Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.7%

Time: 10.6s

Alternatives: 12

Speedup: 1.4×

• 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: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -4 \cdot 10^{+86} \lor \neg \left(t_0 \leq 4 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(t_0 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -4e+86) (not (<= t_0 4e-7)))
     (+ (* (cos (/ K 2.0)) (* t_0 J)) U)
     (+ U (* 2.0 (* l (* J (cos (* K 0.5)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -4e+86) || !(t_0 <= 4e-7)) {
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (2.0 * (l * (J * 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) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if ((t_0 <= (-4d+86)) .or. (.not. (t_0 <= 4d-7))) then
        tmp = (cos((k / 2.0d0)) * (t_0 * j)) + u
    else
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    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) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -4e+86) || !(t_0 <= 4e-7)) {
		tmp = (Math.cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -4e+86) or not (t_0 <= 4e-7):
		tmp = (math.cos((K / 2.0)) * (t_0 * J)) + U
	else:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -4e+86) || !(t_0 <= 4e-7))
		tmp = Float64(Float64(cos(Float64(K / 2.0)) * Float64(t_0 * J)) + U);
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -4e+86) || ~((t_0 <= 4e-7)))
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	else
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -4e+86], N[Not[LessEqual[t$95$0, 4e-7]], $MachinePrecision]], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -4.0000000000000001e86 or 3.9999999999999998e-7 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -4.0000000000000001e86 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 3.9999999999999998e-7

   1. Initial program 72.1%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*l* 99.9%

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

      3. *-commutative 99.9%

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

   4. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -4 \cdot 10^{+86} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 4 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 2: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+116}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(t_0 \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -200:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.0004:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot t_0\right)\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -1.35e+116)
     (+ U (* (pow l 3.0) (* J (* t_0 0.3333333333333333))))
     (if (<= l -200.0)
       t_1
       (if (<= l 0.0004)
         (+ U (* 2.0 (* l (* J t_0))))
         (if (<= l 2.35e+80)
           t_1
           (+
            U
            (*
             (cos (/ K 2.0))
             (* J (+ (* (pow l 3.0) 0.3333333333333333) (* l 2.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -1.35e+116) {
		tmp = U + (pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)));
	} else if (l <= -200.0) {
		tmp = t_1;
	} else if (l <= 0.0004) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else if (l <= 2.35e+80) {
		tmp = t_1;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * ((pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-1.35d+116)) then
        tmp = u + ((l ** 3.0d0) * (j * (t_0 * 0.3333333333333333d0)))
    else if (l <= (-200.0d0)) then
        tmp = t_1
    else if (l <= 0.0004d0) then
        tmp = u + (2.0d0 * (l * (j * t_0)))
    else if (l <= 2.35d+80) then
        tmp = t_1
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (((l ** 3.0d0) * 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 t_0 = Math.cos((K * 0.5));
	double t_1 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -1.35e+116) {
		tmp = U + (Math.pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)));
	} else if (l <= -200.0) {
		tmp = t_1;
	} else if (l <= 0.0004) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else if (l <= 2.35e+80) {
		tmp = t_1;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((Math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -1.35e+116:
		tmp = U + (math.pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)))
	elif l <= -200.0:
		tmp = t_1
	elif l <= 0.0004:
		tmp = U + (2.0 * (l * (J * t_0)))
	elif l <= 2.35e+80:
		tmp = t_1
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * ((math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -1.35e+116)
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * Float64(t_0 * 0.3333333333333333))));
	elseif (l <= -200.0)
		tmp = t_1;
	elseif (l <= 0.0004)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * t_0))));
	elseif (l <= 2.35e+80)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64((l ^ 3.0) * 0.3333333333333333) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -1.35e+116)
		tmp = U + ((l ^ 3.0) * (J * (t_0 * 0.3333333333333333)));
	elseif (l <= -200.0)
		tmp = t_1;
	elseif (l <= 0.0004)
		tmp = U + (2.0 * (l * (J * t_0)));
	elseif (l <= 2.35e+80)
		tmp = t_1;
	else
		tmp = U + (cos((K / 2.0)) * (J * (((l ^ 3.0) * 0.3333333333333333) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.35e+116], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -200.0], t$95$1, If[LessEqual[l, 0.0004], N[(U + N[(2.0 * N[(l * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.35e+80], t$95$1, N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.35e116

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 100.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 100.0%

         \[\leadsto 0.3333333333333333 \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right) \cdot \cos \left(K \cdot 0.5\right)} + U \]

      4. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      5. associate-*r* 100.0%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      6. associate-*l* 100.0%

         \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. associate-*l* 100.0%

         \[\leadsto {\ell}^{3} \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      8. *-commutative 100.0%

         \[\leadsto {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if -1.35e116 < l < -200 or 4.00000000000000019e-4 < l < 2.35000000000000005e80

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 76.4%

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

   if -200 < l < 4.00000000000000019e-4

   1. Initial program 72.3%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   if 2.35000000000000005e80 < 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. Taylor expanded in l around 0 95.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+116}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -200:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.0004:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+80}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{3} \cdot \left(J \cdot \left(t_0 \cdot 0.3333333333333333\right)\right)\\ t_2 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -200:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 0.00022:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot t_0\right)\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 3.0) (* J (* t_0 0.3333333333333333)))))
        (t_2 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -1.35e+116)
     t_1
     (if (<= l -200.0)
       t_2
       (if (<= l 0.00022)
         (+ U (* 2.0 (* l (* J t_0))))
         (if (<= l 2.35e+80) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)));
	double t_2 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -1.35e+116) {
		tmp = t_1;
	} else if (l <= -200.0) {
		tmp = t_2;
	} else if (l <= 0.00022) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else if (l <= 2.35e+80) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((l ** 3.0d0) * (j * (t_0 * 0.3333333333333333d0)))
    t_2 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-1.35d+116)) then
        tmp = t_1
    else if (l <= (-200.0d0)) then
        tmp = t_2
    else if (l <= 0.00022d0) then
        tmp = u + (2.0d0 * (l * (j * t_0)))
    else if (l <= 2.35d+80) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (Math.pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)));
	double t_2 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -1.35e+116) {
		tmp = t_1;
	} else if (l <= -200.0) {
		tmp = t_2;
	} else if (l <= 0.00022) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else if (l <= 2.35e+80) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (math.pow(l, 3.0) * (J * (t_0 * 0.3333333333333333)))
	t_2 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -1.35e+116:
		tmp = t_1
	elif l <= -200.0:
		tmp = t_2
	elif l <= 0.00022:
		tmp = U + (2.0 * (l * (J * t_0)))
	elif l <= 2.35e+80:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64((l ^ 3.0) * Float64(J * Float64(t_0 * 0.3333333333333333))))
	t_2 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -1.35e+116)
		tmp = t_1;
	elseif (l <= -200.0)
		tmp = t_2;
	elseif (l <= 0.00022)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * t_0))));
	elseif (l <= 2.35e+80)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((l ^ 3.0) * (J * (t_0 * 0.3333333333333333)));
	t_2 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -1.35e+116)
		tmp = t_1;
	elseif (l <= -200.0)
		tmp = t_2;
	elseif (l <= 0.00022)
		tmp = U + (2.0 * (l * (J * t_0)));
	elseif (l <= 2.35e+80)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * N[(t$95$0 * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.35e+116], t$95$1, If[LessEqual[l, -200.0], t$95$2, If[LessEqual[l, 0.00022], N[(U + N[(2.0 * N[(l * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.35e+80], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.35e116 or 2.35000000000000005e80 < 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. Taylor expanded in l around 0 97.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 97.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative 97.7%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 97.7%

         \[\leadsto 0.3333333333333333 \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) + U \]

      3. associate-*r* 97.7%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right) \cdot \cos \left(K \cdot 0.5\right)} + U \]

      4. *-commutative 97.7%

         \[\leadsto \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      5. associate-*r* 97.7%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      6. associate-*l* 97.7%

         \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. associate-*l* 97.7%

         \[\leadsto {\ell}^{3} \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      8. *-commutative 97.7%

         \[\leadsto {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]

   5. Simplified 97.7%

      \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if -1.35e116 < l < -200 or 2.20000000000000008e-4 < l < 2.35000000000000005e80

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 76.4%

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

   if -200 < l < 2.20000000000000008e-4

   1. Initial program 72.3%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+116}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -200:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.00022:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+80}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 4: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.94:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.94)
   (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
   (+ U (* J (+ (* (pow l 3.0) 0.3333333333333333) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.94) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else {
		tmp = U + (J * ((pow(l, 3.0) * 0.3333333333333333) + (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)) <= 0.94d0) then
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (((l ** 3.0d0) * 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)) <= 0.94) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * ((Math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.94:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * ((math.pow(l, 3.0) * 0.3333333333333333) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.94)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64((l ^ 3.0) * 0.3333333333333333) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.94)
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	else
		tmp = U + (J * (((l ^ 3.0) * 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], 0.94], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(N[Power[l, 3.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.93999999999999995

   1. Initial program 83.2%

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

   2. Taylor expanded in l around 0 70.3%

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

      1. *-commutative 70.3%

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

      2. associate-*l* 70.3%

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

      3. *-commutative 70.3%

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

   4. Simplified 70.3%

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

   if 0.93999999999999995 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0 86.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in K around 0 86.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]

   4. Taylor expanded in J around 0 86.3%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.94:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left({\ell}^{3} \cdot 0.3333333333333333 + \ell \cdot 2\right)\\ \end{array} \]

Alternative 5: 86.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -200:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 0.00085:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (- (exp l) (exp (- l))) J))))
   (if (<= l -200.0)
     t_0
     (if (<= l 0.00085)
       (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
       (if (<= l 2.35e+80)
         t_0
         (+
          U
          (*
           (pow l 3.0)
           (*
            J
            (+ 0.3333333333333333 (* -0.041666666666666664 (* K K)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((exp(l) - exp(-l)) * J);
	double tmp;
	if (l <= -200.0) {
		tmp = t_0;
	} else if (l <= 0.00085) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 2.35e+80) {
		tmp = t_0;
	} else {
		tmp = U + (pow(l, 3.0) * (J * (0.3333333333333333 + (-0.041666666666666664 * (K * K)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((exp(l) - exp(-l)) * j)
    if (l <= (-200.0d0)) then
        tmp = t_0
    else if (l <= 0.00085d0) then
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    else if (l <= 2.35d+80) then
        tmp = t_0
    else
        tmp = u + ((l ** 3.0d0) * (j * (0.3333333333333333d0 + ((-0.041666666666666664d0) * (k * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double tmp;
	if (l <= -200.0) {
		tmp = t_0;
	} else if (l <= 0.00085) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 2.35e+80) {
		tmp = t_0;
	} else {
		tmp = U + (Math.pow(l, 3.0) * (J * (0.3333333333333333 + (-0.041666666666666664 * (K * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((math.exp(l) - math.exp(-l)) * J)
	tmp = 0
	if l <= -200.0:
		tmp = t_0
	elif l <= 0.00085:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 2.35e+80:
		tmp = t_0
	else:
		tmp = U + (math.pow(l, 3.0) * (J * (0.3333333333333333 + (-0.041666666666666664 * (K * K)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	tmp = 0.0
	if (l <= -200.0)
		tmp = t_0;
	elseif (l <= 0.00085)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 2.35e+80)
		tmp = t_0;
	else
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * Float64(0.3333333333333333 + Float64(-0.041666666666666664 * Float64(K * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((exp(l) - exp(-l)) * J);
	tmp = 0.0;
	if (l <= -200.0)
		tmp = t_0;
	elseif (l <= 0.00085)
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	elseif (l <= 2.35e+80)
		tmp = t_0;
	else
		tmp = U + ((l ^ 3.0) * (J * (0.3333333333333333 + (-0.041666666666666664 * (K * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -200.0], t$95$0, If[LessEqual[l, 0.00085], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.35e+80], t$95$0, N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * N[(0.3333333333333333 + N[(-0.041666666666666664 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -200 or 8.49999999999999953e-4 < l < 2.35000000000000005e80

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 74.4%

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

   if -200 < l < 8.49999999999999953e-4

   1. Initial program 72.3%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   if 2.35000000000000005e80 < 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. Taylor expanded in l around 0 95.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 95.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 95.0%

         \[\leadsto 0.3333333333333333 \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) + U \]

      3. associate-*r* 95.0%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right) \cdot \cos \left(K \cdot 0.5\right)} + U \]

      4. *-commutative 95.0%

         \[\leadsto \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      5. associate-*r* 95.0%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      6. associate-*l* 95.0%

         \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. associate-*l* 95.0%

         \[\leadsto {\ell}^{3} \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      8. *-commutative 95.0%

         \[\leadsto {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]

   5. Simplified 95.0%

      \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0 81.8%

      \[\leadsto {\ell}^{3} \cdot \left(J \cdot \color{blue}{\left(0.3333333333333333 + -0.041666666666666664 \cdot {K}^{2}\right)}\right) + U \]
   7. Step-by-step derivation

      1. unpow2 81.8%

         \[\leadsto {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) + U \]

   8. Simplified 81.8%

      \[\leadsto {\ell}^{3} \cdot \left(J \cdot \color{blue}{\left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)}\right) + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -200:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.00085:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 2.35 \cdot 10^{+80}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 6: 80.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -195 \lor \neg \left(\ell \leq 500\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -195.0) (not (<= l 500.0)))
   (+
    U
    (*
     (pow l 3.0)
     (* J (+ 0.3333333333333333 (* -0.041666666666666664 (* K K))))))
   (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -195.0) || !(l <= 500.0)) {
		tmp = U + (pow(l, 3.0) * (J * (0.3333333333333333 + (-0.041666666666666664 * (K * K)))));
	} else {
		tmp = U + (2.0 * (l * (J * 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 ((l <= (-195.0d0)) .or. (.not. (l <= 500.0d0))) then
        tmp = u + ((l ** 3.0d0) * (j * (0.3333333333333333d0 + ((-0.041666666666666664d0) * (k * k)))))
    else
        tmp = u + (2.0d0 * (l * (j * 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 ((l <= -195.0) || !(l <= 500.0)) {
		tmp = U + (Math.pow(l, 3.0) * (J * (0.3333333333333333 + (-0.041666666666666664 * (K * K)))));
	} else {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -195.0) or not (l <= 500.0):
		tmp = U + (math.pow(l, 3.0) * (J * (0.3333333333333333 + (-0.041666666666666664 * (K * K)))))
	else:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -195.0) || !(l <= 500.0))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * Float64(0.3333333333333333 + Float64(-0.041666666666666664 * Float64(K * K))))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -195.0) || ~((l <= 500.0)))
		tmp = U + ((l ^ 3.0) * (J * (0.3333333333333333 + (-0.041666666666666664 * (K * K)))));
	else
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -195.0], N[Not[LessEqual[l, 500.0]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * N[(0.3333333333333333 + N[(-0.041666666666666664 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -195 or 500 < 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. Taylor expanded in l around 0 73.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 73.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative 73.3%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 73.3%

         \[\leadsto 0.3333333333333333 \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) + U \]

      3. associate-*r* 73.3%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right) \cdot \cos \left(K \cdot 0.5\right)} + U \]

      4. *-commutative 73.3%

         \[\leadsto \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      5. associate-*r* 73.3%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      6. associate-*l* 73.3%

         \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. associate-*l* 73.3%

         \[\leadsto {\ell}^{3} \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      8. *-commutative 73.3%

         \[\leadsto {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]

   5. Simplified 73.3%

      \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0 63.8%

      \[\leadsto {\ell}^{3} \cdot \left(J \cdot \color{blue}{\left(0.3333333333333333 + -0.041666666666666664 \cdot {K}^{2}\right)}\right) + U \]
   7. Step-by-step derivation

      1. unpow2 63.8%

         \[\leadsto {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \color{blue}{\left(K \cdot K\right)}\right)\right) + U \]

   8. Simplified 63.8%

      \[\leadsto {\ell}^{3} \cdot \left(J \cdot \color{blue}{\left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)}\right) + U \]

   if -195 < l < 500

   1. Initial program 72.3%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

   4. Simplified 99.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -195 \lor \neg \left(\ell \leq 500\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 + -0.041666666666666664 \cdot \left(K \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 7: 78.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -5 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 3800:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+86}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (pow l 3.0) (* J 0.3333333333333333)))))
   (if (<= l -5e+39)
     t_0
     (if (<= l 3800.0)
       (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))
       (if (<= l 3e+86) (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25)))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	double tmp;
	if (l <= -5e+39) {
		tmp = t_0;
	} else if (l <= 3800.0) {
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	} else if (l <= 3e+86) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    if (l <= (-5d+39)) then
        tmp = t_0
    else if (l <= 3800.0d0) then
        tmp = u + (2.0d0 * (l * (j * cos((k * 0.5d0)))))
    else if (l <= 3d+86) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    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.pow(l, 3.0) * (J * 0.3333333333333333));
	double tmp;
	if (l <= -5e+39) {
		tmp = t_0;
	} else if (l <= 3800.0) {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	} else if (l <= 3e+86) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	tmp = 0
	if l <= -5e+39:
		tmp = t_0
	elif l <= 3800.0:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	elif l <= 3e+86:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -5e+39)
		tmp = t_0;
	elseif (l <= 3800.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	elseif (l <= 3e+86)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	tmp = 0.0;
	if (l <= -5e+39)
		tmp = t_0;
	elseif (l <= 3800.0)
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	elseif (l <= 3e+86)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5e+39], t$95$0, If[LessEqual[l, 3800.0], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3e+86], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.00000000000000015e39 or 2.99999999999999977e86 < 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. Taylor expanded in l around 0 88.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 88.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative 88.8%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 88.8%

         \[\leadsto 0.3333333333333333 \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) + U \]

      3. associate-*r* 88.8%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right) \cdot \cos \left(K \cdot 0.5\right)} + U \]

      4. *-commutative 88.8%

         \[\leadsto \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      5. associate-*r* 88.8%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      6. associate-*l* 88.8%

         \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. associate-*l* 88.8%

         \[\leadsto {\ell}^{3} \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      8. *-commutative 88.8%

         \[\leadsto {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]

   5. Simplified 88.8%

      \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0 63.2%

      \[\leadsto {\ell}^{3} \cdot \left(J \cdot \color{blue}{0.3333333333333333}\right) + U \]

   if -5.00000000000000015e39 < l < 3800

   1. Initial program 73.8%

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

   2. Taylor expanded in l around 0 93.9%

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

      1. *-commutative 93.9%

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

      2. associate-*l* 94.0%

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

      3. *-commutative 94.0%

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

   4. Simplified 94.0%

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

   if 3800 < l < 2.99999999999999977e86

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 3.5%

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

      1. *-commutative 3.5%

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

      2. associate-*l* 3.5%

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

      3. *-commutative 3.5%

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

   4. Simplified 3.5%

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

   5. Taylor expanded in K around 0 31.2%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right) + \left(-0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right) + U\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 31.2%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right) + U} \]

      2. +-commutative 31.2%

         \[\leadsto \color{blue}{U + \left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} \]

      3. associate-*r* 31.2%

         \[\leadsto U + \left(2 \cdot \left(\ell \cdot J\right) + \color{blue}{\left(-0.25 \cdot {K}^{2}\right) \cdot \left(\ell \cdot J\right)}\right) \]

      4. distribute-rgt-out 31.2%

         \[\leadsto U + \color{blue}{\left(\ell \cdot J\right) \cdot \left(2 + -0.25 \cdot {K}^{2}\right)} \]

      5. *-commutative 31.2%

         \[\leadsto U + \left(\ell \cdot J\right) \cdot \left(2 + \color{blue}{{K}^{2} \cdot -0.25}\right) \]

      6. unpow2 31.2%

         \[\leadsto U + \left(\ell \cdot J\right) \cdot \left(2 + \color{blue}{\left(K \cdot K\right)} \cdot -0.25\right) \]

   7. Simplified 31.2%

      \[\leadsto \color{blue}{U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+39}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq 3800:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{+86}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 8: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 470:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+85}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (pow l 3.0) (* J 0.3333333333333333)))))
   (if (<= l -3.4e-10)
     t_0
     (if (<= l 470.0)
       (+ U (* 2.0 (* l J)))
       (if (<= l 1.06e+85) (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25)))) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	double tmp;
	if (l <= -3.4e-10) {
		tmp = t_0;
	} else if (l <= 470.0) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= 1.06e+85) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    if (l <= (-3.4d-10)) then
        tmp = t_0
    else if (l <= 470.0d0) then
        tmp = u + (2.0d0 * (l * j))
    else if (l <= 1.06d+85) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    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.pow(l, 3.0) * (J * 0.3333333333333333));
	double tmp;
	if (l <= -3.4e-10) {
		tmp = t_0;
	} else if (l <= 470.0) {
		tmp = U + (2.0 * (l * J));
	} else if (l <= 1.06e+85) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	tmp = 0
	if l <= -3.4e-10:
		tmp = t_0
	elif l <= 470.0:
		tmp = U + (2.0 * (l * J))
	elif l <= 1.06e+85:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)))
	tmp = 0.0
	if (l <= -3.4e-10)
		tmp = t_0;
	elseif (l <= 470.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	elseif (l <= 1.06e+85)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	tmp = 0.0;
	if (l <= -3.4e-10)
		tmp = t_0;
	elseif (l <= 470.0)
		tmp = U + (2.0 * (l * J));
	elseif (l <= 1.06e+85)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.4e-10], t$95$0, If[LessEqual[l, 470.0], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.06e+85], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.40000000000000015e-10 or 1.0600000000000001e85 < l

   1. Initial program 99.6%

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

   2. Taylor expanded in l around 0 83.4%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + 0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 82.5%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)} + U \]
   4. Step-by-step derivation

      1. *-commutative 82.5%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 82.5%

         \[\leadsto 0.3333333333333333 \cdot \left(\left({\ell}^{3} \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) + U \]

      3. associate-*r* 82.5%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \left({\ell}^{3} \cdot J\right)\right) \cdot \cos \left(K \cdot 0.5\right)} + U \]

      4. *-commutative 82.5%

         \[\leadsto \color{blue}{\left(\left({\ell}^{3} \cdot J\right) \cdot 0.3333333333333333\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      5. associate-*r* 82.5%

         \[\leadsto \color{blue}{\left({\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(K \cdot 0.5\right) + U \]

      6. associate-*l* 82.5%

         \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)} + U \]

      7. associate-*l* 82.5%

         \[\leadsto {\ell}^{3} \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot \cos \left(K \cdot 0.5\right)\right)\right)} + U \]

      8. *-commutative 82.5%

         \[\leadsto {\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right)\right) + U \]

   5. Simplified 82.5%

      \[\leadsto \color{blue}{{\ell}^{3} \cdot \left(J \cdot \left(0.3333333333333333 \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   6. Taylor expanded in K around 0 58.3%

      \[\leadsto {\ell}^{3} \cdot \left(J \cdot \color{blue}{0.3333333333333333}\right) + U \]

   if -3.40000000000000015e-10 < l < 470

   1. Initial program 72.2%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in K around 0 86.0%

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

   if 470 < l < 1.0600000000000001e85

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 3.5%

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

      1. *-commutative 3.5%

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

      2. associate-*l* 3.5%

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

      3. *-commutative 3.5%

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

   4. Simplified 3.5%

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

   5. Taylor expanded in K around 0 31.2%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right) + \left(-0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right) + U\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 31.2%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right) + U} \]

      2. +-commutative 31.2%

         \[\leadsto \color{blue}{U + \left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} \]

      3. associate-*r* 31.2%

         \[\leadsto U + \left(2 \cdot \left(\ell \cdot J\right) + \color{blue}{\left(-0.25 \cdot {K}^{2}\right) \cdot \left(\ell \cdot J\right)}\right) \]

      4. distribute-rgt-out 31.2%

         \[\leadsto U + \color{blue}{\left(\ell \cdot J\right) \cdot \left(2 + -0.25 \cdot {K}^{2}\right)} \]

      5. *-commutative 31.2%

         \[\leadsto U + \left(\ell \cdot J\right) \cdot \left(2 + \color{blue}{{K}^{2} \cdot -0.25}\right) \]

      6. unpow2 31.2%

         \[\leadsto U + \left(\ell \cdot J\right) \cdot \left(2 + \color{blue}{\left(K \cdot K\right)} \cdot -0.25\right) \]

   7. Simplified 31.2%

      \[\leadsto \color{blue}{U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{-10}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq 470:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 1.06 \cdot 10^{+85}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]

Alternative 9: 59.5% accurate, 18.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -160 \lor \neg \left(\ell \leq 3800\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -160.0) (not (<= l 3800.0)))
   (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
   (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -160.0) || !(l <= 3800.0)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-160.0d0)) .or. (.not. (l <= 3800.0d0))) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -160.0) || !(l <= 3800.0)) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -160.0) or not (l <= 3800.0):
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -160.0) || !(l <= 3800.0))
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -160.0) || ~((l <= 3800.0)))
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -160.0], N[Not[LessEqual[l, 3800.0]], $MachinePrecision]], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -160 or 3800 < 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. Taylor expanded in l around 0 28.4%

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

      1. *-commutative 28.4%

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

      2. associate-*l* 28.4%

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

      3. *-commutative 28.4%

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

   4. Simplified 28.4%

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

   5. Taylor expanded in K around 0 15.8%

      \[\leadsto \color{blue}{2 \cdot \left(\ell \cdot J\right) + \left(-0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right) + U\right)} \]
   6. Step-by-step derivation

      1. associate-+r+ 15.8%

         \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right) + U} \]

      2. +-commutative 15.8%

         \[\leadsto \color{blue}{U + \left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} \]

      3. associate-*r* 15.8%

         \[\leadsto U + \left(2 \cdot \left(\ell \cdot J\right) + \color{blue}{\left(-0.25 \cdot {K}^{2}\right) \cdot \left(\ell \cdot J\right)}\right) \]

      4. distribute-rgt-out 34.3%

         \[\leadsto U + \color{blue}{\left(\ell \cdot J\right) \cdot \left(2 + -0.25 \cdot {K}^{2}\right)} \]

      5. *-commutative 34.3%

         \[\leadsto U + \left(\ell \cdot J\right) \cdot \left(2 + \color{blue}{{K}^{2} \cdot -0.25}\right) \]

      6. unpow2 34.3%

         \[\leadsto U + \left(\ell \cdot J\right) \cdot \left(2 + \color{blue}{\left(K \cdot K\right)} \cdot -0.25\right) \]

   7. Simplified 34.3%

      \[\leadsto \color{blue}{U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)} \]

   if -160 < l < 3800

   1. Initial program 72.3%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.3%

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

      3. *-commutative 99.3%

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

   4. Simplified 99.3%

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

   5. Taylor expanded in K around 0 85.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -160 \lor \neg \left(\ell \leq 3800\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 10: 54.2% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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 * (l * j))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.2%

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

2. Taylor expanded in l around 0 66.3%

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

   1. *-commutative 66.3%

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

   2. associate-*l* 66.3%

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

   3. *-commutative 66.3%

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

4. Simplified 66.3%

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

5. Taylor expanded in K around 0 54.8%

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

6. Final simplification 54.8%

   \[\leadsto U + 2 \cdot \left(\ell \cdot J\right) \]

Alternative 11: 39.5% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 92:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l 92.0) U (* U U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 92.0) {
		tmp = U;
	} else {
		tmp = 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 <= 92.0d0) then
        tmp = u
    else
        tmp = 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 <= 92.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 92.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 92.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 92.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 92.0], U, N[(U * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 92

   1. Initial program 81.3%

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

      1. associate-*l* 81.3%

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

      2. fma-def 81.3%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in J around 0 47.9%

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

   if 92 < l

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 20.4%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 92:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 12: 36.6% 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 85.2%

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

   1. associate-*l* 85.2%

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

   2. fma-def 85.2%

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

3. Simplified 85.2%

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

4. Taylor expanded in J around 0 38.4%

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

5. Final simplification 38.4%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023243 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))