Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.7%

Time: 15.0s

Alternatives: 19

Speedup: 1.4×

• 50.4% 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: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-10)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (fma
      l
      (* (cos (* K 0.5)) (* J (fma 0.3333333333333333 (pow l 2.0) 2.0)))
      U))))

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 <= 2e-10)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = fma(l, (cos((K * 0.5)) * (J * fma(0.3333333333333333, pow(l, 2.0), 2.0))), U);
	}
	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 <= 2e-10))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = fma(l, Float64(cos(Float64(K * 0.5)) * Float64(J * fma(0.3333333333333333, (l ^ 2.0), 2.0))), U);
	end
	return 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, 2e-10]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 2.00000000000000007e-10 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.00000000000000007e-10

   1. Initial program 68.0%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. associate-*r* 99.9%

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

      6. distribute-rgt-out 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-*l* 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. distribute-lft-out 99.9%

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

      12. fma-define 99.9%

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

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\ell, \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right), U\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\ell, \cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{-10}\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-10)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+ U (* l (* (cos (* K 0.5)) (* J 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 <= 2e-10)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * 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 <= 2e-10)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 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 <= 2e-10):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 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 <= 2e-10))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 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 <= 2e-10)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * 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, 2e-10]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 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 2.00000000000000007e-10 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2.00000000000000007e-10

   1. Initial program 68.0%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 100.0%

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

Alternative 3: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(t\_1 \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -0.08:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2 + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+82}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(t\_1 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U)) (t_1 (cos (* K 0.5))))
   (if (<= l -4.5e+116)
     (* (* t_1 (pow l 3.0)) (* J 0.3333333333333333))
     (if (<= l -0.08)
       t_0
       (if (<= l 480000000.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* l (+ (* J 2.0) (* 0.3333333333333333 (* J (pow l 2.0)))))))
         (if (<= l 1.15e+82)
           t_0
           (+ U (* 0.3333333333333333 (* t_1 (* J (pow l 3.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -4.5e+116) {
		tmp = (t_1 * pow(l, 3.0)) * (J * 0.3333333333333333);
	} else if (l <= -0.08) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (cos((K / 2.0)) * (l * ((J * 2.0) + (0.3333333333333333 * (J * pow(l, 2.0))))));
	} else if (l <= 1.15e+82) {
		tmp = t_0;
	} else {
		tmp = U + (0.3333333333333333 * (t_1 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((exp(l) - exp(-l)) * j) + u
    t_1 = cos((k * 0.5d0))
    if (l <= (-4.5d+116)) then
        tmp = (t_1 * (l ** 3.0d0)) * (j * 0.3333333333333333d0)
    else if (l <= (-0.08d0)) then
        tmp = t_0
    else if (l <= 480000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (l * ((j * 2.0d0) + (0.3333333333333333d0 * (j * (l ** 2.0d0))))))
    else if (l <= 1.15d+82) then
        tmp = t_0
    else
        tmp = u + (0.3333333333333333d0 * (t_1 * (j * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -4.5e+116) {
		tmp = (t_1 * Math.pow(l, 3.0)) * (J * 0.3333333333333333);
	} else if (l <= -0.08) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (l * ((J * 2.0) + (0.3333333333333333 * (J * Math.pow(l, 2.0))))));
	} else if (l <= 1.15e+82) {
		tmp = t_0;
	} else {
		tmp = U + (0.3333333333333333 * (t_1 * (J * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = ((math.exp(l) - math.exp(-l)) * J) + U
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -4.5e+116:
		tmp = (t_1 * math.pow(l, 3.0)) * (J * 0.3333333333333333)
	elif l <= -0.08:
		tmp = t_0
	elif l <= 480000000.0:
		tmp = U + (math.cos((K / 2.0)) * (l * ((J * 2.0) + (0.3333333333333333 * (J * math.pow(l, 2.0))))))
	elif l <= 1.15e+82:
		tmp = t_0
	else:
		tmp = U + (0.3333333333333333 * (t_1 * (J * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -4.5e+116)
		tmp = Float64(Float64(t_1 * (l ^ 3.0)) * Float64(J * 0.3333333333333333));
	elseif (l <= -0.08)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(Float64(J * 2.0) + Float64(0.3333333333333333 * Float64(J * (l ^ 2.0)))))));
	elseif (l <= 1.15e+82)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(t_1 * Float64(J * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((exp(l) - exp(-l)) * J) + U;
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -4.5e+116)
		tmp = (t_1 * (l ^ 3.0)) * (J * 0.3333333333333333);
	elseif (l <= -0.08)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = U + (cos((K / 2.0)) * (l * ((J * 2.0) + (0.3333333333333333 * (J * (l ^ 2.0))))));
	elseif (l <= 1.15e+82)
		tmp = t_0;
	else
		tmp = U + (0.3333333333333333 * (t_1 * (J * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.5e+116], N[(N[(t$95$1 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -0.08], t$95$0, If[LessEqual[l, 480000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(N[(J * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.15e+82], t$95$0, N[(U + N[(0.3333333333333333 * N[(t$95$1 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.50000000000000016e116

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in U around -inf 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \color{blue}{-U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)} \]

      2. *-commutative 100.0%

         \[\leadsto -\color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot U} \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot \left(-U\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(-1 - J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right) \cdot \left(-U\right)} \]

   9. Taylor expanded in l around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*l* 100.0%

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

       4. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -4.50000000000000016e116 < l < -0.0800000000000000017 or 4.8e8 < l < 1.14999999999999994e82

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 80.6%

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

   if -0.0800000000000000017 < l < 4.8e8

   1. Initial program 69.0%

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

   3. Taylor expanded in l around 0 98.7%

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

   if 1.14999999999999994e82 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.2%

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

   4. Taylor expanded in l around inf 98.2%

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

      1. associate-*r* 98.2%

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

   6. Simplified 98.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -0.08:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2 + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+82}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(t\_1 \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -9.6 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \ell \cdot \left(t\_1 \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(t\_1 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U)) (t_1 (cos (* K 0.5))))
   (if (<= l -4.5e+116)
     (* (* t_1 (pow l 3.0)) (* J 0.3333333333333333))
     (if (<= l -9.6e-5)
       t_0
       (if (<= l 480000000.0)
         (+ U (* l (* t_1 (* J 2.0))))
         (if (<= l 7.8e+88)
           t_0
           (+ U (* 0.3333333333333333 (* t_1 (* J (pow l 3.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -4.5e+116) {
		tmp = (t_1 * pow(l, 3.0)) * (J * 0.3333333333333333);
	} else if (l <= -9.6e-5) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (l * (t_1 * (J * 2.0)));
	} else if (l <= 7.8e+88) {
		tmp = t_0;
	} else {
		tmp = U + (0.3333333333333333 * (t_1 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((exp(l) - exp(-l)) * j) + u
    t_1 = cos((k * 0.5d0))
    if (l <= (-4.5d+116)) then
        tmp = (t_1 * (l ** 3.0d0)) * (j * 0.3333333333333333d0)
    else if (l <= (-9.6d-5)) then
        tmp = t_0
    else if (l <= 480000000.0d0) then
        tmp = u + (l * (t_1 * (j * 2.0d0)))
    else if (l <= 7.8d+88) then
        tmp = t_0
    else
        tmp = u + (0.3333333333333333d0 * (t_1 * (j * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -4.5e+116) {
		tmp = (t_1 * Math.pow(l, 3.0)) * (J * 0.3333333333333333);
	} else if (l <= -9.6e-5) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (l * (t_1 * (J * 2.0)));
	} else if (l <= 7.8e+88) {
		tmp = t_0;
	} else {
		tmp = U + (0.3333333333333333 * (t_1 * (J * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = ((math.exp(l) - math.exp(-l)) * J) + U
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -4.5e+116:
		tmp = (t_1 * math.pow(l, 3.0)) * (J * 0.3333333333333333)
	elif l <= -9.6e-5:
		tmp = t_0
	elif l <= 480000000.0:
		tmp = U + (l * (t_1 * (J * 2.0)))
	elif l <= 7.8e+88:
		tmp = t_0
	else:
		tmp = U + (0.3333333333333333 * (t_1 * (J * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -4.5e+116)
		tmp = Float64(Float64(t_1 * (l ^ 3.0)) * Float64(J * 0.3333333333333333));
	elseif (l <= -9.6e-5)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = Float64(U + Float64(l * Float64(t_1 * Float64(J * 2.0))));
	elseif (l <= 7.8e+88)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(t_1 * Float64(J * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((exp(l) - exp(-l)) * J) + U;
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -4.5e+116)
		tmp = (t_1 * (l ^ 3.0)) * (J * 0.3333333333333333);
	elseif (l <= -9.6e-5)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = U + (l * (t_1 * (J * 2.0)));
	elseif (l <= 7.8e+88)
		tmp = t_0;
	else
		tmp = U + (0.3333333333333333 * (t_1 * (J * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.5e+116], N[(N[(t$95$1 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -9.6e-5], t$95$0, If[LessEqual[l, 480000000.0], N[(U + N[(l * N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.8e+88], t$95$0, N[(U + N[(0.3333333333333333 * N[(t$95$1 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.50000000000000016e116

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in U around -inf 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \color{blue}{-U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)} \]

      2. *-commutative 100.0%

         \[\leadsto -\color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot U} \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot \left(-U\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(-1 - J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right) \cdot \left(-U\right)} \]

   9. Taylor expanded in l around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*l* 100.0%

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

       4. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -4.50000000000000016e116 < l < -9.6000000000000002e-5 or 4.8e8 < l < 7.8000000000000002e88

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 80.6%

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

   if -9.6000000000000002e-5 < l < 4.8e8

   1. Initial program 69.0%

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

   3. Taylor expanded in l around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.6%

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

      3. associate-*l* 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-*r* 98.6%

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

      6. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   if 7.8000000000000002e88 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.2%

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

   4. Taylor expanded in l around inf 98.2%

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

      1. associate-*r* 98.2%

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

   6. Simplified 98.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -9.6 \cdot 10^{-5}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+88}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(t\_1 \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -0.009:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(t\_1 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U)) (t_1 (cos (* K 0.5))))
   (if (<= l -4.5e+116)
     (* (* t_1 (pow l 3.0)) (* J 0.3333333333333333))
     (if (<= l -0.009)
       t_0
       (if (<= l 480000000.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0)))))))
         (if (<= l 1.55e+87)
           t_0
           (+ U (* 0.3333333333333333 (* t_1 (* J (pow l 3.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -4.5e+116) {
		tmp = (t_1 * pow(l, 3.0)) * (J * 0.3333333333333333);
	} else if (l <= -0.009) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	} else if (l <= 1.55e+87) {
		tmp = t_0;
	} else {
		tmp = U + (0.3333333333333333 * (t_1 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((exp(l) - exp(-l)) * j) + u
    t_1 = cos((k * 0.5d0))
    if (l <= (-4.5d+116)) then
        tmp = (t_1 * (l ** 3.0d0)) * (j * 0.3333333333333333d0)
    else if (l <= (-0.009d0)) then
        tmp = t_0
    else if (l <= 480000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    else if (l <= 1.55d+87) then
        tmp = t_0
    else
        tmp = u + (0.3333333333333333d0 * (t_1 * (j * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -4.5e+116) {
		tmp = (t_1 * Math.pow(l, 3.0)) * (J * 0.3333333333333333);
	} else if (l <= -0.009) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	} else if (l <= 1.55e+87) {
		tmp = t_0;
	} else {
		tmp = U + (0.3333333333333333 * (t_1 * (J * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = ((math.exp(l) - math.exp(-l)) * J) + U
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -4.5e+116:
		tmp = (t_1 * math.pow(l, 3.0)) * (J * 0.3333333333333333)
	elif l <= -0.009:
		tmp = t_0
	elif l <= 480000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	elif l <= 1.55e+87:
		tmp = t_0
	else:
		tmp = U + (0.3333333333333333 * (t_1 * (J * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -4.5e+116)
		tmp = Float64(Float64(t_1 * (l ^ 3.0)) * Float64(J * 0.3333333333333333));
	elseif (l <= -0.009)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	elseif (l <= 1.55e+87)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(t_1 * Float64(J * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((exp(l) - exp(-l)) * J) + U;
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -4.5e+116)
		tmp = (t_1 * (l ^ 3.0)) * (J * 0.3333333333333333);
	elseif (l <= -0.009)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	elseif (l <= 1.55e+87)
		tmp = t_0;
	else
		tmp = U + (0.3333333333333333 * (t_1 * (J * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.5e+116], N[(N[(t$95$1 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -0.009], t$95$0, If[LessEqual[l, 480000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+87], t$95$0, N[(U + N[(0.3333333333333333 * N[(t$95$1 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.50000000000000016e116

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in U around -inf 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \color{blue}{-U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)} \]

      2. *-commutative 100.0%

         \[\leadsto -\color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot U} \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot \left(-U\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(-1 - J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right) \cdot \left(-U\right)} \]

   9. Taylor expanded in l around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*l* 100.0%

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

       4. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -4.50000000000000016e116 < l < -0.00899999999999999932 or 4.8e8 < l < 1.55e87

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 80.6%

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

   if -0.00899999999999999932 < l < 4.8e8

   1. Initial program 69.0%

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

   3. Taylor expanded in l around 0 98.7%

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

   if 1.55e87 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.2%

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

   4. Taylor expanded in l around inf 98.2%

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

      1. associate-*r* 98.2%

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

   6. Simplified 98.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -0.009:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+87}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(t\_1 \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -0.05:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(t\_1 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U)) (t_1 (cos (* K 0.5))))
   (if (<= l -4.5e+116)
     (* (* t_1 (pow l 3.0)) (* J 0.3333333333333333))
     (if (<= l -0.05)
       t_0
       (if (<= l 480000000.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0))))))
         (if (<= l 7.8e+88)
           t_0
           (+ U (* 0.3333333333333333 (* t_1 (* J (pow l 3.0)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K * 0.5));
	double tmp;
	if (l <= -4.5e+116) {
		tmp = (t_1 * pow(l, 3.0)) * (J * 0.3333333333333333);
	} else if (l <= -0.05) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0)))));
	} else if (l <= 7.8e+88) {
		tmp = t_0;
	} else {
		tmp = U + (0.3333333333333333 * (t_1 * (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((exp(l) - exp(-l)) * j) + u
    t_1 = cos((k * 0.5d0))
    if (l <= (-4.5d+116)) then
        tmp = (t_1 * (l ** 3.0d0)) * (j * 0.3333333333333333d0)
    else if (l <= (-0.05d0)) then
        tmp = t_0
    else if (l <= 480000000.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0)))))
    else if (l <= 7.8d+88) then
        tmp = t_0
    else
        tmp = u + (0.3333333333333333d0 * (t_1 * (j * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double t_1 = Math.cos((K * 0.5));
	double tmp;
	if (l <= -4.5e+116) {
		tmp = (t_1 * Math.pow(l, 3.0)) * (J * 0.3333333333333333);
	} else if (l <= -0.05) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0)))));
	} else if (l <= 7.8e+88) {
		tmp = t_0;
	} else {
		tmp = U + (0.3333333333333333 * (t_1 * (J * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = ((math.exp(l) - math.exp(-l)) * J) + U
	t_1 = math.cos((K * 0.5))
	tmp = 0
	if l <= -4.5e+116:
		tmp = (t_1 * math.pow(l, 3.0)) * (J * 0.3333333333333333)
	elif l <= -0.05:
		tmp = t_0
	elif l <= 480000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0)))))
	elif l <= 7.8e+88:
		tmp = t_0
	else:
		tmp = U + (0.3333333333333333 * (t_1 * (J * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (l <= -4.5e+116)
		tmp = Float64(Float64(t_1 * (l ^ 3.0)) * Float64(J * 0.3333333333333333));
	elseif (l <= -0.05)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0))))));
	elseif (l <= 7.8e+88)
		tmp = t_0;
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(t_1 * Float64(J * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((exp(l) - exp(-l)) * J) + U;
	t_1 = cos((K * 0.5));
	tmp = 0.0;
	if (l <= -4.5e+116)
		tmp = (t_1 * (l ^ 3.0)) * (J * 0.3333333333333333);
	elseif (l <= -0.05)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = U + (cos((K / 2.0)) * (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0)))));
	elseif (l <= 7.8e+88)
		tmp = t_0;
	else
		tmp = U + (0.3333333333333333 * (t_1 * (J * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.5e+116], N[(N[(t$95$1 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -0.05], t$95$0, If[LessEqual[l, 480000000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.8e+88], t$95$0, N[(U + N[(0.3333333333333333 * N[(t$95$1 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.50000000000000016e116

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*l* 100.0%

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

      4. unpow2 100.0%

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

      5. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in U around -inf 100.0%

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto \color{blue}{-U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)} \]

      2. *-commutative 100.0%

         \[\leadsto -\color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot U} \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot \left(-U\right)} \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(-1 - J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right) \cdot \left(-U\right)} \]

   9. Taylor expanded in l around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. *-commutative 100.0%

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

       3. associate-*l* 100.0%

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

       4. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -4.50000000000000016e116 < l < -0.050000000000000003 or 4.8e8 < l < 7.8000000000000002e88

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 80.6%

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

   if -0.050000000000000003 < l < 4.8e8

   1. Initial program 69.0%

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

   3. Taylor expanded in l around 0 98.7%

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

      1. distribute-lft-in 98.7%

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

      2. *-commutative 98.7%

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

      3. associate-*l* 98.7%

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

      4. unpow2 98.7%

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

      5. pow3 98.7%

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

   5. Applied egg-rr 98.7%

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

   if 7.8000000000000002e88 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.2%

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

   4. Taylor expanded in l around inf 98.2%

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

      1. associate-*r* 98.2%

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

   6. Simplified 98.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -0.05:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+88}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := \cos \left(K \cdot 0.5\right)\\ t_2 := \left(t\_1 \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq -0.019:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \ell \cdot \left(t\_1 \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U))
        (t_1 (cos (* K 0.5)))
        (t_2 (* (* t_1 (pow l 3.0)) (* J 0.3333333333333333))))
   (if (<= l -4.5e+116)
     t_2
     (if (<= l -0.019)
       t_0
       (if (<= l 480000000.0)
         (+ U (* l (* t_1 (* J 2.0))))
         (if (<= l 7.8e+88) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = cos((K * 0.5));
	double t_2 = (t_1 * pow(l, 3.0)) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -4.5e+116) {
		tmp = t_2;
	} else if (l <= -0.019) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (l * (t_1 * (J * 2.0)));
	} else if (l <= 7.8e+88) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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 = ((exp(l) - exp(-l)) * j) + u
    t_1 = cos((k * 0.5d0))
    t_2 = (t_1 * (l ** 3.0d0)) * (j * 0.3333333333333333d0)
    if (l <= (-4.5d+116)) then
        tmp = t_2
    else if (l <= (-0.019d0)) then
        tmp = t_0
    else if (l <= 480000000.0d0) then
        tmp = u + (l * (t_1 * (j * 2.0d0)))
    else if (l <= 7.8d+88) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double t_1 = Math.cos((K * 0.5));
	double t_2 = (t_1 * Math.pow(l, 3.0)) * (J * 0.3333333333333333);
	double tmp;
	if (l <= -4.5e+116) {
		tmp = t_2;
	} else if (l <= -0.019) {
		tmp = t_0;
	} else if (l <= 480000000.0) {
		tmp = U + (l * (t_1 * (J * 2.0)));
	} else if (l <= 7.8e+88) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = ((math.exp(l) - math.exp(-l)) * J) + U
	t_1 = math.cos((K * 0.5))
	t_2 = (t_1 * math.pow(l, 3.0)) * (J * 0.3333333333333333)
	tmp = 0
	if l <= -4.5e+116:
		tmp = t_2
	elif l <= -0.019:
		tmp = t_0
	elif l <= 480000000.0:
		tmp = U + (l * (t_1 * (J * 2.0)))
	elif l <= 7.8e+88:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = cos(Float64(K * 0.5))
	t_2 = Float64(Float64(t_1 * (l ^ 3.0)) * Float64(J * 0.3333333333333333))
	tmp = 0.0
	if (l <= -4.5e+116)
		tmp = t_2;
	elseif (l <= -0.019)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = Float64(U + Float64(l * Float64(t_1 * Float64(J * 2.0))));
	elseif (l <= 7.8e+88)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((exp(l) - exp(-l)) * J) + U;
	t_1 = cos((K * 0.5));
	t_2 = (t_1 * (l ^ 3.0)) * (J * 0.3333333333333333);
	tmp = 0.0;
	if (l <= -4.5e+116)
		tmp = t_2;
	elseif (l <= -0.019)
		tmp = t_0;
	elseif (l <= 480000000.0)
		tmp = U + (l * (t_1 * (J * 2.0)));
	elseif (l <= 7.8e+88)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+116], t$95$2, If[LessEqual[l, -0.019], t$95$0, If[LessEqual[l, 480000000.0], N[(U + N[(l * N[(t$95$1 * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 7.8e+88], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.50000000000000016e116 or 7.8000000000000002e88 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 99.0%

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

      1. distribute-lft-in 99.0%

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

      2. *-commutative 99.0%

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

      3. associate-*l* 99.0%

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

      4. unpow2 99.0%

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

      5. pow3 99.0%

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

   5. Applied egg-rr 99.0%

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

   6. Taylor expanded in U around -inf 99.0%

      \[\leadsto \color{blue}{-1 \cdot \left(U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.0%

         \[\leadsto \color{blue}{-U \cdot \left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right)} \]

      2. *-commutative 99.0%

         \[\leadsto -\color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot U} \]

      3. distribute-rgt-neg-in 99.0%

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)}{U} - 1\right) \cdot \left(-U\right)} \]

   8. Simplified 99.0%

      \[\leadsto \color{blue}{\left(-1 - J \cdot \left(\mathsf{fma}\left(0.3333333333333333, {\ell}^{3}, \ell \cdot 2\right) \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}\right)\right) \cdot \left(-U\right)} \]

   9. Taylor expanded in l around inf 99.0%

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

       1. *-commutative 99.0%

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

       2. *-commutative 99.0%

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

       3. associate-*l* 99.0%

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

       4. *-commutative 99.0%

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

   11. Simplified 99.0%

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

   if -4.50000000000000016e116 < l < -0.0189999999999999995 or 4.8e8 < l < 7.8000000000000002e88

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 80.6%

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

   if -0.0189999999999999995 < l < 4.8e8

   1. Initial program 69.0%

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

   3. Taylor expanded in l around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.6%

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

      3. associate-*l* 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-*r* 98.6%

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

      6. *-commutative 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+116}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{elif}\;\ell \leq -0.019:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 480000000:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{+88}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(K \cdot 0.5\right) \cdot {\ell}^{3}\right) \cdot \left(J \cdot 0.3333333333333333\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.0023 \lor \neg \left(\ell \leq 480000000\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.0023) (not (<= l 480000000.0)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0023) || !(l <= 480000000.0)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.0023d0)) .or. (.not. (l <= 480000000.0d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.0023) || !(l <= 480000000.0)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.0023) or not (l <= 480000000.0):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.0023) || !(l <= 480000000.0))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.0023) || ~((l <= 480000000.0)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.0023], N[Not[LessEqual[l, 480000000.0]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.0023 or 4.8e8 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 76.2%

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

   if -0.0023 < l < 4.8e8

   1. Initial program 69.0%

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

   3. Taylor expanded in l around 0 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.6%

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

      3. associate-*l* 98.6%

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

      4. *-commutative 98.6%

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

      5. associate-*r* 98.6%

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

      6. *-commutative 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.0023 \lor \neg \left(\ell \leq 480000000\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+98}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 0.3333333333333333 (pow l 3.0)))))
   (if (<= l -5.3e+45)
     t_0
     (if (<= l -8.5e+16)
       (pow U -4.0)
       (if (<= l 5e+98)
         (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
         (+ U t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (0.3333333333333333 * pow(l, 3.0));
	double tmp;
	if (l <= -5.3e+45) {
		tmp = t_0;
	} else if (l <= -8.5e+16) {
		tmp = pow(U, -4.0);
	} else if (l <= 5e+98) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (0.3333333333333333d0 * (l ** 3.0d0))
    if (l <= (-5.3d+45)) then
        tmp = t_0
    else if (l <= (-8.5d+16)) then
        tmp = u ** (-4.0d0)
    else if (l <= 5d+98) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + 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 * (0.3333333333333333 * Math.pow(l, 3.0));
	double tmp;
	if (l <= -5.3e+45) {
		tmp = t_0;
	} else if (l <= -8.5e+16) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 5e+98) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (0.3333333333333333 * math.pow(l, 3.0))
	tmp = 0
	if l <= -5.3e+45:
		tmp = t_0
	elif l <= -8.5e+16:
		tmp = math.pow(U, -4.0)
	elif l <= 5e+98:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))
	tmp = 0.0
	if (l <= -5.3e+45)
		tmp = t_0;
	elseif (l <= -8.5e+16)
		tmp = U ^ -4.0;
	elseif (l <= 5e+98)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (0.3333333333333333 * (l ^ 3.0));
	tmp = 0.0;
	if (l <= -5.3e+45)
		tmp = t_0;
	elseif (l <= -8.5e+16)
		tmp = U ^ -4.0;
	elseif (l <= 5e+98)
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.3e+45], t$95$0, If[LessEqual[l, -8.5e+16], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 5e+98], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -5.29999999999999991e45

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 89.9%

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

   4. Taylor expanded in K around 0 71.4%

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

   5. Taylor expanded in U around inf 71.4%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   6. Taylor expanded in l around inf 71.4%

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

      1. associate-*r* 71.4%

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

      2. *-commutative 71.4%

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

      3. associate-*r* 71.4%

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

   8. Simplified 71.4%

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

   if -5.29999999999999991e45 < l < -8.5e16

   1. Initial program 100.0%

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

   4. Applied egg-rr 56.1%

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

   if -8.5e16 < l < 4.9999999999999998e98

   1. Initial program 72.9%

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

   3. Taylor expanded in l around 0 88.6%

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

   if 4.9999999999999998e98 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in K around 0 77.1%

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

   5. Taylor expanded in l around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 77.1%

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

   7. Simplified 77.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.3 \cdot 10^{+45}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+98}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+98}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 0.3333333333333333 (pow l 3.0)))))
   (if (<= l -1.05e+47)
     t_0
     (if (<= l -7.8e+18)
       (pow U -4.0)
       (if (<= l 5e+98)
         (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))
         (+ U t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (0.3333333333333333 * pow(l, 3.0));
	double tmp;
	if (l <= -1.05e+47) {
		tmp = t_0;
	} else if (l <= -7.8e+18) {
		tmp = pow(U, -4.0);
	} else if (l <= 5e+98) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (0.3333333333333333d0 * (l ** 3.0d0))
    if (l <= (-1.05d+47)) then
        tmp = t_0
    else if (l <= (-7.8d+18)) then
        tmp = u ** (-4.0d0)
    else if (l <= 5d+98) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else
        tmp = u + 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 * (0.3333333333333333 * Math.pow(l, 3.0));
	double tmp;
	if (l <= -1.05e+47) {
		tmp = t_0;
	} else if (l <= -7.8e+18) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 5e+98) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (0.3333333333333333 * math.pow(l, 3.0))
	tmp = 0
	if l <= -1.05e+47:
		tmp = t_0
	elif l <= -7.8e+18:
		tmp = math.pow(U, -4.0)
	elif l <= 5e+98:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	else:
		tmp = U + t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))
	tmp = 0.0
	if (l <= -1.05e+47)
		tmp = t_0;
	elseif (l <= -7.8e+18)
		tmp = U ^ -4.0;
	elseif (l <= 5e+98)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	else
		tmp = Float64(U + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (0.3333333333333333 * (l ^ 3.0));
	tmp = 0.0;
	if (l <= -1.05e+47)
		tmp = t_0;
	elseif (l <= -7.8e+18)
		tmp = U ^ -4.0;
	elseif (l <= 5e+98)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	else
		tmp = U + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.05e+47], t$95$0, If[LessEqual[l, -7.8e+18], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 5e+98], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.05e47

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 89.9%

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

   4. Taylor expanded in K around 0 71.4%

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

   5. Taylor expanded in U around inf 71.4%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   6. Taylor expanded in l around inf 71.4%

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

      1. associate-*r* 71.4%

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

      2. *-commutative 71.4%

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

      3. associate-*r* 71.4%

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

   8. Simplified 71.4%

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

   if -1.05e47 < l < -7.8e18

   1. Initial program 100.0%

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

   4. Applied egg-rr 56.1%

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

   if -7.8e18 < l < 4.9999999999999998e98

   1. Initial program 72.9%

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

   3. Taylor expanded in l around 0 88.6%

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

      1. *-commutative 88.6%

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

      2. associate-*r* 88.7%

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

      3. associate-*l* 88.7%

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

      4. *-commutative 88.7%

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

      5. associate-*r* 88.7%

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

      6. *-commutative 88.7%

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

   5. Simplified 88.7%

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

   if 4.9999999999999998e98 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in K around 0 77.1%

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

   5. Taylor expanded in l around inf 77.1%

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

      1. associate-*r* 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-*r* 77.1%

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

   7. Simplified 77.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+47}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{+18}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+98}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 120:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -8.5e+16)
   (* U (+ 1.0 (* 0.3333333333333333 (* J (/ (pow l 3.0) U)))))
   (if (<= l 120.0)
     (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))
     (*
      U
      (+ 1.0 (/ (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) U))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e+16) {
		tmp = U * (1.0 + (0.3333333333333333 * (J * (pow(l, 3.0) / U))));
	} else if (l <= 120.0) {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) / 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 <= (-8.5d+16)) then
        tmp = u * (1.0d0 + (0.3333333333333333d0 * (j * ((l ** 3.0d0) / u))))
    else if (l <= 120.0d0) then
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    else
        tmp = u * (1.0d0 + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) / u))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -8.5e+16) {
		tmp = U * (1.0 + (0.3333333333333333 * (J * (Math.pow(l, 3.0) / U))));
	} else if (l <= 120.0) {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -8.5e+16:
		tmp = U * (1.0 + (0.3333333333333333 * (J * (math.pow(l, 3.0) / U))))
	elif l <= 120.0:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	else:
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) / U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -8.5e+16)
		tmp = Float64(U * Float64(1.0 + Float64(0.3333333333333333 * Float64(J * Float64((l ^ 3.0) / U)))));
	elseif (l <= 120.0)
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -8.5e+16)
		tmp = U * (1.0 + (0.3333333333333333 * (J * ((l ^ 3.0) / U))));
	elseif (l <= 120.0)
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	else
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) / U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -8.5e+16], N[(U * N[(1.0 + N[(0.3333333333333333 * N[(J * N[(N[Power[l, 3.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 120.0], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -8.5e16

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 79.6%

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

   4. Taylor expanded in K around 0 62.0%

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

   5. Taylor expanded in U around inf 66.3%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   6. Taylor expanded in l around inf 66.3%

      \[\leadsto U \cdot \left(1 + \color{blue}{0.3333333333333333 \cdot \frac{J \cdot {\ell}^{3}}{U}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 67.9%

         \[\leadsto U \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(J \cdot \frac{{\ell}^{3}}{U}\right)}\right) \]

   8. Simplified 67.9%

      \[\leadsto U \cdot \left(1 + \color{blue}{0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)}\right) \]

   if -8.5e16 < l < 120

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*r* 97.0%

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

      3. associate-*l* 97.0%

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

      4. *-commutative 97.0%

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

      5. associate-*r* 97.0%

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

      6. *-commutative 97.0%

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

   5. Simplified 97.0%

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

   if 120 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 82.4%

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

   4. Taylor expanded in K around 0 62.3%

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

   5. Taylor expanded in U around inf 66.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+16}:\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \mathbf{elif}\;\ell \leq 120:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+16} \lor \neg \left(\ell \leq 62\right):\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -8.5e+16) (not (<= l 62.0)))
   (* U (+ 1.0 (* 0.3333333333333333 (* J (/ (pow l 3.0) U)))))
   (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -8.5e+16) || !(l <= 62.0)) {
		tmp = U * (1.0 + (0.3333333333333333 * (J * (pow(l, 3.0) / U))));
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-8.5d+16)) .or. (.not. (l <= 62.0d0))) then
        tmp = u * (1.0d0 + (0.3333333333333333d0 * (j * ((l ** 3.0d0) / u))))
    else
        tmp = u + (l * (cos((k * 0.5d0)) * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -8.5e+16) || !(l <= 62.0)) {
		tmp = U * (1.0 + (0.3333333333333333 * (J * (Math.pow(l, 3.0) / U))));
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -8.5e+16) or not (l <= 62.0):
		tmp = U * (1.0 + (0.3333333333333333 * (J * (math.pow(l, 3.0) / U))))
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -8.5e+16) || !(l <= 62.0))
		tmp = Float64(U * Float64(1.0 + Float64(0.3333333333333333 * Float64(J * Float64((l ^ 3.0) / U)))));
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -8.5e+16) || ~((l <= 62.0)))
		tmp = U * (1.0 + (0.3333333333333333 * (J * ((l ^ 3.0) / U))));
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -8.5e+16], N[Not[LessEqual[l, 62.0]], $MachinePrecision]], N[(U * N[(1.0 + N[(0.3333333333333333 * N[(J * N[(N[Power[l, 3.0], $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8.5e16 or 62 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 81.0%

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

   4. Taylor expanded in K around 0 62.2%

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

   5. Taylor expanded in U around inf 66.5%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   6. Taylor expanded in l around inf 66.5%

      \[\leadsto U \cdot \left(1 + \color{blue}{0.3333333333333333 \cdot \frac{J \cdot {\ell}^{3}}{U}}\right) \]
   7. Step-by-step derivation

      1. associate-/l* 67.3%

         \[\leadsto U \cdot \left(1 + 0.3333333333333333 \cdot \color{blue}{\left(J \cdot \frac{{\ell}^{3}}{U}\right)}\right) \]

   8. Simplified 67.3%

      \[\leadsto U \cdot \left(1 + \color{blue}{0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)}\right) \]

   if -8.5e16 < l < 62

   1. Initial program 69.3%

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

   3. Taylor expanded in l around 0 97.0%

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

      1. *-commutative 97.0%

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

      2. associate-*r* 97.0%

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

      3. associate-*l* 97.0%

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

      4. *-commutative 97.0%

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

      5. associate-*r* 97.0%

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

      6. *-commutative 97.0%

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

   5. Simplified 97.0%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{+16} \lor \neg \left(\ell \leq 62\right):\\ \;\;\;\;U \cdot \left(1 + 0.3333333333333333 \cdot \left(J \cdot \frac{{\ell}^{3}}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -1.46 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\ell \cdot \frac{2}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* J (* 0.3333333333333333 (pow l 3.0)))))
   (if (<= l -1.46e+49)
     t_0
     (if (<= l 1.4e+33) (* U (+ 1.0 (* J (* l (/ 2.0 U))))) (+ U t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = J * (0.3333333333333333 * pow(l, 3.0));
	double tmp;
	if (l <= -1.46e+49) {
		tmp = t_0;
	} else if (l <= 1.4e+33) {
		tmp = U * (1.0 + (J * (l * (2.0 / U))));
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = j * (0.3333333333333333d0 * (l ** 3.0d0))
    if (l <= (-1.46d+49)) then
        tmp = t_0
    else if (l <= 1.4d+33) then
        tmp = u * (1.0d0 + (j * (l * (2.0d0 / u))))
    else
        tmp = u + 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 * (0.3333333333333333 * Math.pow(l, 3.0));
	double tmp;
	if (l <= -1.46e+49) {
		tmp = t_0;
	} else if (l <= 1.4e+33) {
		tmp = U * (1.0 + (J * (l * (2.0 / U))));
	} else {
		tmp = U + t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = J * (0.3333333333333333 * math.pow(l, 3.0))
	tmp = 0
	if l <= -1.46e+49:
		tmp = t_0
	elif l <= 1.4e+33:
		tmp = U * (1.0 + (J * (l * (2.0 / U))))
	else:
		tmp = U + t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)))
	tmp = 0.0
	if (l <= -1.46e+49)
		tmp = t_0;
	elseif (l <= 1.4e+33)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(l * Float64(2.0 / U)))));
	else
		tmp = Float64(U + t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = J * (0.3333333333333333 * (l ^ 3.0));
	tmp = 0.0;
	if (l <= -1.46e+49)
		tmp = t_0;
	elseif (l <= 1.4e+33)
		tmp = U * (1.0 + (J * (l * (2.0 / U))));
	else
		tmp = U + t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.46e+49], t$95$0, If[LessEqual[l, 1.4e+33], N[(U * N[(1.0 + N[(J * N[(l * N[(2.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.46000000000000008e49

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 91.4%

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

   4. Taylor expanded in K around 0 72.7%

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

   5. Taylor expanded in U around inf 72.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   6. Taylor expanded in l around inf 72.8%

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

      1. associate-*r* 72.8%

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

      2. *-commutative 72.8%

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

      3. associate-*r* 72.8%

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

   8. Simplified 72.8%

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

   if -1.46000000000000008e49 < l < 1.4e33

   1. Initial program 72.9%

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

   3. Taylor expanded in l around 0 88.2%

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

   4. Taylor expanded in K around 0 73.4%

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

   5. Taylor expanded in U around inf 75.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   6. Taylor expanded in l around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. associate-/l* 75.8%

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

      3. associate-*l* 75.8%

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

      4. associate-*l/ 75.8%

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

      5. associate-/l* 75.8%

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

   8. Simplified 75.8%

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

   if 1.4e33 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 88.5%

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

   4. Taylor expanded in K around 0 66.1%

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

   5. Taylor expanded in l around inf 66.1%

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

      1. associate-*r* 66.1%

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

      2. *-commutative 66.1%

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

      3. associate-*r* 66.1%

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

   7. Simplified 66.1%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.46 \cdot 10^{+49}:\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+33}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\ell \cdot \frac{2}{U}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 70.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+48} \lor \neg \left(\ell \leq 2.7 \cdot 10^{+30}\right):\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\ell \cdot \frac{2}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.95e+48) (not (<= l 2.7e+30)))
   (* J (* 0.3333333333333333 (pow l 3.0)))
   (* U (+ 1.0 (* J (* l (/ 2.0 U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.95e+48) || !(l <= 2.7e+30)) {
		tmp = J * (0.3333333333333333 * pow(l, 3.0));
	} else {
		tmp = U * (1.0 + (J * (l * (2.0 / U))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.95d+48)) .or. (.not. (l <= 2.7d+30))) then
        tmp = j * (0.3333333333333333d0 * (l ** 3.0d0))
    else
        tmp = u * (1.0d0 + (j * (l * (2.0d0 / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.95e+48) || !(l <= 2.7e+30)) {
		tmp = J * (0.3333333333333333 * Math.pow(l, 3.0));
	} else {
		tmp = U * (1.0 + (J * (l * (2.0 / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.95e+48) or not (l <= 2.7e+30):
		tmp = J * (0.3333333333333333 * math.pow(l, 3.0))
	else:
		tmp = U * (1.0 + (J * (l * (2.0 / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.95e+48) || !(l <= 2.7e+30))
		tmp = Float64(J * Float64(0.3333333333333333 * (l ^ 3.0)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(l * Float64(2.0 / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.95e+48) || ~((l <= 2.7e+30)))
		tmp = J * (0.3333333333333333 * (l ^ 3.0));
	else
		tmp = U * (1.0 + (J * (l * (2.0 / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.95e+48], N[Not[LessEqual[l, 2.7e+30]], $MachinePrecision]], N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(J * N[(l * N[(2.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.95e48 or 2.6999999999999999e30 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 89.9%

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

   4. Taylor expanded in K around 0 69.3%

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

   5. Taylor expanded in U around inf 70.1%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   6. Taylor expanded in l around inf 69.3%

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

      1. associate-*r* 69.3%

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

      2. *-commutative 69.3%

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

      3. associate-*r* 69.3%

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

   8. Simplified 69.3%

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

   if -1.95e48 < l < 2.6999999999999999e30

   1. Initial program 72.9%

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

   3. Taylor expanded in l around 0 88.2%

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

   4. Taylor expanded in K around 0 73.4%

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

   5. Taylor expanded in U around inf 75.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

   6. Taylor expanded in l around 0 75.2%

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

      1. *-commutative 75.2%

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

      2. associate-/l* 75.8%

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

      3. associate-*l* 75.8%

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

      4. associate-*l/ 75.8%

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

      5. associate-/l* 75.8%

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

   8. Simplified 75.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.95 \cdot 10^{+48} \lor \neg \left(\ell \leq 2.7 \cdot 10^{+30}\right):\\ \;\;\;\;J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + J \cdot \left(\ell \cdot \frac{2}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.4% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -950 \lor \neg \left(\ell \leq 660\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -950.0) (not (<= l 660.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -950.0) || !(l <= 660.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-950.0d0)) .or. (.not. (l <= 660.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -950.0) || !(l <= 660.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -950.0) or not (l <= 660.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -950.0) || !(l <= 660.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -950.0) || ~((l <= 660.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -950.0], N[Not[LessEqual[l, 660.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -950 or 660 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 18.3%

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

   if -950 < l < 660

   1. Initial program 68.8%

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

   3. Taylor expanded in J around 0 67.4%

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

4. Final simplification 42.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -950 \lor \neg \left(\ell \leq 660\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 50.8% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.42 \cdot 10^{+56}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l 1.42e+56) (+ U (* J (* l 2.0))) (* U U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 1.42e+56) {
		tmp = U + (J * (l * 2.0));
	} 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 <= 1.42d+56) then
        tmp = u + (j * (l * 2.0d0))
    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 <= 1.42e+56) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 1.42e+56:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 1.42e+56)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 1.42e+56)
		tmp = U + (J * (l * 2.0));
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 1.42e+56], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.42e56

   1. Initial program 80.7%

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

   3. Taylor expanded in l around 0 87.4%

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

   4. Taylor expanded in K around 0 71.8%

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

   5. Taylor expanded in l around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. associate-*r* 59.5%

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

      3. *-commutative 59.5%

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

   7. Simplified 59.5%

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

   if 1.42e56 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 29.0%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.42 \cdot 10^{+56}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 59.2% accurate, 28.4× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + (J * (l * (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 = u * (1.0d0 + (j * (l * (2.0d0 / u))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in l around 0 89.0%

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

4. Taylor expanded in K around 0 71.6%

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

5. Taylor expanded in U around inf 73.0%

   \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]

6. Taylor expanded in l around 0 56.5%

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

   1. *-commutative 56.5%

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

   2. associate-/l* 60.6%

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

   3. associate-*l* 60.6%

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

   4. associate-*l/ 60.6%

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

   5. associate-/l* 60.6%

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

8. Simplified 60.6%

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

9. Final simplification 60.6%

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

Alternative 18: 2.8% 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 84.8%

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

4. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

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

6. Simplified 2.8%

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

7. Final simplification 2.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 19: 36.0% 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 84.8%

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

3. Taylor expanded in J around 0 34.0%

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

4. Final simplification 34.0%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))