Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.4%

Time: 13.8s

Alternatives: 19

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (cos (/ K 2.0)) (* t_0 J)) U)
     (+ U (* J (* (cos (* K 0.5)) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0)) {
		tmp = (cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0)) {
		tmp = (Math.cos((K / 2.0)) * (t_0 * J)) + U;
	} else {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0):
		tmp = (math.cos((K / 2.0)) * (t_0 * J)) + U
	else:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 74.7%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 100.0%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.66:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.056:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + \left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.66)
     (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
     (if (<= t_0 -0.056)
       (+
        U
        (*
         (* J (pow l 3.0))
         (+ 0.3333333333333333 (* (* K K) -0.041666666666666664))))
       (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (t_0 <= -0.056) {
		tmp = U + ((J * pow(l, 3.0)) * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.66d0)) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else if (t_0 <= (-0.056d0)) then
        tmp = u + ((j * (l ** 3.0d0)) * (0.3333333333333333d0 + ((k * k) * (-0.041666666666666664d0))))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (t_0 <= -0.056) {
		tmp = U + ((J * Math.pow(l, 3.0)) * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.66:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif t_0 <= -0.056:
		tmp = U + ((J * math.pow(l, 3.0)) * (0.3333333333333333 + ((K * K) * -0.041666666666666664)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.66)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (t_0 <= -0.056)
		tmp = Float64(U + Float64(Float64(J * (l ^ 3.0)) * Float64(0.3333333333333333 + Float64(Float64(K * K) * -0.041666666666666664))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.66)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	elseif (t_0 <= -0.056)
		tmp = U + ((J * (l ^ 3.0)) * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.66], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.056], N[(U + N[(N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 + N[(N[(K * K), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.660000000000000031

   1. Initial program 85.0%

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

   2. Taylor expanded in l around 0 77.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 77.9%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 77.9%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 77.9%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 77.9%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 77.9%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 77.9%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 77.9%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 77.9%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 77.9%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 77.9%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 77.9%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 77.9%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 77.9%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. associate-*r* 60.5%

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

   7. Simplified 60.5%

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

   if -0.660000000000000031 < (cos.f64 (/.f64 K 2)) < -0.0560000000000000012

   1. Initial program 98.1%

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

   2. Taylor expanded in l around 0 83.4%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 83.4%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 83.4%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 83.4%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 83.4%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 83.4%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 83.4%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 83.4%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 83.4%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 83.4%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 83.4%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 83.4%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 83.4%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 83.4%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 81.5%

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

      1. associate-*r* 81.5%

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

   7. Simplified 81.5%

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

   8. Taylor expanded in K around 0 77.4%

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

      1. unpow2 77.4%

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

   10. Simplified 77.4%

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

   if -0.0560000000000000012 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.2%

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

   2. Taylor expanded in l around 0 88.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 88.9%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 88.9%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 88.9%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 88.9%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 88.9%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 88.9%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 88.9%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 88.9%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 88.9%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 88.9%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 88.9%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 88.9%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 88.9%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in K around 0 84.8%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.66:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.056:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + \left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 3: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.66:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.015:\\ \;\;\;\;U + K \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot J\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.66)
     (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
     (if (<= t_0 -0.015)
       (+ U (* K (* -0.25 (* K (* l J)))))
       (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (t_0 <= -0.015) {
		tmp = U + (K * (-0.25 * (K * (l * J))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.66d0)) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else if (t_0 <= (-0.015d0)) then
        tmp = u + (k * ((-0.25d0) * (k * (l * j))))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.66) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (t_0 <= -0.015) {
		tmp = U + (K * (-0.25 * (K * (l * J))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.66:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif t_0 <= -0.015:
		tmp = U + (K * (-0.25 * (K * (l * J))))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.66)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (t_0 <= -0.015)
		tmp = Float64(U + Float64(K * Float64(-0.25 * Float64(K * Float64(l * J)))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.66)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	elseif (t_0 <= -0.015)
		tmp = U + (K * (-0.25 * (K * (l * J))));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.66], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.015], N[(U + N[(K * N[(-0.25 * N[(K * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.660000000000000031

   1. Initial program 85.0%

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

   2. Taylor expanded in l around 0 77.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 77.9%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 77.9%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 77.9%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 77.9%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 77.9%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 77.9%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 77.9%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 77.9%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 77.9%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 77.9%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 77.9%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 77.9%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 77.9%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. associate-*r* 60.5%

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

   7. Simplified 60.5%

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

   if -0.660000000000000031 < (cos.f64 (/.f64 K 2)) < -0.014999999999999999

   1. Initial program 98.2%

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

   2. Taylor expanded in l around 0 40.3%

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

   3. Taylor expanded in K around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. associate-*r* 45.3%

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

      3. distribute-rgt-out 53.5%

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

      4. *-commutative 53.5%

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

      5. unpow2 53.5%

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

   5. Simplified 53.5%

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

   6. Taylor expanded in K around inf 53.5%

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

      1. associate-*r* 53.5%

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

      2. *-commutative 53.5%

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

      3. unpow2 53.5%

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

      4. associate-*r* 53.5%

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

      5. associate-*l* 61.7%

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

      6. *-commutative 61.7%

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

      7. associate-*l* 61.7%

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

   8. Simplified 61.7%

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

   if -0.014999999999999999 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.1%

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

   2. Taylor expanded in l around 0 88.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 88.8%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 88.8%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 88.8%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 88.8%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 88.8%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 88.8%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 88.8%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 88.8%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 88.8%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 88.8%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 88.8%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 88.8%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 88.8%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in K around 0 85.2%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.66:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.015:\\ \;\;\;\;U + K \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot J\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 4: 94.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + \left(t_0 \cdot 0.3333333333333333\right) \cdot \left(J \cdot {\ell}^{3}\right)\\ t_2 := U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 11.2:\\ \;\;\;\;U + J \cdot \left(t_0 \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (* t_0 0.3333333333333333) (* J (pow l 3.0)))))
        (t_2
         (+ U (* (+ (* -0.125 (* K K)) 1.0) (* (- (exp l) (exp (- l))) J)))))
   (if (<= l -5.6e+102)
     t_1
     (if (<= l -1.35e+19)
       t_2
       (if (<= l 11.2)
         (+ U (* J (* t_0 (* l 2.0))))
         (if (<= l 5.6e+102) t_2 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + ((t_0 * 0.3333333333333333) * (J * pow(l, 3.0)));
	double t_2 = U + (((-0.125 * (K * K)) + 1.0) * ((exp(l) - exp(-l)) * J));
	double tmp;
	if (l <= -5.6e+102) {
		tmp = t_1;
	} else if (l <= -1.35e+19) {
		tmp = t_2;
	} else if (l <= 11.2) {
		tmp = U + (J * (t_0 * (l * 2.0)));
	} else if (l <= 5.6e+102) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((t_0 * 0.3333333333333333d0) * (j * (l ** 3.0d0)))
    t_2 = u + ((((-0.125d0) * (k * k)) + 1.0d0) * ((exp(l) - exp(-l)) * j))
    if (l <= (-5.6d+102)) then
        tmp = t_1
    else if (l <= (-1.35d+19)) then
        tmp = t_2
    else if (l <= 11.2d0) then
        tmp = u + (j * (t_0 * (l * 2.0d0)))
    else if (l <= 5.6d+102) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + ((t_0 * 0.3333333333333333) * (J * Math.pow(l, 3.0)));
	double t_2 = U + (((-0.125 * (K * K)) + 1.0) * ((Math.exp(l) - Math.exp(-l)) * J));
	double tmp;
	if (l <= -5.6e+102) {
		tmp = t_1;
	} else if (l <= -1.35e+19) {
		tmp = t_2;
	} else if (l <= 11.2) {
		tmp = U + (J * (t_0 * (l * 2.0)));
	} else if (l <= 5.6e+102) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + ((t_0 * 0.3333333333333333) * (J * math.pow(l, 3.0)))
	t_2 = U + (((-0.125 * (K * K)) + 1.0) * ((math.exp(l) - math.exp(-l)) * J))
	tmp = 0
	if l <= -5.6e+102:
		tmp = t_1
	elif l <= -1.35e+19:
		tmp = t_2
	elif l <= 11.2:
		tmp = U + (J * (t_0 * (l * 2.0)))
	elif l <= 5.6e+102:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(Float64(t_0 * 0.3333333333333333) * Float64(J * (l ^ 3.0))))
	t_2 = Float64(U + Float64(Float64(Float64(-0.125 * Float64(K * K)) + 1.0) * Float64(Float64(exp(l) - exp(Float64(-l))) * J)))
	tmp = 0.0
	if (l <= -5.6e+102)
		tmp = t_1;
	elseif (l <= -1.35e+19)
		tmp = t_2;
	elseif (l <= 11.2)
		tmp = Float64(U + Float64(J * Float64(t_0 * Float64(l * 2.0))));
	elseif (l <= 5.6e+102)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((t_0 * 0.3333333333333333) * (J * (l ^ 3.0)));
	t_2 = U + (((-0.125 * (K * K)) + 1.0) * ((exp(l) - exp(-l)) * J));
	tmp = 0.0;
	if (l <= -5.6e+102)
		tmp = t_1;
	elseif (l <= -1.35e+19)
		tmp = t_2;
	elseif (l <= 11.2)
		tmp = U + (J * (t_0 * (l * 2.0)));
	elseif (l <= 5.6e+102)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(t$95$0 * 0.3333333333333333), $MachinePrecision] * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(U + N[(N[(N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.6e+102], t$95$1, If[LessEqual[l, -1.35e+19], t$95$2, If[LessEqual[l, 11.2], N[(U + N[(J * N[(t$95$0 * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+102], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.60000000000000037e102 or 5.60000000000000037e102 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 100.0%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if -5.60000000000000037e102 < l < -1.35e19 or 11.199999999999999 < l < 5.60000000000000037e102

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 0.0%

      \[\leadsto \color{blue}{\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J + -0.125 \cdot \left({K}^{2} \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 0.0%

         \[\leadsto \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J + \color{blue}{\left(-0.125 \cdot {K}^{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)}\right) + U \]

      2. distribute-rgt1-in 90.5%

         \[\leadsto \color{blue}{\left(-0.125 \cdot {K}^{2} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

      3. unpow2 90.5%

         \[\leadsto \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U \]

   4. Simplified 90.5%

      \[\leadsto \color{blue}{\left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)} + U \]

   if -1.35e19 < l < 11.199999999999999

   1. Initial program 75.3%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 100.0%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;U + \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right) \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \mathbf{elif}\;\ell \leq 11.2:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;U + \left(-0.125 \cdot \left(K \cdot K\right) + 1\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right) \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 5: 92.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + \left(t_0 \cdot 0.3333333333333333\right) \cdot \left(J \cdot {\ell}^{3}\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;U + J \cdot \left(t_0 \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+67}:\\ \;\;\;\;U + t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (* t_0 0.3333333333333333) (* J (pow l 3.0)))))
        (t_2 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -4e+120)
     t_1
     (if (<= l -1.35e+19)
       t_2
       (if (<= l 1.5e-38)
         (+ U (* J (* t_0 (* l 2.0))))
         (if (<= l 1.15e+67) (+ U t_2) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + ((t_0 * 0.3333333333333333) * (J * pow(l, 3.0)));
	double t_2 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -4e+120) {
		tmp = t_1;
	} else if (l <= -1.35e+19) {
		tmp = t_2;
	} else if (l <= 1.5e-38) {
		tmp = U + (J * (t_0 * (l * 2.0)));
	} else if (l <= 1.15e+67) {
		tmp = U + t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((t_0 * 0.3333333333333333d0) * (j * (l ** 3.0d0)))
    t_2 = (exp(l) - exp(-l)) * j
    if (l <= (-4d+120)) then
        tmp = t_1
    else if (l <= (-1.35d+19)) then
        tmp = t_2
    else if (l <= 1.5d-38) then
        tmp = u + (j * (t_0 * (l * 2.0d0)))
    else if (l <= 1.15d+67) then
        tmp = u + t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + ((t_0 * 0.3333333333333333) * (J * Math.pow(l, 3.0)));
	double t_2 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -4e+120) {
		tmp = t_1;
	} else if (l <= -1.35e+19) {
		tmp = t_2;
	} else if (l <= 1.5e-38) {
		tmp = U + (J * (t_0 * (l * 2.0)));
	} else if (l <= 1.15e+67) {
		tmp = U + t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + ((t_0 * 0.3333333333333333) * (J * math.pow(l, 3.0)))
	t_2 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -4e+120:
		tmp = t_1
	elif l <= -1.35e+19:
		tmp = t_2
	elif l <= 1.5e-38:
		tmp = U + (J * (t_0 * (l * 2.0)))
	elif l <= 1.15e+67:
		tmp = U + t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(Float64(t_0 * 0.3333333333333333) * Float64(J * (l ^ 3.0))))
	t_2 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -4e+120)
		tmp = t_1;
	elseif (l <= -1.35e+19)
		tmp = t_2;
	elseif (l <= 1.5e-38)
		tmp = Float64(U + Float64(J * Float64(t_0 * Float64(l * 2.0))));
	elseif (l <= 1.15e+67)
		tmp = Float64(U + t_2);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((t_0 * 0.3333333333333333) * (J * (l ^ 3.0)));
	t_2 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -4e+120)
		tmp = t_1;
	elseif (l <= -1.35e+19)
		tmp = t_2;
	elseif (l <= 1.5e-38)
		tmp = U + (J * (t_0 * (l * 2.0)));
	elseif (l <= 1.15e+67)
		tmp = U + t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(t$95$0 * 0.3333333333333333), $MachinePrecision] * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -4e+120], t$95$1, If[LessEqual[l, -1.35e+19], t$95$2, If[LessEqual[l, 1.5e-38], N[(U + N[(J * N[(t$95$0 * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.15e+67], N[(U + t$95$2), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.9999999999999999e120 or 1.1499999999999999e67 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 95.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 95.7%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 95.7%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 95.7%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 95.7%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 95.7%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 95.7%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 95.7%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 95.7%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 95.7%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 95.7%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 95.7%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 95.7%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 95.7%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 95.7%

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

      1. associate-*r* 95.7%

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

   7. Simplified 95.7%

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

   if -3.9999999999999999e120 < l < -1.35e19

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 71.4%

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

   3. Taylor expanded in J around inf 71.4%

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

   if -1.35e19 < l < 1.49999999999999994e-38

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 100.0%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if 1.49999999999999994e-38 < l < 1.1499999999999999e67

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 75.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+120}:\\ \;\;\;\;U + \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right) \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+67}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \left(\cos \left(K \cdot 0.5\right) \cdot 0.3333333333333333\right) \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 6: 82.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot {\ell}^{3}\\ t_1 := U + t_0 \cdot \left(0.3333333333333333 + \left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -1 \cdot 10^{+136}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.95 \cdot 10^{+25}:\\ \;\;\;\;U + t_2\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+199} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+267}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (pow l 3.0)))
        (t_1
         (+
          U
          (* t_0 (+ 0.3333333333333333 (* (* K K) -0.041666666666666664)))))
        (t_2 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -1e+136)
     t_1
     (if (<= l -1.35e+19)
       t_2
       (if (<= l 1.5e-38)
         (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
         (if (<= l 2.95e+25)
           (+ U t_2)
           (if (or (<= l 2e+199) (not (<= l 1.7e+267)))
             t_1
             (+ U (* 0.3333333333333333 t_0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * pow(l, 3.0);
	double t_1 = U + (t_0 * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	double t_2 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -1e+136) {
		tmp = t_1;
	} else if (l <= -1.35e+19) {
		tmp = t_2;
	} else if (l <= 1.5e-38) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 2.95e+25) {
		tmp = U + t_2;
	} else if ((l <= 2e+199) || !(l <= 1.7e+267)) {
		tmp = t_1;
	} else {
		tmp = U + (0.3333333333333333 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = j * (l ** 3.0d0)
    t_1 = u + (t_0 * (0.3333333333333333d0 + ((k * k) * (-0.041666666666666664d0))))
    t_2 = (exp(l) - exp(-l)) * j
    if (l <= (-1d+136)) then
        tmp = t_1
    else if (l <= (-1.35d+19)) then
        tmp = t_2
    else if (l <= 1.5d-38) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else if (l <= 2.95d+25) then
        tmp = u + t_2
    else if ((l <= 2d+199) .or. (.not. (l <= 1.7d+267))) then
        tmp = t_1
    else
        tmp = u + (0.3333333333333333d0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * Math.pow(l, 3.0);
	double t_1 = U + (t_0 * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	double t_2 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -1e+136) {
		tmp = t_1;
	} else if (l <= -1.35e+19) {
		tmp = t_2;
	} else if (l <= 1.5e-38) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 2.95e+25) {
		tmp = U + t_2;
	} else if ((l <= 2e+199) || !(l <= 1.7e+267)) {
		tmp = t_1;
	} else {
		tmp = U + (0.3333333333333333 * t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * math.pow(l, 3.0)
	t_1 = U + (t_0 * (0.3333333333333333 + ((K * K) * -0.041666666666666664)))
	t_2 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -1e+136:
		tmp = t_1
	elif l <= -1.35e+19:
		tmp = t_2
	elif l <= 1.5e-38:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif l <= 2.95e+25:
		tmp = U + t_2
	elif (l <= 2e+199) or not (l <= 1.7e+267):
		tmp = t_1
	else:
		tmp = U + (0.3333333333333333 * t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * (l ^ 3.0))
	t_1 = Float64(U + Float64(t_0 * Float64(0.3333333333333333 + Float64(Float64(K * K) * -0.041666666666666664))))
	t_2 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -1e+136)
		tmp = t_1;
	elseif (l <= -1.35e+19)
		tmp = t_2;
	elseif (l <= 1.5e-38)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (l <= 2.95e+25)
		tmp = Float64(U + t_2);
	elseif ((l <= 2e+199) || !(l <= 1.7e+267))
		tmp = t_1;
	else
		tmp = Float64(U + Float64(0.3333333333333333 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l ^ 3.0);
	t_1 = U + (t_0 * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	t_2 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -1e+136)
		tmp = t_1;
	elseif (l <= -1.35e+19)
		tmp = t_2;
	elseif (l <= 1.5e-38)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	elseif (l <= 2.95e+25)
		tmp = U + t_2;
	elseif ((l <= 2e+199) || ~((l <= 1.7e+267)))
		tmp = t_1;
	else
		tmp = U + (0.3333333333333333 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(0.3333333333333333 + N[(N[(K * K), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -1e+136], t$95$1, If[LessEqual[l, -1.35e+19], t$95$2, If[LessEqual[l, 1.5e-38], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.95e+25], N[(U + t$95$2), $MachinePrecision], If[Or[LessEqual[l, 2e+199], N[Not[LessEqual[l, 1.7e+267]], $MachinePrecision]], t$95$1, N[(U + N[(0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.00000000000000006e136 or 2.95e25 < l < 2.00000000000000019e199 or 1.69999999999999991e267 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 85.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 85.3%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 85.3%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 85.3%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 85.3%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 85.3%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 85.3%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 85.3%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 85.3%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 85.3%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 85.3%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 85.3%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 85.3%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 85.3%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 85.3%

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

      1. associate-*r* 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in K around 0 77.3%

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

      1. unpow2 77.3%

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

   10. Simplified 77.3%

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

   if -1.00000000000000006e136 < l < -1.35e19

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 68.8%

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

   3. Taylor expanded in J around inf 68.8%

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

   if -1.35e19 < l < 1.49999999999999994e-38

   1. Initial program 73.9%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 100.0%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if 1.49999999999999994e-38 < l < 2.95e25

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 100.0%

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

   if 2.00000000000000019e199 < l < 1.69999999999999991e267

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 100.0%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in K around 0 84.6%

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \left({\ell}^{3} \cdot J\right) + U \]
3. Recombined 5 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{+136}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + \left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-38}:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.95 \cdot 10^{+25}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+199} \lor \neg \left(\ell \leq 1.7 \cdot 10^{+267}\right):\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + \left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 7: 83.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot {\ell}^{3}\\ t_1 := U + t_0 \cdot \left(0.3333333333333333 + \left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 250:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.95 \cdot 10^{+25}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 10^{+195} \lor \neg \left(\ell \leq 9.2 \cdot 10^{+268}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (pow l 3.0)))
        (t_1
         (+
          U
          (* t_0 (+ 0.3333333333333333 (* (* K K) -0.041666666666666664)))))
        (t_2 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -1.7e+132)
     t_1
     (if (<= l -1.35e+19)
       t_2
       (if (<= l 250.0)
         (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
         (if (<= l 2.95e+25)
           t_2
           (if (or (<= l 1e+195) (not (<= l 9.2e+268)))
             t_1
             (+ U (* 0.3333333333333333 t_0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * pow(l, 3.0);
	double t_1 = U + (t_0 * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	double t_2 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -1.7e+132) {
		tmp = t_1;
	} else if (l <= -1.35e+19) {
		tmp = t_2;
	} else if (l <= 250.0) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 2.95e+25) {
		tmp = t_2;
	} else if ((l <= 1e+195) || !(l <= 9.2e+268)) {
		tmp = t_1;
	} else {
		tmp = U + (0.3333333333333333 * t_0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = j * (l ** 3.0d0)
    t_1 = u + (t_0 * (0.3333333333333333d0 + ((k * k) * (-0.041666666666666664d0))))
    t_2 = (exp(l) - exp(-l)) * j
    if (l <= (-1.7d+132)) then
        tmp = t_1
    else if (l <= (-1.35d+19)) then
        tmp = t_2
    else if (l <= 250.0d0) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else if (l <= 2.95d+25) then
        tmp = t_2
    else if ((l <= 1d+195) .or. (.not. (l <= 9.2d+268))) then
        tmp = t_1
    else
        tmp = u + (0.3333333333333333d0 * t_0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = J * Math.pow(l, 3.0);
	double t_1 = U + (t_0 * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	double t_2 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -1.7e+132) {
		tmp = t_1;
	} else if (l <= -1.35e+19) {
		tmp = t_2;
	} else if (l <= 250.0) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else if (l <= 2.95e+25) {
		tmp = t_2;
	} else if ((l <= 1e+195) || !(l <= 9.2e+268)) {
		tmp = t_1;
	} else {
		tmp = U + (0.3333333333333333 * t_0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * math.pow(l, 3.0)
	t_1 = U + (t_0 * (0.3333333333333333 + ((K * K) * -0.041666666666666664)))
	t_2 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -1.7e+132:
		tmp = t_1
	elif l <= -1.35e+19:
		tmp = t_2
	elif l <= 250.0:
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	elif l <= 2.95e+25:
		tmp = t_2
	elif (l <= 1e+195) or not (l <= 9.2e+268):
		tmp = t_1
	else:
		tmp = U + (0.3333333333333333 * t_0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * (l ^ 3.0))
	t_1 = Float64(U + Float64(t_0 * Float64(0.3333333333333333 + Float64(Float64(K * K) * -0.041666666666666664))))
	t_2 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -1.7e+132)
		tmp = t_1;
	elseif (l <= -1.35e+19)
		tmp = t_2;
	elseif (l <= 250.0)
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	elseif (l <= 2.95e+25)
		tmp = t_2;
	elseif ((l <= 1e+195) || !(l <= 9.2e+268))
		tmp = t_1;
	else
		tmp = Float64(U + Float64(0.3333333333333333 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (l ^ 3.0);
	t_1 = U + (t_0 * (0.3333333333333333 + ((K * K) * -0.041666666666666664)));
	t_2 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -1.7e+132)
		tmp = t_1;
	elseif (l <= -1.35e+19)
		tmp = t_2;
	elseif (l <= 250.0)
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	elseif (l <= 2.95e+25)
		tmp = t_2;
	elseif ((l <= 1e+195) || ~((l <= 9.2e+268)))
		tmp = t_1;
	else
		tmp = U + (0.3333333333333333 * t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(0.3333333333333333 + N[(N[(K * K), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -1.7e+132], t$95$1, If[LessEqual[l, -1.35e+19], t$95$2, If[LessEqual[l, 250.0], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.95e+25], t$95$2, If[Or[LessEqual[l, 1e+195], N[Not[LessEqual[l, 9.2e+268]], $MachinePrecision]], t$95$1, N[(U + N[(0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.70000000000000013e132 or 2.95e25 < l < 9.99999999999999977e194 or 9.20000000000000049e268 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 85.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 85.3%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 85.3%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 85.3%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 85.3%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 85.3%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 85.3%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 85.3%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 85.3%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 85.3%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 85.3%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 85.3%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 85.3%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 85.3%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 85.3%

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

      1. associate-*r* 85.3%

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

   7. Simplified 85.3%

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

   8. Taylor expanded in K around 0 77.3%

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

      1. unpow2 77.3%

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

   10. Simplified 77.3%

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

   if -1.70000000000000013e132 < l < -1.35e19 or 250 < l < 2.95e25

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 73.0%

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

   3. Taylor expanded in J around inf 73.0%

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

   if -1.35e19 < l < 250

   1. Initial program 75.3%

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

   2. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 100.0%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around 0 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   if 9.99999999999999977e194 < l < 9.20000000000000049e268

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 100.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 100.0%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 100.0%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 100.0%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 100.0%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 100.0%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 100.0%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 100.0%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 100.0%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 100.0%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in K around 0 84.6%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+132}:\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + \left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ \mathbf{elif}\;\ell \leq -1.35 \cdot 10^{+19}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 250:\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.95 \cdot 10^{+25}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 10^{+195} \lor \neg \left(\ell \leq 9.2 \cdot 10^{+268}\right):\\ \;\;\;\;U + \left(J \cdot {\ell}^{3}\right) \cdot \left(0.3333333333333333 + \left(K \cdot K\right) \cdot -0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 8: 69.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.015:\\ \;\;\;\;U + K \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot J\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.015)
   (+ U (* K (* -0.25 (* K (* l J)))))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.015) {
		tmp = U + (K * (-0.25 * (K * (l * J))));
	} else {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.015d0)) then
        tmp = u + (k * ((-0.25d0) * (k * (l * j))))
    else
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.015) {
		tmp = U + (K * (-0.25 * (K * (l * J))));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.015:
		tmp = U + (K * (-0.25 * (K * (l * J))))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.015)
		tmp = Float64(U + Float64(K * Float64(-0.25 * Float64(K * Float64(l * J)))));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.015)
		tmp = U + (K * (-0.25 * (K * (l * J))));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.015], N[(U + N[(K * N[(-0.25 * N[(K * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.014999999999999999

   1. Initial program 91.4%

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

   2. Taylor expanded in l around 0 50.7%

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

   3. Taylor expanded in K around 0 35.1%

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

      1. +-commutative 35.1%

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

      2. associate-*r* 35.1%

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

      3. distribute-rgt-out 45.7%

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

      4. *-commutative 45.7%

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

      5. unpow2 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in K around inf 45.7%

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

      1. associate-*r* 45.7%

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

      2. *-commutative 45.7%

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

      3. unpow2 45.7%

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

      4. associate-*r* 45.7%

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

      5. associate-*l* 53.7%

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

      6. *-commutative 53.7%

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

      7. associate-*l* 53.7%

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

   8. Simplified 53.7%

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

   if -0.014999999999999999 < (cos.f64 (/.f64 K 2))

   1. Initial program 86.1%

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

   2. Taylor expanded in l around 0 88.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 88.8%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 88.8%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 88.8%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 88.8%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 88.8%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 88.8%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 88.8%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 88.8%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 88.8%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 88.8%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 88.8%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 88.8%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 88.8%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 75.0%

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

      1. associate-*r* 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in K around 0 75.0%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.015:\\ \;\;\;\;U + K \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot J\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 9: 74.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -3.2 \cdot 10^{-18} \lor \neg \left(J \leq 7.2 \cdot 10^{+96}\right):\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J -3.2e-18) (not (<= J 7.2e+96)))
   (+ U (* J (* (cos (* K 0.5)) (* l 2.0))))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -3.2e-18) || !(J <= 7.2e+96)) {
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	} else {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((j <= (-3.2d-18)) .or. (.not. (j <= 7.2d+96))) then
        tmp = u + (j * (cos((k * 0.5d0)) * (l * 2.0d0)))
    else
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -3.2e-18) || !(J <= 7.2e+96)) {
		tmp = U + (J * (Math.cos((K * 0.5)) * (l * 2.0)));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= -3.2e-18) or not (J <= 7.2e+96):
		tmp = U + (J * (math.cos((K * 0.5)) * (l * 2.0)))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= -3.2e-18) || !(J <= 7.2e+96))
		tmp = Float64(U + Float64(J * Float64(cos(Float64(K * 0.5)) * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= -3.2e-18) || ~((J <= 7.2e+96)))
		tmp = U + (J * (cos((K * 0.5)) * (l * 2.0)));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, -3.2e-18], N[Not[LessEqual[J, 7.2e+96]], $MachinePrecision]], N[(U + N[(J * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -3.1999999999999999e-18 or 7.20000000000000026e96 < J

   1. Initial program 72.8%

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

   2. Taylor expanded in l around 0 96.3%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 96.4%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 96.4%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 96.4%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 96.4%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 96.4%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 96.4%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 96.4%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 96.4%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 96.4%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 96.4%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 96.4%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 96.4%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 96.4%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around 0 88.1%

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

      1. *-commutative 88.1%

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

      2. associate-*r* 88.1%

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

   7. Simplified 88.1%

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

   if -3.1999999999999999e-18 < J < 7.20000000000000026e96

   1. Initial program 97.7%

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

   2. Taylor expanded in l around 0 79.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. associate-*r* 79.8%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) \cdot J\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      2. associate-*r* 79.8%

         \[\leadsto \left(\color{blue}{\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left({\ell}^{3} \cdot J\right)\right)\right) + U \]

      3. associate-*r* 79.8%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + 0.3333333333333333 \cdot \color{blue}{\left(\left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right) \cdot J\right)}\right) + U \]

      4. associate-*r* 79.8%

         \[\leadsto \left(\left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)\right) \cdot J + \color{blue}{\left(0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) \cdot J}\right) + U \]

      5. distribute-rgt-out 79.8%

         \[\leadsto \color{blue}{J \cdot \left(2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \ell\right) + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right)} + U \]

      6. *-commutative 79.8%

         \[\leadsto J \cdot \left(2 \cdot \color{blue}{\left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      7. associate-*r* 79.8%

         \[\leadsto J \cdot \left(\color{blue}{\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)} + 0.3333333333333333 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + U \]

      8. *-commutative 79.8%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + 0.3333333333333333 \cdot \color{blue}{\left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)}\right) + U \]

      9. associate-*r* 79.8%

         \[\leadsto J \cdot \left(\left(2 \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      10. distribute-rgt-out 79.8%

          \[\leadsto J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(2 \cdot \ell + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)} + U \]

      11. +-commutative 79.8%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)}\right) + U \]

      12. fma-def 79.8%

          \[\leadsto J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)}\right) + U \]

   4. Simplified 79.8%

      \[\leadsto \color{blue}{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, 2 \cdot \ell\right)\right)} + U \]

   5. Taylor expanded in l around inf 77.5%

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

      1. associate-*r* 77.5%

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

   7. Simplified 77.5%

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

   8. Taylor expanded in K around 0 64.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.2 \cdot 10^{-18} \lor \neg \left(J \leq 7.2 \cdot 10^{+96}\right):\\ \;\;\;\;U + J \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 10: 58.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+202} \lor \neg \left(\ell \leq 1.1 \cdot 10^{+263}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= l -6.5e+20)
     t_0
     (if (<= l 1e+25)
       (fma l (* J 2.0) U)
       (if (or (<= l 1.75e+202) (not (<= l 1.1e+263)))
         t_0
         (+ U (* J (+ (* (* K K) 0.0625) -0.5))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -6.5e+20) {
		tmp = t_0;
	} else if (l <= 1e+25) {
		tmp = fma(l, (J * 2.0), U);
	} else if ((l <= 1.75e+202) || !(l <= 1.1e+263)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (((K * K) * 0.0625) + -0.5));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (l <= -6.5e+20)
		tmp = t_0;
	elseif (l <= 1e+25)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif ((l <= 1.75e+202) || !(l <= 1.1e+263))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(Float64(Float64(K * K) * 0.0625) + -0.5)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -6.5e+20], t$95$0, If[LessEqual[l, 1e+25], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[Or[LessEqual[l, 1.75e+202], N[Not[LessEqual[l, 1.1e+263]], $MachinePrecision]], t$95$0, N[(U + N[(J * N[(N[(N[(K * K), $MachinePrecision] * 0.0625), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6.5e20 or 1.00000000000000009e25 < l < 1.74999999999999994e202 or 1.1e263 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 23.2%

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

   3. Taylor expanded in K around 0 20.6%

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

      1. +-commutative 20.6%

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

      2. associate-*r* 20.6%

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

      3. distribute-rgt-out 34.3%

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

      4. *-commutative 34.3%

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

      5. unpow2 34.3%

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

   5. Simplified 34.3%

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

   if -6.5e20 < l < 1.00000000000000009e25

   1. Initial program 76.4%

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

   2. Taylor expanded in l around 0 95.6%

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

   3. Taylor expanded in K around 0 85.8%

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

      1. *-commutative 85.8%

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

      2. associate-*r* 85.8%

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

      3. fma-def 85.8%

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

      4. *-commutative 85.8%

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

   5. Simplified 85.8%

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

   if 1.74999999999999994e202 < l < 1.1e263

   1. Initial program 100.0%

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

   3. Applied egg-rr 1.5%

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

   4. Taylor expanded in K around 0 47.1%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \left({K}^{2} \cdot J\right) + -0.5 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 47.1%

         \[\leadsto \left(\color{blue}{\left(0.0625 \cdot {K}^{2}\right) \cdot J} + -0.5 \cdot J\right) + U \]

      2. distribute-rgt-out 47.1%

         \[\leadsto \color{blue}{J \cdot \left(0.0625 \cdot {K}^{2} + -0.5\right)} + U \]

      3. *-commutative 47.1%

         \[\leadsto J \cdot \left(\color{blue}{{K}^{2} \cdot 0.0625} + -0.5\right) + U \]

      4. unpow2 47.1%

         \[\leadsto J \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot 0.0625 + -0.5\right) + U \]

   6. Simplified 47.1%

      \[\leadsto \color{blue}{J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+20}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+202} \lor \neg \left(\ell \leq 1.1 \cdot 10^{+263}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)\\ \end{array} \]

Alternative 11: 58.1% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 7.3 \cdot 10^{+24}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.72 \cdot 10^{+202} \lor \neg \left(\ell \leq 1.1 \cdot 10^{+263}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= l -8.2e+20)
     t_0
     (if (<= l 7.3e+24)
       (+ U (* l (* J 2.0)))
       (if (or (<= l 1.72e+202) (not (<= l 1.1e+263)))
         t_0
         (+ U (* J (+ (* (* K K) 0.0625) -0.5))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -8.2e+20) {
		tmp = t_0;
	} else if (l <= 7.3e+24) {
		tmp = U + (l * (J * 2.0));
	} else if ((l <= 1.72e+202) || !(l <= 1.1e+263)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (((K * K) * 0.0625) + -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    if (l <= (-8.2d+20)) then
        tmp = t_0
    else if (l <= 7.3d+24) then
        tmp = u + (l * (j * 2.0d0))
    else if ((l <= 1.72d+202) .or. (.not. (l <= 1.1d+263))) then
        tmp = t_0
    else
        tmp = u + (j * (((k * k) * 0.0625d0) + (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -8.2e+20) {
		tmp = t_0;
	} else if (l <= 7.3e+24) {
		tmp = U + (l * (J * 2.0));
	} else if ((l <= 1.72e+202) || !(l <= 1.1e+263)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (((K * K) * 0.0625) + -0.5));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	tmp = 0
	if l <= -8.2e+20:
		tmp = t_0
	elif l <= 7.3e+24:
		tmp = U + (l * (J * 2.0))
	elif (l <= 1.72e+202) or not (l <= 1.1e+263):
		tmp = t_0
	else:
		tmp = U + (J * (((K * K) * 0.0625) + -0.5))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (l <= -8.2e+20)
		tmp = t_0;
	elseif (l <= 7.3e+24)
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	elseif ((l <= 1.72e+202) || !(l <= 1.1e+263))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(Float64(Float64(K * K) * 0.0625) + -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	tmp = 0.0;
	if (l <= -8.2e+20)
		tmp = t_0;
	elseif (l <= 7.3e+24)
		tmp = U + (l * (J * 2.0));
	elseif ((l <= 1.72e+202) || ~((l <= 1.1e+263)))
		tmp = t_0;
	else
		tmp = U + (J * (((K * K) * 0.0625) + -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -8.2e+20], t$95$0, If[LessEqual[l, 7.3e+24], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 1.72e+202], N[Not[LessEqual[l, 1.1e+263]], $MachinePrecision]], t$95$0, N[(U + N[(J * N[(N[(N[(K * K), $MachinePrecision] * 0.0625), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.2e20 or 7.29999999999999982e24 < l < 1.71999999999999996e202 or 1.1e263 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 23.2%

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

   3. Taylor expanded in K around 0 20.6%

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

      1. +-commutative 20.6%

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

      2. associate-*r* 20.6%

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

      3. distribute-rgt-out 34.3%

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

      4. *-commutative 34.3%

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

      5. unpow2 34.3%

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

   5. Simplified 34.3%

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

   if -8.2e20 < l < 7.29999999999999982e24

   1. Initial program 76.4%

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

   2. Taylor expanded in l around 0 95.6%

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

   3. Taylor expanded in K around 0 85.8%

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

      1. *-commutative 85.8%

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

      2. associate-*r* 85.8%

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

      3. *-commutative 85.8%

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

   5. Simplified 85.8%

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

   if 1.71999999999999996e202 < l < 1.1e263

   1. Initial program 100.0%

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

   3. Applied egg-rr 1.5%

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

   4. Taylor expanded in K around 0 47.1%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \left({K}^{2} \cdot J\right) + -0.5 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 47.1%

         \[\leadsto \left(\color{blue}{\left(0.0625 \cdot {K}^{2}\right) \cdot J} + -0.5 \cdot J\right) + U \]

      2. distribute-rgt-out 47.1%

         \[\leadsto \color{blue}{J \cdot \left(0.0625 \cdot {K}^{2} + -0.5\right)} + U \]

      3. *-commutative 47.1%

         \[\leadsto J \cdot \left(\color{blue}{{K}^{2} \cdot 0.0625} + -0.5\right) + U \]

      4. unpow2 47.1%

         \[\leadsto J \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot 0.0625 + -0.5\right) + U \]

   6. Simplified 47.1%

      \[\leadsto \color{blue}{J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.2 \cdot 10^{+20}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 7.3 \cdot 10^{+24}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 1.72 \cdot 10^{+202} \lor \neg \left(\ell \leq 1.1 \cdot 10^{+263}\right):\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)\\ \end{array} \]

Alternative 12: 54.1% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq 1.1 \cdot 10^{-308} \lor \neg \left(J \leq 1.7 \cdot 10^{-232}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J 1.1e-308) (not (<= J 1.7e-232)))
   (+ U (* l (* J 2.0)))
   (+ U (* J (+ (* (* K K) 0.0625) -0.5)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= 1.1e-308) || !(J <= 1.7e-232)) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (J * (((K * K) * 0.0625) + -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 ((j <= 1.1d-308) .or. (.not. (j <= 1.7d-232))) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u + (j * (((k * k) * 0.0625d0) + (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= 1.1e-308) || !(J <= 1.7e-232)) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (J * (((K * K) * 0.0625) + -0.5));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= 1.1e-308) or not (J <= 1.7e-232):
		tmp = U + (l * (J * 2.0))
	else:
		tmp = U + (J * (((K * K) * 0.0625) + -0.5))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= 1.1e-308) || !(J <= 1.7e-232))
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(Float64(K * K) * 0.0625) + -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= 1.1e-308) || ~((J <= 1.7e-232)))
		tmp = U + (l * (J * 2.0));
	else
		tmp = U + (J * (((K * K) * 0.0625) + -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, 1.1e-308], N[Not[LessEqual[J, 1.7e-232]], $MachinePrecision]], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(N[(K * K), $MachinePrecision] * 0.0625), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < 1.1000000000000001e-308 or 1.7000000000000001e-232 < J

   1. Initial program 86.7%

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

   2. Taylor expanded in l around 0 64.9%

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

   3. Taylor expanded in K around 0 55.7%

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

      1. *-commutative 55.7%

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

      2. associate-*r* 55.7%

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

      3. *-commutative 55.7%

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

   5. Simplified 55.7%

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

   if 1.1000000000000001e-308 < J < 1.7000000000000001e-232

   1. Initial program 100.0%

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

   3. Applied egg-rr 19.0%

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

   4. Taylor expanded in K around 0 57.1%

      \[\leadsto \color{blue}{\left(0.0625 \cdot \left({K}^{2} \cdot J\right) + -0.5 \cdot J\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 57.1%

         \[\leadsto \left(\color{blue}{\left(0.0625 \cdot {K}^{2}\right) \cdot J} + -0.5 \cdot J\right) + U \]

      2. distribute-rgt-out 57.1%

         \[\leadsto \color{blue}{J \cdot \left(0.0625 \cdot {K}^{2} + -0.5\right)} + U \]

      3. *-commutative 57.1%

         \[\leadsto J \cdot \left(\color{blue}{{K}^{2} \cdot 0.0625} + -0.5\right) + U \]

      4. unpow2 57.1%

         \[\leadsto J \cdot \left(\color{blue}{\left(K \cdot K\right)} \cdot 0.0625 + -0.5\right) + U \]

   6. Simplified 57.1%

      \[\leadsto \color{blue}{J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq 1.1 \cdot 10^{-308} \lor \neg \left(J \leq 1.7 \cdot 10^{-232}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\left(K \cdot K\right) \cdot 0.0625 + -0.5\right)\\ \end{array} \]

Alternative 13: 53.4% accurate, 20.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -9 \cdot 10^{-210} \lor \neg \left(J \leq 5.4 \cdot 10^{+59}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + K \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot J\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J -9e-210) (not (<= J 5.4e+59)))
   (+ U (* l (* J 2.0)))
   (+ U (* K (* -0.25 (* K (* l J)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -9e-210) || !(J <= 5.4e+59)) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (K * (-0.25 * (K * (l * J))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((j <= (-9d-210)) .or. (.not. (j <= 5.4d+59))) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = u + (k * ((-0.25d0) * (k * (l * j))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -9e-210) || !(J <= 5.4e+59)) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = U + (K * (-0.25 * (K * (l * J))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= -9e-210) or not (J <= 5.4e+59):
		tmp = U + (l * (J * 2.0))
	else:
		tmp = U + (K * (-0.25 * (K * (l * J))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= -9e-210) || !(J <= 5.4e+59))
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64(U + Float64(K * Float64(-0.25 * Float64(K * Float64(l * J)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((J <= -9e-210) || ~((J <= 5.4e+59)))
		tmp = U + (l * (J * 2.0));
	else
		tmp = U + (K * (-0.25 * (K * (l * J))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, -9e-210], N[Not[LessEqual[J, 5.4e+59]], $MachinePrecision]], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(K * N[(-0.25 * N[(K * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if J < -9.00000000000000039e-210 or 5.4000000000000002e59 < J

   1. Initial program 80.6%

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

   2. Taylor expanded in l around 0 76.5%

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

   3. Taylor expanded in K around 0 63.7%

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

      1. *-commutative 63.7%

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

      2. associate-*r* 63.7%

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

      3. *-commutative 63.7%

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

   5. Simplified 63.7%

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

   if -9.00000000000000039e-210 < J < 5.4000000000000002e59

   1. Initial program 97.7%

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

   2. Taylor expanded in l around 0 40.7%

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

   3. Taylor expanded in K around 0 44.7%

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

      1. +-commutative 44.7%

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

      2. associate-*r* 44.7%

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

      3. distribute-rgt-out 44.7%

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

      4. *-commutative 44.7%

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

      5. unpow2 44.7%

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

   5. Simplified 44.7%

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

   6. Taylor expanded in K around inf 44.5%

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

      1. associate-*r* 44.5%

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

      2. *-commutative 44.5%

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

      3. unpow2 44.5%

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

      4. associate-*r* 44.5%

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

      5. associate-*l* 50.2%

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

      6. *-commutative 50.2%

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

      7. associate-*l* 50.2%

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

   8. Simplified 50.2%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -9 \cdot 10^{-210} \lor \neg \left(J \leq 5.4 \cdot 10^{+59}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + K \cdot \left(-0.25 \cdot \left(K \cdot \left(\ell \cdot J\right)\right)\right)\\ \end{array} \]

Alternative 14: 43.4% accurate, 28.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -980 \lor \neg \left(\ell \leq 700\right):\\ \;\;\;\;U + \left(U \cdot U\right) \cdot 18014398509481984\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -980.0) (not (<= l 700.0)))
   (+ U (* (* U U) 18014398509481984.0))
   U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -980.0) || !(l <= 700.0)) {
		tmp = U + ((U * U) * 18014398509481984.0);
	} 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 <= (-980.0d0)) .or. (.not. (l <= 700.0d0))) then
        tmp = u + ((u * u) * 18014398509481984.0d0)
    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 <= -980.0) || !(l <= 700.0)) {
		tmp = U + ((U * U) * 18014398509481984.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -980.0) or not (l <= 700.0):
		tmp = U + ((U * U) * 18014398509481984.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -980.0) || !(l <= 700.0))
		tmp = Float64(U + Float64(Float64(U * U) * 18014398509481984.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -980.0) || ~((l <= 700.0)))
		tmp = U + ((U * U) * 18014398509481984.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -980.0], N[Not[LessEqual[l, 700.0]], $MachinePrecision]], N[(U + N[(N[(U * U), $MachinePrecision] * 18014398509481984.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -980 or 700 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 63.6%

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

   4. Applied egg-rr 15.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-134217728 \cdot U, -134217728 \cdot U, U\right)} \]
   5. Step-by-step derivation

      1. fma-udef 15.0%

         \[\leadsto \color{blue}{\left(-134217728 \cdot U\right) \cdot \left(-134217728 \cdot U\right) + U} \]

      2. +-commutative 15.0%

         \[\leadsto \color{blue}{U + \left(-134217728 \cdot U\right) \cdot \left(-134217728 \cdot U\right)} \]

      3. swap-sqr 15.0%

         \[\leadsto U + \color{blue}{\left(-134217728 \cdot -134217728\right) \cdot \left(U \cdot U\right)} \]

      4. *-commutative 15.0%

         \[\leadsto U + \color{blue}{\left(U \cdot U\right) \cdot \left(-134217728 \cdot -134217728\right)} \]

      5. metadata-eval 15.0%

         \[\leadsto U + \left(U \cdot U\right) \cdot \color{blue}{18014398509481984} \]

   6. Simplified 15.0%

      \[\leadsto \color{blue}{U + \left(U \cdot U\right) \cdot 18014398509481984} \]

   if -980 < l < 700

   1. Initial program 75.1%

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

   2. Taylor expanded in J around 0 75.1%

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

4. Final simplification 44.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -980 \lor \neg \left(\ell \leq 700\right):\\ \;\;\;\;U + \left(U \cdot U\right) \cdot 18014398509481984\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 15: 42.9% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 70000000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1000.0) (* U U) (if (<= l 70000000000.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1000.0) {
		tmp = U * U;
	} else if (l <= 70000000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1000.0d0)) then
        tmp = u * u
    else if (l <= 70000000000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1000.0) {
		tmp = U * U;
	} else if (l <= 70000000000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1000.0:
		tmp = U * U
	elif l <= 70000000000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1000.0)
		tmp = Float64(U * U);
	elseif (l <= 70000000000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1000.0)
		tmp = U * U;
	elseif (l <= 70000000000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1000.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 70000000000.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -1e3 or 7e10 < l

   1. Initial program 100.0%

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

   3. Applied egg-rr 13.7%

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

   if -1e3 < l < 7e10

   1. Initial program 75.5%

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

   2. Taylor expanded in J around 0 74.0%

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

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 70000000000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 16: 54.7% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.7%

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

2. Taylor expanded in l around 0 61.6%

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

3. Taylor expanded in K around 0 53.1%

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

   1. *-commutative 53.1%

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

   2. associate-*r* 53.1%

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

   3. *-commutative 53.1%

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

5. Simplified 53.1%

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

6. Final simplification 53.1%

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

Alternative 17: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -3.725290312339702 \cdot 10^{-9} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -3.725290312339702e-9)

C

double code(double J, double l, double K, double U) {
	return -3.725290312339702e-9;
}

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 = -3.725290312339702d-9
end function

Java

public static double code(double J, double l, double K, double U) {
	return -3.725290312339702e-9;
}

Python

def code(J, l, K, U):
	return -3.725290312339702e-9

Julia

function code(J, l, K, U)
	return -3.725290312339702e-9
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -3.725290312339702e-9;
end

Wolfram

code[J_, l_, K_, U_] := -3.725290312339702e-9

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. Taylor expanded in K around 0 69.3%

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

4. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{\frac{U}{U + \left(-134217728 \cdot U + -134217728 \cdot U\right)}} \]
5. Step-by-step derivation

   1. associate-+r+ 2.7%

      \[\leadsto \frac{U}{\color{blue}{\left(U + -134217728 \cdot U\right) + -134217728 \cdot U}} \]

   2. distribute-rgt1-in 2.7%

      \[\leadsto \frac{U}{\color{blue}{\left(-134217728 + 1\right) \cdot U} + -134217728 \cdot U} \]

   3. distribute-rgt-out 2.7%

      \[\leadsto \frac{U}{\color{blue}{U \cdot \left(\left(-134217728 + 1\right) + -134217728\right)}} \]

   4. associate-/r* 2.7%

      \[\leadsto \color{blue}{\frac{\frac{U}{U}}{\left(-134217728 + 1\right) + -134217728}} \]

   5. *-inverses 2.7%

      \[\leadsto \frac{\color{blue}{1}}{\left(-134217728 + 1\right) + -134217728} \]

   6. metadata-eval 2.7%

      \[\leadsto \frac{1}{\color{blue}{-134217727} + -134217728} \]

   7. metadata-eval 2.7%

      \[\leadsto \frac{1}{\color{blue}{-268435455}} \]

   8. metadata-eval 2.7%

      \[\leadsto \color{blue}{-3.725290312339702 \cdot 10^{-9}} \]

6. Simplified 2.7%

   \[\leadsto \color{blue}{-3.725290312339702 \cdot 10^{-9}} \]

7. Final simplification 2.7%

   \[\leadsto -3.725290312339702 \cdot 10^{-9} \]

Alternative 18: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 1.0)

C

double code(double J, double l, double K, double U) {
	return 1.0;
}

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 = 1.0d0
end function

Java

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

Python

def code(J, l, K, U):
	return 1.0

Julia

function code(J, l, K, U)
	return 1.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 1.0;
end

Wolfram

code[J_, l_, K_, U_] := 1.0

Derivation

1. Initial program 87.7%

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

3. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

      \[\leadsto \color{blue}{1} \]

5. Simplified 2.8%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 2.8%

   \[\leadsto 1 \]

Alternative 19: 37.3% 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. Taylor expanded in J around 0 38.4%

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

3. Final simplification 38.4%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023252 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))