Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.7% → 99.2%

Time: 10.2s

Alternatives: 21

Speedup: 2.6×

• 48.9% 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: 85.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := e^{\ell} - t\_0\\ t_2 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;t\_2 \cdot \left(J \cdot \left(27 - t\_0\right)\right) + U\\ \mathbf{elif}\;t\_1 \leq 10^{-15}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_2 \cdot \left(J \cdot \left(e^{\ell} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (- (exp l) t_0)) (t_2 (cos (/ K 2.0))))
   (if (<= t_1 (- INFINITY))
     (+ (* t_2 (* J (- 27.0 t_0))) U)
     (if (<= t_1 1e-15)
       (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
       (+ U (* t_2 (* J (+ (exp l) -1.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = exp(l) - t_0;
	double t_2 = cos((K / 2.0));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = (t_2 * (J * (27.0 - t_0))) + U;
	} else if (t_1 <= 1e-15) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else {
		tmp = U + (t_2 * (J * (exp(l) + -1.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.exp(l) - t_0;
	double t_2 = Math.cos((K / 2.0));
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = (t_2 * (J * (27.0 - t_0))) + U;
	} else if (t_1 <= 1e-15) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else {
		tmp = U + (t_2 * (J * (Math.exp(l) + -1.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.exp(l) - t_0
	t_2 = math.cos((K / 2.0))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = (t_2 * (J * (27.0 - t_0))) + U
	elif t_1 <= 1e-15:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	else:
		tmp = U + (t_2 * (J * (math.exp(l) + -1.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = Float64(exp(l) - t_0)
	t_2 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(Float64(t_2 * Float64(J * Float64(27.0 - t_0))) + U);
	elseif (t_1 <= 1e-15)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U + Float64(t_2 * Float64(J * Float64(exp(l) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = exp(l) - t_0;
	t_2 = cos((K / 2.0));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = (t_2 * (J * (27.0 - t_0))) + U;
	elseif (t_1 <= 1e-15)
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	else
		tmp = U + (t_2 * (J * (exp(l) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t$95$2 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$1, 1e-15], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$2 * N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.0000000000000001e-15

   1. Initial program 73.3%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if 1.0000000000000001e-15 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 10^{-15}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 95.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{-\ell}\\ \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+106}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5.2:\\ \;\;\;\;U + J \cdot \left(27 - t\_1\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;U + \left(e^{\ell} - t\_1\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (exp (- l))))
   (if (<= l -1.25e+106)
     (+ U (* t_0 (* J (* l (+ 2.0 (* l (+ 0.5 (* l 0.16666666666666666))))))))
     (if (<= l -5.2)
       (+ U (* J (- 27.0 t_1)))
       (if (<= l 114000000.0)
         (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
         (if (<= l 2.6e+77)
           (+ U (* (- (exp l) t_1) J))
           (+
            U
            (*
             t_0
             (*
              J
              (*
               l
               (+
                2.0
                (*
                 l
                 (+
                  0.5
                  (*
                   l
                   (+
                    0.16666666666666666
                    (* l 0.041666666666666664))))))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(-l);
	double tmp;
	if (l <= -1.25e+106) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))));
	} else if (l <= -5.2) {
		tmp = U + (J * (27.0 - t_1));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else if (l <= 2.6e+77) {
		tmp = U + ((exp(l) - t_1) * J);
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664)))))))));
	}
	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 / 2.0d0))
    t_1 = exp(-l)
    if (l <= (-1.25d+106)) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (l * (0.5d0 + (l * 0.16666666666666666d0)))))))
    else if (l <= (-5.2d0)) then
        tmp = u + (j * (27.0d0 - t_1))
    else if (l <= 114000000.0d0) then
        tmp = u + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
    else if (l <= 2.6d+77) then
        tmp = u + ((exp(l) - t_1) * j)
    else
        tmp = u + (t_0 * (j * (l * (2.0d0 + (l * (0.5d0 + (l * (0.16666666666666666d0 + (l * 0.041666666666666664d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(-l);
	double tmp;
	if (l <= -1.25e+106) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))));
	} else if (l <= -5.2) {
		tmp = U + (J * (27.0 - t_1));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else if (l <= 2.6e+77) {
		tmp = U + ((Math.exp(l) - t_1) * J);
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664)))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(-l)
	tmp = 0
	if l <= -1.25e+106:
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))))
	elif l <= -5.2:
		tmp = U + (J * (27.0 - t_1))
	elif l <= 114000000.0:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	elif l <= 2.6e+77:
		tmp = U + ((math.exp(l) - t_1) * J)
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664)))))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = exp(Float64(-l))
	tmp = 0.0
	if (l <= -1.25e+106)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666))))))));
	elseif (l <= -5.2)
		tmp = Float64(U + Float64(J * Float64(27.0 - t_1)));
	elseif (l <= 114000000.0)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	elseif (l <= 2.6e+77)
		tmp = Float64(U + Float64(Float64(exp(l) - t_1) * J));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * Float64(0.16666666666666666 + Float64(l * 0.041666666666666664))))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(-l);
	tmp = 0.0;
	if (l <= -1.25e+106)
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))));
	elseif (l <= -5.2)
		tmp = U + (J * (27.0 - t_1));
	elseif (l <= 114000000.0)
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	elseif (l <= 2.6e+77)
		tmp = U + ((exp(l) - t_1) * J);
	else
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664)))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[l, -1.25e+106], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5.2], N[(U + N[(J * N[(27.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 114000000.0], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+77], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - t$95$1), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * N[(0.16666666666666666 + N[(l * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.25e106

   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 39.4%

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

      1. neg-mul-1 39.4%

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

      2. unsub-neg 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + 0.16666666666666666 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -1.25e106 < l < -5.20000000000000018

   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 80.0%

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

      1. *-commutative 80.0%

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

   5. Simplified 80.0%

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

   7. Applied egg-rr 80.0%

      \[\leadsto \left(\color{blue}{27} - e^{-\ell}\right) \cdot J + U \]

   if -5.20000000000000018 < l < 1.14e8

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r* 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. *-commutative 99.2%

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

      7. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1.14e8 < l < 2.6000000000000002e77

   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 76.5%

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

      1. *-commutative 76.5%

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

   5. Simplified 76.5%

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

   if 2.6000000000000002e77 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot \ell\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \color{blue}{\ell \cdot 0.041666666666666664}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 5 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+106}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -5.2:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -10.5:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(e^{\ell} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= l -1.4e+106)
     (+ U (* t_0 (* J (* l (+ 2.0 (* l (+ 0.5 (* l 0.16666666666666666))))))))
     (if (<= l -10.5)
       (+ U (* J (- 27.0 (exp (- l)))))
       (if (<= l 5.6e-10)
         (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
         (+ U (* t_0 (* J (+ (exp l) -1.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (l <= -1.4e+106) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))));
	} else if (l <= -10.5) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 5.6e-10) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else {
		tmp = U + (t_0 * (J * (exp(l) + -1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (l <= (-1.4d+106)) then
        tmp = u + (t_0 * (j * (l * (2.0d0 + (l * (0.5d0 + (l * 0.16666666666666666d0)))))))
    else if (l <= (-10.5d0)) then
        tmp = u + (j * (27.0d0 - exp(-l)))
    else if (l <= 5.6d-10) then
        tmp = u + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
    else
        tmp = u + (t_0 * (j * (exp(l) + (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -1.4e+106) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))));
	} else if (l <= -10.5) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 5.6e-10) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else {
		tmp = U + (t_0 * (J * (Math.exp(l) + -1.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if l <= -1.4e+106:
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))))
	elif l <= -10.5:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 5.6e-10:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	else:
		tmp = U + (t_0 * (J * (math.exp(l) + -1.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -1.4e+106)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666))))))));
	elseif (l <= -10.5)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 5.6e-10)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(exp(l) + -1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -1.4e+106)
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))));
	elseif (l <= -10.5)
		tmp = U + (J * (27.0 - exp(-l)));
	elseif (l <= 5.6e-10)
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	else
		tmp = U + (t_0 * (J * (exp(l) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.4e+106], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -10.5], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e-10], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[Exp[l], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.39999999999999996e106

   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 39.4%

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

      1. neg-mul-1 39.4%

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

      2. unsub-neg 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + 0.16666666666666666 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -1.39999999999999996e106 < l < -10.5

   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

   3. Taylor expanded in K around 0 80.0%

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

      1. *-commutative 80.0%

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

   5. Simplified 80.0%

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

   7. Applied egg-rr 80.0%

      \[\leadsto \left(\color{blue}{27} - e^{-\ell}\right) \cdot J + U \]

   if -10.5 < l < 5.60000000000000031e-10

   1. Initial program 73.3%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if 5.60000000000000031e-10 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \left(e^{\ell} - \color{blue}{1}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -10.5:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{-10}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+106}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4.2:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= l -1.25e+106)
     (+ U (* t_0 (* J (* l (+ 2.0 (* l (+ 0.5 (* l 0.16666666666666666))))))))
     (if (<= l -4.2)
       (+ U (* J (- 27.0 (exp (- l)))))
       (if (<= l 114000000.0)
         (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
         (if (<= l 2.6e+77)
           (+ U (* J (+ l (expm1 l))))
           (+
            U
            (*
             t_0
             (*
              J
              (*
               l
               (+
                2.0
                (*
                 l
                 (+
                  0.5
                  (*
                   l
                   (+
                    0.16666666666666666
                    (* l 0.041666666666666664))))))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (l <= -1.25e+106) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))));
	} else if (l <= -4.2) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else if (l <= 2.6e+77) {
		tmp = U + (J * (l + expm1(l)));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664)))))))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -1.25e+106) {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))));
	} else if (l <= -4.2) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else if (l <= 2.6e+77) {
		tmp = U + (J * (l + Math.expm1(l)));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664)))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if l <= -1.25e+106:
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666)))))))
	elif l <= -4.2:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 114000000.0:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	elif l <= 2.6e+77:
		tmp = U + (J * (l + math.expm1(l)))
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664)))))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -1.25e+106)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666))))))));
	elseif (l <= -4.2)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 114000000.0)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	elseif (l <= 2.6e+77)
		tmp = Float64(U + Float64(J * Float64(l + expm1(l))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * Float64(0.16666666666666666 + Float64(l * 0.041666666666666664))))))))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -1.25e+106], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4.2], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 114000000.0], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.6e+77], N[(U + N[(J * N[(l + N[(Exp[l] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * N[(0.16666666666666666 + N[(l * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.25e106

   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 39.4%

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

      1. neg-mul-1 39.4%

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

      2. unsub-neg 39.4%

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

   5. Simplified 39.4%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + 0.16666666666666666 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -1.25e106 < l < -4.20000000000000018

   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 80.0%

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

      1. *-commutative 80.0%

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

   5. Simplified 80.0%

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

   7. Applied egg-rr 80.0%

      \[\leadsto \left(\color{blue}{27} - e^{-\ell}\right) \cdot J + U \]

   if -4.20000000000000018 < l < 1.14e8

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r* 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. *-commutative 99.2%

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

      7. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1.14e8 < l < 2.6000000000000002e77

   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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 76.5%

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

      1. associate--l+ 76.5%

         \[\leadsto J \cdot \color{blue}{\left(\ell + \left(e^{\ell} - 1\right)\right)} + U \]

      2. expm1-define 76.5%

         \[\leadsto J \cdot \left(\ell + \color{blue}{\mathsf{expm1}\left(\ell\right)}\right) + U \]

   8. Simplified 76.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)} + U \]

   if 2.6000000000000002e77 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot \ell\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \color{blue}{\ell \cdot 0.041666666666666664}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 5 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.25 \cdot 10^{+106}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -4.2:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -9.2:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (* l (+ 2.0 (* l (+ 0.5 (* l 0.16666666666666666))))))))))
   (if (<= l -1.4e+106)
     t_0
     (if (<= l -9.2)
       (+ U (* J (- 27.0 (exp (- l)))))
       (if (<= l 114000000.0)
         (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
         (if (<= l 1.05e+103) (+ U (* J (+ l (expm1 l)))) 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 + (l * (0.5 + (l * 0.16666666666666666)))))));
	double tmp;
	if (l <= -1.4e+106) {
		tmp = t_0;
	} else if (l <= -9.2) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else if (l <= 1.05e+103) {
		tmp = U + (J * (l + expm1(l)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

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 + (l * (0.5 + (l * 0.16666666666666666)))))));
	double tmp;
	if (l <= -1.4e+106) {
		tmp = t_0;
	} else if (l <= -9.2) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else if (l <= 1.05e+103) {
		tmp = U + (J * (l + Math.expm1(l)));
	} 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 + (l * (0.5 + (l * 0.16666666666666666)))))))
	tmp = 0
	if l <= -1.4e+106:
		tmp = t_0
	elif l <= -9.2:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 114000000.0:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	elif l <= 1.05e+103:
		tmp = U + (J * (l + math.expm1(l)))
	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(l * Float64(0.5 + Float64(l * 0.16666666666666666))))))))
	tmp = 0.0
	if (l <= -1.4e+106)
		tmp = t_0;
	elseif (l <= -9.2)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 114000000.0)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	elseif (l <= 1.05e+103)
		tmp = Float64(U + Float64(J * Float64(l + expm1(l))));
	else
		tmp = t_0;
	end
	return 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[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.4e+106], t$95$0, If[LessEqual[l, -9.2], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 114000000.0], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.05e+103], N[(U + N[(J * N[(l + N[(Exp[l] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.39999999999999996e106 or 1.0500000000000001e103 < 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 68.0%

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

      1. neg-mul-1 68.0%

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

      2. unsub-neg 68.0%

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

   5. Simplified 68.0%

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

   6. Taylor expanded in l around 0 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + 0.16666666666666666 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -1.39999999999999996e106 < l < -9.1999999999999993

   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 80.0%

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

      1. *-commutative 80.0%

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

   5. Simplified 80.0%

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

   7. Applied egg-rr 80.0%

      \[\leadsto \left(\color{blue}{27} - e^{-\ell}\right) \cdot J + U \]

   if -9.1999999999999993 < l < 1.14e8

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r* 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. *-commutative 99.2%

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

      7. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1.14e8 < l < 1.0500000000000001e103

   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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 79.2%

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

      1. associate--l+ 79.2%

         \[\leadsto J \cdot \color{blue}{\left(\ell + \left(e^{\ell} - 1\right)\right)} + U \]

      2. expm1-define 79.2%

         \[\leadsto J \cdot \left(\ell + \color{blue}{\mathsf{expm1}\left(\ell\right)}\right) + U \]

   8. Simplified 79.2%

      \[\leadsto \color{blue}{J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -9.2:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+103}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.6:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.6)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 114000000.0)
     (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
     (if (<= l 1.9e+154)
       (+ U (* J (+ l (expm1 l))))
       (+ U (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* l 0.5))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.6) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else if (l <= 1.9e+154) {
		tmp = U + (J * (l + expm1(l)));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (l * 0.5)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.6) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else if (l <= 1.9e+154) {
		tmp = U + (J * (l + Math.expm1(l)));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (l * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.6:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 114000000.0:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	elif l <= 1.9e+154:
		tmp = U + (J * (l + math.expm1(l)))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (l * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.6)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 114000000.0)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	elseif (l <= 1.9e+154)
		tmp = Float64(U + Float64(J * Float64(l + expm1(l))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(l * 0.5))))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.6], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 114000000.0], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.9e+154], N[(U + N[(J * N[(l + N[(Exp[l] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.5999999999999996

   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 73.2%

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

      1. *-commutative 73.2%

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

   5. Simplified 73.2%

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

   7. Applied egg-rr 73.2%

      \[\leadsto \left(\color{blue}{27} - e^{-\ell}\right) \cdot J + U \]

   if -4.5999999999999996 < l < 1.14e8

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r* 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. *-commutative 99.2%

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

      7. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1.14e8 < l < 1.8999999999999999e154

   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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 83.3%

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

      1. associate--l+ 83.3%

         \[\leadsto J \cdot \color{blue}{\left(\ell + \left(e^{\ell} - 1\right)\right)} + U \]

      2. expm1-define 83.3%

         \[\leadsto J \cdot \left(\ell + \color{blue}{\mathsf{expm1}\left(\ell\right)}\right) + U \]

   8. Simplified 83.3%

      \[\leadsto \color{blue}{J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)} + U \]

   if 1.8999999999999999e154 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

   8. Simplified 100.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot 0.5\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.6:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 87.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.2)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 114000000.0)
     (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
     (+ U (* J (+ l (expm1 l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.2) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else {
		tmp = U + (J * (l + expm1(l)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.2) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else {
		tmp = U + (J * (l + Math.expm1(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.2:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 114000000.0:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	else:
		tmp = U + (J * (l + math.expm1(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.2)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 114000000.0)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U + Float64(J * Float64(l + expm1(l))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.2], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 114000000.0], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l + N[(Exp[l] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.20000000000000018

   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 73.2%

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

      1. *-commutative 73.2%

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

   5. Simplified 73.2%

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

   7. Applied egg-rr 73.2%

      \[\leadsto \left(\color{blue}{27} - e^{-\ell}\right) \cdot J + U \]

   if -5.20000000000000018 < l < 1.14e8

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*r* 99.2%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. *-commutative 99.2%

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

      7. *-commutative 99.2%

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

   5. Simplified 99.2%

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

   if 1.14e8 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 81.5%

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

      1. associate--l+ 81.5%

         \[\leadsto J \cdot \color{blue}{\left(\ell + \left(e^{\ell} - 1\right)\right)} + U \]

      2. expm1-define 81.5%

         \[\leadsto J \cdot \left(\ell + \color{blue}{\mathsf{expm1}\left(\ell\right)}\right) + U \]

   8. Simplified 81.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 87.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.9:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.9)
   (+ U (* J (- 27.0 (exp (- l)))))
   (if (<= l 114000000.0)
     (+ U (* J (* 2.0 (* l (cos (* K 0.5))))))
     (+ U (* J (+ l (expm1 l)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.9) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + (J * (2.0 * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l + expm1(l)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.9) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else if (l <= 114000000.0) {
		tmp = U + (J * (2.0 * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l + Math.expm1(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.9:
		tmp = U + (J * (27.0 - math.exp(-l)))
	elif l <= 114000000.0:
		tmp = U + (J * (2.0 * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (l + math.expm1(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.9)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	elseif (l <= 114000000.0)
		tmp = Float64(U + Float64(J * Float64(2.0 * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(l + expm1(l))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.9], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 114000000.0], 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[(Exp[l] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.9000000000000004

   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 73.2%

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

      1. *-commutative 73.2%

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

   5. Simplified 73.2%

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

   7. Applied egg-rr 73.2%

      \[\leadsto \left(\color{blue}{27} - e^{-\ell}\right) \cdot J + U \]

   if -4.9000000000000004 < l < 1.14e8

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. *-commutative 99.1%

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

   5. Simplified 99.1%

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

   if 1.14e8 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 81.5%

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

      1. associate--l+ 81.5%

         \[\leadsto J \cdot \color{blue}{\left(\ell + \left(e^{\ell} - 1\right)\right)} + U \]

      2. expm1-define 81.5%

         \[\leadsto J \cdot \left(\ell + \color{blue}{\mathsf{expm1}\left(\ell\right)}\right) + U \]

   8. Simplified 81.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.9:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{elif}\;\ell \leq 114000000:\\ \;\;\;\;U + J \cdot \left(2 \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.45:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.45)
   (+ U (* J (- 27.0 (exp (- l)))))
   (+ U (* J (+ l (expm1 l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45) {
		tmp = U + (J * (27.0 - exp(-l)));
	} else {
		tmp = U + (J * (l + expm1(l)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.45) {
		tmp = U + (J * (27.0 - Math.exp(-l)));
	} else {
		tmp = U + (J * (l + Math.expm1(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.45:
		tmp = U + (J * (27.0 - math.exp(-l)))
	else:
		tmp = U + (J * (l + math.expm1(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.45)
		tmp = Float64(U + Float64(J * Float64(27.0 - exp(Float64(-l)))));
	else
		tmp = Float64(U + Float64(J * Float64(l + expm1(l))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.45], N[(U + N[(J * N[(27.0 - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l + N[(Exp[l] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.4500000000000002

   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 73.2%

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

      1. *-commutative 73.2%

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

   5. Simplified 73.2%

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

   7. Applied egg-rr 73.2%

      \[\leadsto \left(\color{blue}{27} - e^{-\ell}\right) \cdot J + U \]

   if -2.4500000000000002 < l

   1. Initial program 82.9%

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

   3. Taylor expanded in l around 0 82.9%

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

      1. neg-mul-1 82.9%

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

      2. unsub-neg 82.9%

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

   5. Simplified 82.9%

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

   6. Taylor expanded in K around 0 75.6%

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

      1. associate--l+ 77.8%

         \[\leadsto J \cdot \color{blue}{\left(\ell + \left(e^{\ell} - 1\right)\right)} + U \]

      2. expm1-define 85.7%

         \[\leadsto J \cdot \left(\ell + \color{blue}{\mathsf{expm1}\left(\ell\right)}\right) + U \]

   8. Simplified 85.7%

      \[\leadsto \color{blue}{J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.45:\\ \;\;\;\;U + J \cdot \left(27 - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 114000000:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 114000000.0)
   (+ U (* J (* l (+ 2.0 (* l (+ 0.5 (* l 0.16666666666666666)))))))
   (+ U (* J (+ l (expm1 l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 114000000.0) {
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	} else {
		tmp = U + (J * (l + expm1(l)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 114000000.0) {
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	} else {
		tmp = U + (J * (l + Math.expm1(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 114000000.0:
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))))
	else:
		tmp = U + (J * (l + math.expm1(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 114000000.0)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666)))))));
	else
		tmp = Float64(U + Float64(J * Float64(l + expm1(l))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 114000000.0], N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l + N[(Exp[l] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.14e8

   1. Initial program 83.5%

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

   3. Taylor expanded in K around 0 72.7%

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

      1. *-commutative 72.7%

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

   5. Simplified 72.7%

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

   6. Taylor expanded in l around 0 52.5%

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

      1. neg-mul-1 57.7%

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

      2. unsub-neg 57.7%

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

   8. Simplified 52.5%

      \[\leadsto \left(e^{\ell} - \color{blue}{\left(1 - \ell\right)}\right) \cdot J + U \]

   9. Taylor expanded in l around 0 74.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + 0.16666666666666666 \cdot \ell\right)\right)\right)} \cdot J + U \]
   10. Step-by-step derivation

       1. *-commutative 90.1%

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. Simplified 74.9%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)} \cdot J + U \]

   if 1.14e8 < 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%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in K around 0 81.5%

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

      1. associate--l+ 81.5%

         \[\leadsto J \cdot \color{blue}{\left(\ell + \left(e^{\ell} - 1\right)\right)} + U \]

      2. expm1-define 81.5%

         \[\leadsto J \cdot \left(\ell + \color{blue}{\mathsf{expm1}\left(\ell\right)}\right) + U \]

   8. Simplified 81.5%

      \[\leadsto \color{blue}{J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 114000000:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell + \mathsf{expm1}\left(\ell\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 3.8 \cdot 10^{-12}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 3.8e-12)
   (+ U (* J (* l (+ 2.0 (* l (+ 0.5 (* l 0.16666666666666666)))))))
   (+
    U
    (*
     J
     (*
      l
      (+
       2.0
       (*
        l
        (+ 0.5 (* l (+ 0.16666666666666666 (* l 0.041666666666666664)))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 3.8e-12) {
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	} else {
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664))))))));
	}
	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 <= 3.8d-12) then
        tmp = u + (j * (l * (2.0d0 + (l * (0.5d0 + (l * 0.16666666666666666d0))))))
    else
        tmp = u + (j * (l * (2.0d0 + (l * (0.5d0 + (l * (0.16666666666666666d0 + (l * 0.041666666666666664d0))))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 3.8e-12) {
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	} else {
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664))))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 3.8e-12:
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))))
	else:
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664))))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 3.8e-12)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666)))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * Float64(0.16666666666666666 + Float64(l * 0.041666666666666664)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 3.8e-12)
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	else
		tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * (0.16666666666666666 + (l * 0.041666666666666664))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 3.8e-12], N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * N[(0.16666666666666666 + N[(l * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 3.79999999999999996e-12

   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 K around 0 76.2%

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

      1. *-commutative 76.2%

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

   5. Simplified 76.2%

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

   6. Taylor expanded in l around 0 56.7%

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

      1. neg-mul-1 64.5%

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

      2. unsub-neg 64.5%

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

   8. Simplified 56.7%

      \[\leadsto \left(e^{\ell} - \color{blue}{\left(1 - \ell\right)}\right) \cdot J + U \]

   9. Taylor expanded in l around 0 74.5%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + 0.16666666666666666 \cdot \ell\right)\right)\right)} \cdot J + U \]
   10. Step-by-step derivation

       1. *-commutative 86.3%

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. Simplified 74.5%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)} \cdot J + U \]

   if 3.79999999999999996e-12 < K

   1. Initial program 93.0%

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

   3. Taylor expanded in K around 0 71.8%

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

      1. *-commutative 71.8%

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

   5. Simplified 71.8%

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

   6. Taylor expanded in l around 0 67.9%

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

      1. neg-mul-1 78.5%

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

      2. unsub-neg 78.5%

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

   8. Simplified 67.9%

      \[\leadsto \left(e^{\ell} - \color{blue}{\left(1 - \ell\right)}\right) \cdot J + U \]

   9. Taylor expanded in l around 0 74.4%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + 0.041666666666666664 \cdot \ell\right)\right)\right)\right)} \cdot J + U \]
   10. Step-by-step derivation

       1. *-commutative 70.9%

          \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \color{blue}{\ell \cdot 0.041666666666666664}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   11. Simplified 74.4%

       \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)} \cdot J + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 3.8 \cdot 10^{-12}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot \left(0.16666666666666666 + \ell \cdot 0.041666666666666664\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.6% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+119}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 5e+119)
   (* U (+ 1.0 (* 2.0 (* J (/ l U)))))
   (+ U (* J (* l (+ 2.0 (* l 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 5e+119) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (J * (l * (2.0 + (l * 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 <= 5d+119) then
        tmp = u * (1.0d0 + (2.0d0 * (j * (l / u))))
    else
        tmp = u + (j * (l * (2.0d0 + (l * 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 <= 5e+119) {
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	} else {
		tmp = U + (J * (l * (2.0 + (l * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 5e+119:
		tmp = U * (1.0 + (2.0 * (J * (l / U))))
	else:
		tmp = U + (J * (l * (2.0 + (l * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 5e+119)
		tmp = Float64(U * Float64(1.0 + Float64(2.0 * Float64(J * Float64(l / U)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 5e+119)
		tmp = U * (1.0 + (2.0 * (J * (l / U))));
	else
		tmp = U + (J * (l * (2.0 + (l * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 5e+119], N[(U * N[(1.0 + N[(2.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.9999999999999999e119

   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 K around 0 73.5%

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

      1. *-commutative 73.5%

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

   5. Simplified 73.5%

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

   6. Taylor expanded in l around 0 56.1%

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

   7. Taylor expanded in U around inf 59.4%

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

      1. associate-/l* 61.6%

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

   9. Simplified 61.6%

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

   if 4.9999999999999999e119 < 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 K around 0 83.8%

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

      1. *-commutative 83.8%

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

   5. Simplified 83.8%

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

   6. Taylor expanded in l around 0 83.8%

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

      1. neg-mul-1 100.0%

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

      2. unsub-neg 100.0%

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

   8. Simplified 83.8%

      \[\leadsto \left(e^{\ell} - \color{blue}{\left(1 - \ell\right)}\right) \cdot J + U \]

   9. Taylor expanded in l around 0 76.1%

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

       1. *-commutative 92.3%

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

   11. Simplified 76.1%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5 \cdot 10^{+119}:\\ \;\;\;\;U \cdot \left(1 + 2 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 41.1% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-15} \lor \neg \left(\ell \leq 3.9 \cdot 10^{+58}\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.2e-15) (not (<= l 3.9e+58))) (* U (- U -4.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.2e-15) || !(l <= 3.9e+58)) {
		tmp = U * (U - -4.0);
	} 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 <= (-1.2d-15)) .or. (.not. (l <= 3.9d+58))) then
        tmp = u * (u - (-4.0d0))
    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 <= -1.2e-15) || !(l <= 3.9e+58)) {
		tmp = U * (U - -4.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.2e-15) or not (l <= 3.9e+58):
		tmp = U * (U - -4.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.2e-15) || !(l <= 3.9e+58))
		tmp = Float64(U * Float64(U - -4.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.2e-15) || ~((l <= 3.9e+58)))
		tmp = U * (U - -4.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.2e-15], N[Not[LessEqual[l, 3.9e+58]], $MachinePrecision]], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.19999999999999997e-15 or 3.9000000000000001e58 < l

   1. Initial program 99.1%

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

      1. associate-*l* 99.1%

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

      2. fma-define 99.1%

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

   3. Simplified 99.1%

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

   6. Applied egg-rr 11.9%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if -1.19999999999999997e-15 < l < 3.9000000000000001e58

   1. Initial program 76.0%

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

      1. associate-*l* 76.0%

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

      2. fma-define 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in J around 0 68.1%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{-15} \lor \neg \left(\ell \leq 3.9 \cdot 10^{+58}\right):\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.5% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ U (* J (* l (+ 2.0 (* l (+ 0.5 (* l 0.16666666666666666))))))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))));
}

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 + (j * (l * (2.0d0 + (l * (0.5d0 + (l * 0.16666666666666666d0))))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * (2.0 + (l * (0.5 + (l * 0.16666666666666666))))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * N[(2.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 K around 0 75.0%

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

   1. *-commutative 75.0%

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

5. Simplified 75.0%

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

6. Taylor expanded in l around 0 59.9%

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

   1. neg-mul-1 68.5%

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

   2. unsub-neg 68.5%

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

8. Simplified 59.9%

   \[\leadsto \left(e^{\ell} - \color{blue}{\left(1 - \ell\right)}\right) \cdot J + U \]

9. Taylor expanded in l around 0 71.8%

   \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + 0.16666666666666666 \cdot \ell\right)\right)\right)} \cdot J + U \]
10. Step-by-step derivation

    1. *-commutative 86.3%

       \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \color{blue}{\ell \cdot 0.16666666666666666}\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

11. Simplified 71.8%

    \[\leadsto \color{blue}{\left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)} \cdot J + U \]

12. Final simplification 71.8%

    \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)\right) \]
13. Add Preprocessing

Alternative 15: 55.5% accurate, 22.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 2.5e-18) (+ U (* J (* l 2.0))) (* J (+ (* l 2.0) (/ U J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.5e-18) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = J * ((l * 2.0) + (U / 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 <= 2.5d-18) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = j * ((l * 2.0d0) + (u / j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 2.5e-18) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = J * ((l * 2.0) + (U / J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 2.5e-18:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = J * ((l * 2.0) + (U / J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 2.5e-18)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(J * Float64(Float64(l * 2.0) + Float64(U / J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 2.5e-18)
		tmp = U + (J * (l * 2.0));
	else
		tmp = J * ((l * 2.0) + (U / J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 2.5e-18], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(U / J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 2.50000000000000018e-18

   1. Initial program 83.6%

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

   3. Taylor expanded in K around 0 73.2%

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

      1. *-commutative 73.2%

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

   5. Simplified 73.2%

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

   6. Taylor expanded in l around 0 62.7%

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

   if 2.50000000000000018e-18 < l

   1. Initial program 98.9%

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

   3. Taylor expanded in K around 0 79.8%

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

      1. *-commutative 79.8%

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

   5. Simplified 79.8%

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

   6. Taylor expanded in l around 0 18.9%

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

   7. Taylor expanded in J around inf 25.9%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;J \cdot \left(\ell \cdot 2 + \frac{U}{J}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.2% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-9} \lor \neg \left(\ell \leq 7 \cdot 10^{+73}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -5.2e-9) (not (<= l 7e+73))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.2e-9) || !(l <= 7e+73)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-5.2d-9)) .or. (.not. (l <= 7d+73))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -5.2e-9) || !(l <= 7e+73)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -5.2e-9) or not (l <= 7e+73):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -5.2e-9) || !(l <= 7e+73))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -5.2e-9) || ~((l <= 7e+73)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -5.2e-9], N[Not[LessEqual[l, 7e+73]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -5.2000000000000002e-9 or 7.00000000000000004e73 < l

   1. Initial program 99.7%

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

      1. associate-*l* 99.7%

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

      2. fma-define 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 12.5%

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

   if -5.2000000000000002e-9 < l < 7.00000000000000004e73

   1. Initial program 76.9%

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

      1. associate-*l* 76.9%

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

      2. fma-define 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in J around 0 64.2%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-9} \lor \neg \left(\ell \leq 7 \cdot 10^{+73}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 60.2% 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 87.7%

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

3. Taylor expanded in K around 0 75.0%

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

   1. *-commutative 75.0%

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

5. Simplified 75.0%

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

6. Taylor expanded in l around 0 51.0%

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

7. Taylor expanded in U around inf 54.2%

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

   1. associate-/l* 57.2%

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

9. Simplified 57.2%

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

Alternative 18: 54.3% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 K around 0 75.0%

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

   1. *-commutative 75.0%

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

5. Simplified 75.0%

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

6. Taylor expanded in l around 0 51.0%

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

7. Final simplification 51.0%

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

Alternative 19: 48.1% accurate, 62.4× speedup

Math

\[\begin{array}{l} \\ U + \ell \cdot J \end{array} \]

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U + (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 + (l * j)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 K around 0 75.0%

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

   1. *-commutative 75.0%

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

5. Simplified 75.0%

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

6. Taylor expanded in l around 0 59.9%

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

   1. neg-mul-1 68.5%

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

   2. unsub-neg 68.5%

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

8. Simplified 59.9%

   \[\leadsto \left(e^{\ell} - \color{blue}{\left(1 - \ell\right)}\right) \cdot J + U \]

9. Taylor expanded in l around inf 45.3%

   \[\leadsto \color{blue}{\ell} \cdot J + U \]

10. Final simplification 45.3%

    \[\leadsto U + \ell \cdot J \]
11. Add Preprocessing

Alternative 20: 37.1% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

   1. associate-*l* 87.7%

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

   2. fma-define 87.7%

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

3. Simplified 87.7%

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

5. Taylor expanded in J around 0 34.8%

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

Alternative 21: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -3.0)

C

double code(double J, double l, double K, double U) {
	return -3.0;
}

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 = -3.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -3.0;
}

Python

def code(J, l, K, U):
	return -3.0

Julia

function code(J, l, K, U)
	return -3.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -3.0;
end

Wolfram

code[J_, l_, K_, U_] := -3.0

Derivation

1. Initial program 87.7%

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

   1. associate-*l* 87.7%

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

   2. fma-define 87.7%

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

3. Simplified 87.7%

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

6. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{\sqrt[3]{U}} \]

8. Applied egg-rr 2.6%

   \[\leadsto \color{blue}{-3} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))