5))) * Float64(l * 2.0)));
	elseif (l <= 5.3e+88)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	t_1 = U + (J * (exp(l) - exp(-l)));
	tmp = 0.0;
	if (l <= -2.5e+124)
		tmp = t_0;
	elseif (l <= -0.000108)
		tmp = t_1;
	elseif (l <= 2.8e-11)
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	elseif (l <= 5.3e+88)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.5e+124], t$95$0, If[LessEqual[l, -0.000108], t$95$1, If[LessEqual[l, 2.8e-11], N[(U + N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.3e+88], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.4999999999999998e124 or 5.29999999999999987e88 < 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 0 100.0%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

   if -2.4999999999999998e124 < l < -1.08e-4 or 2.8e-11 < l < 5.29999999999999987e88

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 86.9%

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

   if -1.08e-4 < l < 2.8e-11

   1. Initial program 72.4%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   5. Simplified 99.9%

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

      1. log1p-expm1-u 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+124}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -0.000108:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{+88}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -3.5:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{+88}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -3.5)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (<= l 2.8e-11)
       (+ U (* (* J (cos (* K 0.5))) (* l 2.0)))
       (if (<= l 5.3e+88)
         (+ U (* J (- (exp l) t_0)))
         (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -3.5) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 2.8e-11) {
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	} else if (l <= 5.3e+88) {
		tmp = U + (J * (exp(l) - t_0));
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-3.5d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if (l <= 2.8d-11) then
        tmp = u + ((j * cos((k * 0.5d0))) * (l * 2.0d0))
    else if (l <= 5.3d+88) then
        tmp = u + (j * (exp(l) - t_0))
    else
        tmp = u + (t_1 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -3.5) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 2.8e-11) {
		tmp = U + ((J * Math.cos((K * 0.5))) * (l * 2.0));
	} else if (l <= 5.3e+88) {
		tmp = U + (J * (Math.exp(l) - t_0));
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -3.5:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif l <= 2.8e-11:
		tmp = U + ((J * math.cos((K * 0.5))) * (l * 2.0))
	elif l <= 5.3e+88:
		tmp = U + (J * (math.exp(l) - t_0))
	else:
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -3.5)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif (l <= 2.8e-11)
		tmp = Float64(U + Float64(Float64(J * cos(Float64(K * 0.5))) * Float64(l * 2.0)));
	elseif (l <= 5.3e+88)
		tmp = Float64(U + Float64(J * Float64(exp(l) - t_0)));
	else
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -3.5)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif (l <= 2.8e-11)
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	elseif (l <= 5.3e+88)
		tmp = U + (J * (exp(l) - t_0));
	else
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.5], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.8e-11], N[(U + N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.3e+88], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.5

   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 -3.5 < l < 2.8e-11

   1. Initial program 72.4%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

   5. Simplified 99.9%

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

      1. log1p-expm1-u 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in J around 0 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*l* 100.0%

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

   10. Simplified 100.0%

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

   if 2.8e-11 < l < 5.29999999999999987e88

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 83.3%

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

   if 5.29999999999999987e88 < 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 0 100.0%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.5:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{-11}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 5.3 \cdot 10^{+88}:\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.75:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.75)
   (* 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)) <= 0.75) {
		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)) <= 0.75d0) 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)) <= 0.75) {
		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)) <= 0.75:
		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)) <= 0.75)
		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)) <= 0.75)
		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], 0.75], 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))) < 0.75

   1. Initial program 85.6%

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

   3. Taylor expanded in l around 0 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*l* 65.9%

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

   5. Simplified 65.9%

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

      1. log1p-expm1-u 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in U around inf 68.2%

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

      1. associate-*r* 68.2%

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

      2. *-commutative 68.2%

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

      3. associate-*r/ 68.2%

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

      4. associate-*r* 68.2%

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

      5. *-commutative 68.2%

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

   10. Simplified 68.2%

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

   11. Taylor expanded in J around 0 68.2%

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

       1. associate-/l* 74.1%

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

       2. *-commutative 74.1%

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

   13. Simplified 74.1%

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

   if 0.75 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.5%

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

   3. Taylor expanded in l around 0 90.1%

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

   4. Taylor expanded in K around 0 89.7%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.75:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.05)
   (+ 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)) <= -0.05) {
		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)) <= (-0.05d0)) 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)) <= -0.05) {
		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)) <= -0.05:
		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)) <= -0.05)
		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)) <= -0.05)
		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], -0.05], 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))) < -0.050000000000000003

   1. Initial program 86.7%

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

   3. Taylor expanded in l around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. associate-*l* 62.5%

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

      3. *-commutative 62.5%

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

      4. *-commutative 62.5%

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

   5. Simplified 62.5%

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

   if -0.050000000000000003 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.7%

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

   3. Taylor expanded in l around 0 89.5%

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

   4. Taylor expanded in K around 0 86.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.05:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.54:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.54)
   (+ U (* (* J (cos (* K 0.5))) (* l 2.0)))
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.54) {
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.54d0) then
        tmp = u + ((j * cos((k * 0.5d0))) * (l * 2.0d0))
    else
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.54) {
		tmp = U + ((J * Math.cos((K * 0.5))) * (l * 2.0));
	} else {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.54:
		tmp = U + ((J * math.cos((K * 0.5))) * (l * 2.0))
	else:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.54)
		tmp = Float64(U + Float64(Float64(J * cos(Float64(K * 0.5))) * Float64(l * 2.0)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.54)
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	else
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.54], N[(U + N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $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))) < 0.54000000000000004

   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 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-*l* 65.7%

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

   5. Simplified 65.7%

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

      1. log1p-expm1-u 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in J around 0 65.7%

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

      1. associate-*r* 65.7%

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

      2. *-commutative 65.7%

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

      3. *-commutative 65.7%

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

      4. associate-*r* 65.7%

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

      5. *-commutative 65.7%

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

      6. associate-*r* 65.7%

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

      7. *-commutative 65.7%

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

      8. associate-*l* 65.7%

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

   10. Simplified 65.7%

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

   if 0.54000000000000004 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 88.5%

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

   3. Taylor expanded in l around 0 61.2%

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

      1. *-commutative 61.2%

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

      2. associate-*l* 61.2%

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

   5. Simplified 61.2%

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

   6. Taylor expanded in K around 0 60.3%

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

   7. Taylor expanded in U around inf 67.9%

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

      1. associate-/l* 74.2%

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

   9. Simplified 74.2%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.54:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 89.5%

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

   1. unpow2 89.5%

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

5. Applied egg-rr 89.5%

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

6. Final simplification 89.5%

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

Alternative 8: 63.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.365 \cdot 10^{+20}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 1.365e+20)
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))
   (+ U (* J (* 2.0 (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1.365e+20) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (k <= 1.365d+20) then
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    else
        tmp = u + (j * (2.0d0 * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 1.365e+20) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (J * (2.0 * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 1.365e+20:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	else:
		tmp = U + (J * (2.0 * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 1.365e+20)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 1.365e+20)
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	else
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 1.365e+20], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.365e20

   1. Initial program 86.6%

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

   3. Taylor expanded in l around 0 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l* 60.4%

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

   5. Simplified 60.4%

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

   6. Taylor expanded in K around 0 55.5%

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

   7. Taylor expanded in U around inf 62.1%

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

      1. associate-/l* 68.4%

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

   9. Simplified 68.4%

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

   if 1.365e20 < K

   1. Initial program 90.8%

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

   3. Taylor expanded in l around 0 69.2%

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

      1. *-commutative 69.2%

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

      2. associate-*l* 69.2%

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

      3. *-commutative 69.2%

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

      4. *-commutative 69.2%

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

   5. Simplified 69.2%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1.365 \cdot 10^{+20}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 56.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+68} \lor \neg \left(\ell \leq 9 \cdot 10^{+33}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4e+68) (not (<= l 9e+33)))
   (* J (+ (* l 2.0) (/ U J)))
   (+ U (* 2.0 (* J l)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4e+68) || !(l <= 9e+33)) {
		tmp = J * ((l * 2.0) + (U / J));
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4d+68)) .or. (.not. (l <= 9d+33))) then
        tmp = j * ((l * 2.0d0) + (u / j))
    else
        tmp = u + (2.0d0 * (j * l))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4e+68) || !(l <= 9e+33)) {
		tmp = J * ((l * 2.0) + (U / J));
	} else {
		tmp = U + (2.0 * (J * l));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4e+68) or not (l <= 9e+33):
		tmp = J * ((l * 2.0) + (U / J))
	else:
		tmp = U + (2.0 * (J * l))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4e+68) || !(l <= 9e+33))
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(U / J)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4e+68) || ~((l <= 9e+33)))
		tmp = J * ((l * 2.0) + (U / J));
	else
		tmp = U + (2.0 * (J * l));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4e+68], N[Not[LessEqual[l, 9e+33]], $MachinePrecision]], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.99999999999999981e68 or 9.0000000000000001e33 < 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 33.0%

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

      1. *-commutative 33.0%

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

      2. associate-*l* 33.0%

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

   5. Simplified 33.0%

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

   6. Taylor expanded in K around 0 26.8%

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

   7. Taylor expanded in J around inf 35.4%

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

   if -3.99999999999999981e68 < l < 9.0000000000000001e33

   1. Initial program 76.8%

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

   3. Taylor expanded in l around 0 88.0%

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

      1. *-commutative 88.0%

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

      2. associate-*l* 88.0%

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

   5. Simplified 88.0%

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

   6. Taylor expanded in K around 0 79.4%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+68} \lor \neg \left(\ell \leq 9 \cdot 10^{+33}\right):\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{-79} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+33}\right):\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.65e-79) (not (<= l 1.7e+33))) (* l (* J 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.65e-79) || !(l <= 1.7e+33)) {
		tmp = l * (J * 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 <= (-1.65d-79)) .or. (.not. (l <= 1.7d+33))) then
        tmp = l * (j * 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 <= -1.65e-79) || !(l <= 1.7e+33)) {
		tmp = l * (J * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.65e-79) or not (l <= 1.7e+33):
		tmp = l * (J * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.65e-79) || !(l <= 1.7e+33))
		tmp = Float64(l * Float64(J * 2.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.65e-79) || ~((l <= 1.7e+33)))
		tmp = l * (J * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.65e-79], N[Not[LessEqual[l, 1.7e+33]], $MachinePrecision]], N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.6499999999999999e-79 or 1.7e33 < l

   1. Initial program 94.6%

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

   3. Taylor expanded in l around 0 37.3%

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

      1. *-commutative 37.3%

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

      2. associate-*l* 37.3%

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

   5. Simplified 37.3%

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

   6. Taylor expanded in K around 0 31.0%

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

   7. Taylor expanded in l around inf 30.3%

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

   8. Taylor expanded in J around inf 28.7%

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

   if -1.6499999999999999e-79 < l < 1.7e33

   1. Initial program 78.6%

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

   4. Applied egg-rr 47.5%

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

   5. Taylor expanded in U around inf 73.1%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.65 \cdot 10^{-79} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+33}\right):\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.7% accurate, 28.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 62.4%

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

   1. *-commutative 62.4%

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

   2. associate-*l* 62.4%

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

5. Simplified 62.4%

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

6. Taylor expanded in K around 0 55.0%

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

7. Taylor expanded in U around inf 60.7%

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

   1. associate-/l* 66.0%

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

9. Simplified 66.0%

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

Alternative 12: 53.7% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

3. Taylor expanded in l around 0 62.4%

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

   1. *-commutative 62.4%

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

   2. associate-*l* 62.4%

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

5. Simplified 62.4%

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

6. Taylor expanded in K around 0 55.0%

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

7. Final simplification 55.0%

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

Alternative 13: 36.5% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

4. Applied egg-rr 23.0%

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

5. Taylor expanded in U around inf 34.5%

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

Alternative 14: 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 87.6%

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

4. Applied egg-rr 23.1%

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

5. Taylor expanded in U around 0 2.8%

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

Alternative 15: 2.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.6%

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

4. Applied egg-rr 23.0%

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

5. Taylor expanded in U around 0 2.6%

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

Reproduce

herbie shell --seed 2024184 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))