Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.4% → 99.9%

Time: 14.1s

Alternatives: 15

Speedup: 1.4×

• 49.8% 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: 86.4% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -10000 \lor \neg \left(t_0 \leq 0.002\right):\\ \;\;\;\;J \cdot \left(t_0 \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -10000.0) (not (<= t_0 0.002)))
     (+ (* J (* t_0 (cos (* 0.5 K)))) U)
     (fma
      J
      (*
       (+
        (* 0.016666666666666666 (pow l 5.0))
        (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))
       (cos (/ K 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 <= -10000.0) || !(t_0 <= 0.002)) {
		tmp = (J * (t_0 * cos((0.5 * K)))) + U;
	} else {
		tmp = fma(J, (((0.016666666666666666 * pow(l, 5.0)) + ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))) * cos((K / 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 <= -10000.0) || !(t_0 <= 0.002))
		tmp = Float64(Float64(J * Float64(t_0 * cos(Float64(0.5 * K)))) + U);
	else
		tmp = fma(J, Float64(Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))) * cos(Float64(K / 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, -10000.0], N[Not[LessEqual[t$95$0, 0.002]], $MachinePrecision]], N[(N[(J * N[(t$95$0 * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(J * N[(N[(N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 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))) < -1e4 or 2e-3 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   3. Taylor expanded in J around 0 100.0%

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

   if -1e4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2e-3

   1. Initial program 72.6%

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

      1. associate-*l* 72.6%

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

      2. fma-def 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in l around 0 99.9%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -10000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.002\right):\\ \;\;\;\;J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -10000 \lor \neg \left(t_1 \leq 0.002\right):\\ \;\;\;\;J \cdot \left(t_1 \cdot t_0\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(t_0 \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -10000.0) (not (<= t_1 0.002)))
     (+ (* J (* t_1 t_0)) U)
     (+ U (* J (* t_0 (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -10000.0) || !(t_1 <= 0.002)) {
		tmp = (J * (t_1 * t_0)) + U;
	} else {
		tmp = U + (J * (t_0 * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((0.5d0 * k))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-10000.0d0)) .or. (.not. (t_1 <= 0.002d0))) then
        tmp = (j * (t_1 * t_0)) + u
    else
        tmp = u + (j * (t_0 * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -10000.0) || !(t_1 <= 0.002)) {
		tmp = (J * (t_1 * t_0)) + U;
	} else {
		tmp = U + (J * (t_0 * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((0.5 * K))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -10000.0) or not (t_1 <= 0.002):
		tmp = (J * (t_1 * t_0)) + U
	else:
		tmp = U + (J * (t_0 * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -10000.0) || !(t_1 <= 0.002))
		tmp = Float64(Float64(J * Float64(t_1 * t_0)) + U);
	else
		tmp = Float64(U + Float64(J * Float64(t_0 * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((0.5 * K));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -10000.0) || ~((t_1 <= 0.002)))
		tmp = (J * (t_1 * t_0)) + U;
	else
		tmp = U + (J * (t_0 * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -10000.0], N[Not[LessEqual[t$95$1, 0.002]], $MachinePrecision]], N[(N[(J * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(t$95$0 * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1e4 or 2e-3 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   3. Taylor expanded in J around 0 100.0%

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

   if -1e4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2e-3

   1. Initial program 72.6%

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

   3. Taylor expanded in J around 0 72.6%

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

   4. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. associate-*r* 99.9%

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

      9. distribute-rgt-out 99.9%

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

      10. *-commutative 99.9%

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

      11. *-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -10000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.002\right):\\ \;\;\;\;J \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(0.5 \cdot K\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -10000 \lor \neg \left(t_0 \leq 0.002\right):\\ \;\;\;\;U + t_0 \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -10000.0) (not (<= t_0 0.002)))
     (+ U (* t_0 J))
     (+ U (* 2.0 (* J (* l (cos (* 0.5 K)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -10000.0) || !(t_0 <= 0.002)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(l) - exp(-l)
    if ((t_0 <= (-10000.0d0)) .or. (.not. (t_0 <= 0.002d0))) then
        tmp = u + (t_0 * j)
    else
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -10000.0) || !(t_0 <= 0.002)) {
		tmp = U + (t_0 * J);
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -10000.0) or not (t_0 <= 0.002):
		tmp = U + (t_0 * J)
	else:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -10000.0) || !(t_0 <= 0.002))
		tmp = Float64(U + Float64(t_0 * J));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	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 <= -10000.0) || ~((t_0 <= 0.002)))
		tmp = U + (t_0 * J);
	else
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	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, -10000.0], N[Not[LessEqual[t$95$0, 0.002]], $MachinePrecision]], N[(U + N[(t$95$0 * J), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1e4 or 2e-3 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 73.3%

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

   if -1e4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2e-3

   1. Initial program 72.6%

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

   3. Taylor expanded in l around 0 99.4%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -10000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.002\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_1 \leq -0.9:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t_0\right)\right)\\ \mathbf{elif}\;t_1 \leq -0.58:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{elif}\;t_1 \leq 0.15:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))) (t_1 (cos (/ K 2.0))))
   (if (<= t_1 -0.9)
     (+ U (* 2.0 (* J (* l t_0))))
     (if (<= t_1 -0.58)
       (+ U (* J (* l (+ 2.0 (* (pow K 2.0) -0.25)))))
       (if (<= t_1 0.15)
         (+ U (* 2.0 (* l (* J t_0))))
         (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.9) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (t_1 <= -0.58) {
		tmp = U + (J * (l * (2.0 + (pow(K, 2.0) * -0.25))));
	} else if (t_1 <= 0.15) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((0.5d0 * k))
    t_1 = cos((k / 2.0d0))
    if (t_1 <= (-0.9d0)) then
        tmp = u + (2.0d0 * (j * (l * t_0)))
    else if (t_1 <= (-0.58d0)) then
        tmp = u + (j * (l * (2.0d0 + ((k ** 2.0d0) * (-0.25d0)))))
    else if (t_1 <= 0.15d0) then
        tmp = u + (2.0d0 * (l * (j * t_0)))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_1 <= -0.9) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (t_1 <= -0.58) {
		tmp = U + (J * (l * (2.0 + (Math.pow(K, 2.0) * -0.25))));
	} else if (t_1 <= 0.15) {
		tmp = U + (2.0 * (l * (J * t_0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((0.5 * K))
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_1 <= -0.9:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif t_1 <= -0.58:
		tmp = U + (J * (l * (2.0 + (math.pow(K, 2.0) * -0.25))))
	elif t_1 <= 0.15:
		tmp = U + (2.0 * (l * (J * t_0)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_1 <= -0.9)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif (t_1 <= -0.58)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((K ^ 2.0) * -0.25)))));
	elseif (t_1 <= 0.15)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * t_0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((0.5 * K));
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_1 <= -0.9)
		tmp = U + (2.0 * (J * (l * t_0)));
	elseif (t_1 <= -0.58)
		tmp = U + (J * (l * (2.0 + ((K ^ 2.0) * -0.25))));
	elseif (t_1 <= 0.15)
		tmp = U + (2.0 * (l * (J * t_0)));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.9], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.58], N[(U + N[(J * N[(l * N[(2.0 + N[(N[Power[K, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.15], N[(U + N[(2.0 * N[(l * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.900000000000000022

   1. Initial program 76.4%

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

   3. Taylor expanded in l around 0 71.5%

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

   if -0.900000000000000022 < (cos.f64 (/.f64 K 2)) < -0.57999999999999996

   1. Initial program 95.2%

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

   3. Taylor expanded in l around 0 42.7%

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

      1. associate-*r* 42.8%

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

      2. *-commutative 42.8%

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

      3. associate-*l* 42.7%

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

   5. Simplified 42.7%

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

   6. Taylor expanded in K around 0 70.4%

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

      1. +-commutative 70.4%

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

      2. metadata-eval 70.4%

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

      3. metadata-eval 70.4%

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

      4. distribute-lft-neg-in 70.4%

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

      5. associate-*l* 70.4%

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

      6. *-commutative 70.4%

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

      7. associate-*r* 70.4%

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

      8. distribute-lft-neg-in 70.4%

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

      9. *-commutative 70.4%

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

      10. associate-*l* 70.4%

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

      11. *-commutative 70.4%

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

      12. distribute-neg-in 70.4%

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

      13. distribute-lft-in 80.4%

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

      14. distribute-rgt-neg-in 80.4%

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

      15. associate-*r* 80.4%

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

      16. distribute-rgt-out 80.4%

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

   8. Simplified 80.4%

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

   if -0.57999999999999996 < (cos.f64 (/.f64 K 2)) < 0.149999999999999994

   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 65.8%

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

      1. associate-*r* 65.8%

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

      2. *-commutative 65.8%

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

      3. associate-*l* 65.9%

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

   5. Simplified 65.9%

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

   if 0.149999999999999994 < (cos.f64 (/.f64 K 2))

   1. Initial program 87.9%

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

   3. Taylor expanded in J around 0 87.9%

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

   4. Taylor expanded in l around 0 86.2%

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

      1. *-commutative 86.2%

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

      2. associate-*r* 86.2%

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

      3. *-commutative 86.2%

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

      4. associate-*r* 86.2%

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

      5. distribute-rgt-out 86.2%

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

      6. +-commutative 86.2%

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

      7. associate-*r* 86.2%

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

      8. associate-*r* 86.2%

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

      9. distribute-rgt-out 86.2%

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

      10. *-commutative 86.2%

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

      11. *-commutative 86.2%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in K around 0 82.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.9:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.58:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq 0.15:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := t_1 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -0.235:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 0.0028:\\ \;\;\;\;U + J \cdot \left(t_1 \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (- (exp l) (exp (- l))) J)))
        (t_1 (cos (* 0.5 K)))
        (t_2 (* t_1 (* J (* 0.016666666666666666 (pow l 5.0))))))
   (if (<= l -1.3e+64)
     t_2
     (if (<= l -0.235)
       t_0
       (if (<= l 0.0028)
         (+ U (* J (* t_1 (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 2e+52) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((exp(l) - exp(-l)) * J);
	double t_1 = cos((0.5 * K));
	double t_2 = t_1 * (J * (0.016666666666666666 * pow(l, 5.0)));
	double tmp;
	if (l <= -1.3e+64) {
		tmp = t_2;
	} else if (l <= -0.235) {
		tmp = t_0;
	} else if (l <= 0.0028) {
		tmp = U + (J * (t_1 * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2e+52) {
		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 = u + ((exp(l) - exp(-l)) * j)
    t_1 = cos((0.5d0 * k))
    t_2 = t_1 * (j * (0.016666666666666666d0 * (l ** 5.0d0)))
    if (l <= (-1.3d+64)) then
        tmp = t_2
    else if (l <= (-0.235d0)) then
        tmp = t_0
    else if (l <= 0.0028d0) then
        tmp = u + (j * (t_1 * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 2d+52) 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 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double t_1 = Math.cos((0.5 * K));
	double t_2 = t_1 * (J * (0.016666666666666666 * Math.pow(l, 5.0)));
	double tmp;
	if (l <= -1.3e+64) {
		tmp = t_2;
	} else if (l <= -0.235) {
		tmp = t_0;
	} else if (l <= 0.0028) {
		tmp = U + (J * (t_1 * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2e+52) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((math.exp(l) - math.exp(-l)) * J)
	t_1 = math.cos((0.5 * K))
	t_2 = t_1 * (J * (0.016666666666666666 * math.pow(l, 5.0)))
	tmp = 0
	if l <= -1.3e+64:
		tmp = t_2
	elif l <= -0.235:
		tmp = t_0
	elif l <= 0.0028:
		tmp = U + (J * (t_1 * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 2e+52:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(t_1 * Float64(J * Float64(0.016666666666666666 * (l ^ 5.0))))
	tmp = 0.0
	if (l <= -1.3e+64)
		tmp = t_2;
	elseif (l <= -0.235)
		tmp = t_0;
	elseif (l <= 0.0028)
		tmp = Float64(U + Float64(J * Float64(t_1 * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 2e+52)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((exp(l) - exp(-l)) * J);
	t_1 = cos((0.5 * K));
	t_2 = t_1 * (J * (0.016666666666666666 * (l ^ 5.0)));
	tmp = 0.0;
	if (l <= -1.3e+64)
		tmp = t_2;
	elseif (l <= -0.235)
		tmp = t_0;
	elseif (l <= 0.0028)
		tmp = U + (J * (t_1 * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 2e+52)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(J * N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.3e+64], t$95$2, If[LessEqual[l, -0.235], t$95$0, If[LessEqual[l, 0.0028], N[(U + N[(J * N[(t$95$1 * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2e+52], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.29999999999999998e64 or 2e52 < l

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in l around 0 98.2%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   6. Taylor expanded in l around inf 98.3%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 98.3%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)} \]

      2. *-commutative 98.3%

         \[\leadsto \left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \]

      3. associate-*r* 98.3%

         \[\leadsto \color{blue}{\left(\left(0.016666666666666666 \cdot J\right) \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)} \]

      4. *-commutative 98.3%

         \[\leadsto \left(\color{blue}{\left(J \cdot 0.016666666666666666\right)} \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right) \]

      5. associate-*r* 98.3%

         \[\leadsto \color{blue}{\left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)} \cdot \cos \left(K \cdot 0.5\right) \]

      6. *-commutative 98.3%

         \[\leadsto \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)} \]

   8. Simplified 98.3%

      \[\leadsto \color{blue}{\left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right) \cdot \cos \left(0.5 \cdot K\right)} \]

   if -1.29999999999999998e64 < l < -0.23499999999999999 or 0.00279999999999999997 < l < 2e52

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 78.7%

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

   if -0.23499999999999999 < l < 0.00279999999999999997

   1. Initial program 72.6%

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

   3. Taylor expanded in J around 0 72.6%

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

   4. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. associate-*r* 99.9%

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

      5. distribute-rgt-out 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-*r* 99.9%

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

      8. associate-*r* 99.9%

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

      9. distribute-rgt-out 99.9%

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

      10. *-commutative 99.9%

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

      11. *-commutative 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+64}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.235:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.0028:\\ \;\;\;\;U + J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{+52}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := t_1 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+64}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -0.28:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 0.003:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t_1\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (- (exp l) (exp (- l))) J)))
        (t_1 (cos (* 0.5 K)))
        (t_2 (* t_1 (* J (* 0.016666666666666666 (pow l 5.0))))))
   (if (<= l -1.3e+64)
     t_2
     (if (<= l -0.28)
       t_0
       (if (<= l 0.003)
         (+ U (* 2.0 (* J (* l t_1))))
         (if (<= l 1.05e+54) t_0 t_2))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((exp(l) - exp(-l)) * J);
	double t_1 = cos((0.5 * K));
	double t_2 = t_1 * (J * (0.016666666666666666 * pow(l, 5.0)));
	double tmp;
	if (l <= -1.3e+64) {
		tmp = t_2;
	} else if (l <= -0.28) {
		tmp = t_0;
	} else if (l <= 0.003) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 1.05e+54) {
		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 = u + ((exp(l) - exp(-l)) * j)
    t_1 = cos((0.5d0 * k))
    t_2 = t_1 * (j * (0.016666666666666666d0 * (l ** 5.0d0)))
    if (l <= (-1.3d+64)) then
        tmp = t_2
    else if (l <= (-0.28d0)) then
        tmp = t_0
    else if (l <= 0.003d0) then
        tmp = u + (2.0d0 * (j * (l * t_1)))
    else if (l <= 1.05d+54) 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 = U + ((Math.exp(l) - Math.exp(-l)) * J);
	double t_1 = Math.cos((0.5 * K));
	double t_2 = t_1 * (J * (0.016666666666666666 * Math.pow(l, 5.0)));
	double tmp;
	if (l <= -1.3e+64) {
		tmp = t_2;
	} else if (l <= -0.28) {
		tmp = t_0;
	} else if (l <= 0.003) {
		tmp = U + (2.0 * (J * (l * t_1)));
	} else if (l <= 1.05e+54) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((math.exp(l) - math.exp(-l)) * J)
	t_1 = math.cos((0.5 * K))
	t_2 = t_1 * (J * (0.016666666666666666 * math.pow(l, 5.0)))
	tmp = 0
	if l <= -1.3e+64:
		tmp = t_2
	elif l <= -0.28:
		tmp = t_0
	elif l <= 0.003:
		tmp = U + (2.0 * (J * (l * t_1)))
	elif l <= 1.05e+54:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J))
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(t_1 * Float64(J * Float64(0.016666666666666666 * (l ^ 5.0))))
	tmp = 0.0
	if (l <= -1.3e+64)
		tmp = t_2;
	elseif (l <= -0.28)
		tmp = t_0;
	elseif (l <= 0.003)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_1))));
	elseif (l <= 1.05e+54)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((exp(l) - exp(-l)) * J);
	t_1 = cos((0.5 * K));
	t_2 = t_1 * (J * (0.016666666666666666 * (l ^ 5.0)));
	tmp = 0.0;
	if (l <= -1.3e+64)
		tmp = t_2;
	elseif (l <= -0.28)
		tmp = t_0;
	elseif (l <= 0.003)
		tmp = U + (2.0 * (J * (l * t_1)));
	elseif (l <= 1.05e+54)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(J * N[(0.016666666666666666 * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.3e+64], t$95$2, If[LessEqual[l, -0.28], t$95$0, If[LessEqual[l, 0.003], N[(U + N[(2.0 * N[(J * N[(l * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.05e+54], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.29999999999999998e64 or 1.04999999999999993e54 < l

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in l around 0 98.2%

      \[\leadsto \mathsf{fma}\left(J, \color{blue}{\left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right), U\right) \]

   6. Taylor expanded in l around inf 98.3%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 98.3%

         \[\leadsto \color{blue}{\left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)} \]

      2. *-commutative 98.3%

         \[\leadsto \left(0.016666666666666666 \cdot J\right) \cdot \left({\ell}^{5} \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \]

      3. associate-*r* 98.3%

         \[\leadsto \color{blue}{\left(\left(0.016666666666666666 \cdot J\right) \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)} \]

      4. *-commutative 98.3%

         \[\leadsto \left(\color{blue}{\left(J \cdot 0.016666666666666666\right)} \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right) \]

      5. associate-*r* 98.3%

         \[\leadsto \color{blue}{\left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)} \cdot \cos \left(K \cdot 0.5\right) \]

      6. *-commutative 98.3%

         \[\leadsto \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right) \cdot \cos \color{blue}{\left(0.5 \cdot K\right)} \]

   8. Simplified 98.3%

      \[\leadsto \color{blue}{\left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right) \cdot \cos \left(0.5 \cdot K\right)} \]

   if -1.29999999999999998e64 < l < -0.28000000000000003 or 0.0030000000000000001 < l < 1.04999999999999993e54

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 78.7%

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

   if -0.28000000000000003 < l < 0.0030000000000000001

   1. Initial program 72.6%

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

   3. Taylor expanded in l around 0 99.4%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.3 \cdot 10^{+64}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \mathbf{elif}\;\ell \leq -0.28:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.003:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+54}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+164}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -60000000000000:\\ \;\;\;\;U + {K}^{2} \cdot \left(J \cdot -0.03125\right)\\ \mathbf{elif}\;\ell \leq -2700000000 \lor \neg \left(\ell \leq 3.5 \cdot 10^{+15}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* l (+ 2.0 (* (pow K 2.0) -0.25)))))))
   (if (<= l -1.9e+164)
     t_0
     (if (<= l -60000000000000.0)
       (+ U (* (pow K 2.0) (* J -0.03125)))
       (if (or (<= l -2700000000.0) (not (<= l 3.5e+15)))
         t_0
         (+ U (* 2.0 (* J (* l (cos (* 0.5 K)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * (2.0 + (pow(K, 2.0) * -0.25))));
	double tmp;
	if (l <= -1.9e+164) {
		tmp = t_0;
	} else if (l <= -60000000000000.0) {
		tmp = U + (pow(K, 2.0) * (J * -0.03125));
	} else if ((l <= -2700000000.0) || !(l <= 3.5e+15)) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (j * (l * (2.0d0 + ((k ** 2.0d0) * (-0.25d0)))))
    if (l <= (-1.9d+164)) then
        tmp = t_0
    else if (l <= (-60000000000000.0d0)) then
        tmp = u + ((k ** 2.0d0) * (j * (-0.03125d0)))
    else if ((l <= (-2700000000.0d0)) .or. (.not. (l <= 3.5d+15))) then
        tmp = t_0
    else
        tmp = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * (2.0 + (Math.pow(K, 2.0) * -0.25))));
	double tmp;
	if (l <= -1.9e+164) {
		tmp = t_0;
	} else if (l <= -60000000000000.0) {
		tmp = U + (Math.pow(K, 2.0) * (J * -0.03125));
	} else if ((l <= -2700000000.0) || !(l <= 3.5e+15)) {
		tmp = t_0;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((0.5 * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (l * (2.0 + (math.pow(K, 2.0) * -0.25))))
	tmp = 0
	if l <= -1.9e+164:
		tmp = t_0
	elif l <= -60000000000000.0:
		tmp = U + (math.pow(K, 2.0) * (J * -0.03125))
	elif (l <= -2700000000.0) or not (l <= 3.5e+15):
		tmp = t_0
	else:
		tmp = U + (2.0 * (J * (l * math.cos((0.5 * K)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((K ^ 2.0) * -0.25)))))
	tmp = 0.0
	if (l <= -1.9e+164)
		tmp = t_0;
	elseif (l <= -60000000000000.0)
		tmp = Float64(U + Float64((K ^ 2.0) * Float64(J * -0.03125)));
	elseif ((l <= -2700000000.0) || !(l <= 3.5e+15))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(0.5 * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (l * (2.0 + ((K ^ 2.0) * -0.25))));
	tmp = 0.0;
	if (l <= -1.9e+164)
		tmp = t_0;
	elseif (l <= -60000000000000.0)
		tmp = U + ((K ^ 2.0) * (J * -0.03125));
	elseif ((l <= -2700000000.0) || ~((l <= 3.5e+15)))
		tmp = t_0;
	else
		tmp = U + (2.0 * (J * (l * cos((0.5 * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(l * N[(2.0 + N[(N[Power[K, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.9e+164], t$95$0, If[LessEqual[l, -60000000000000.0], N[(U + N[(N[Power[K, 2.0], $MachinePrecision] * N[(J * -0.03125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -2700000000.0], N[Not[LessEqual[l, 3.5e+15]], $MachinePrecision]], t$95$0, N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.90000000000000011e164 or -6e13 < l < -2.7e9 or 3.5e15 < 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 35.8%

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

      1. associate-*r* 35.8%

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

      2. *-commutative 35.8%

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

      3. associate-*l* 35.8%

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

   5. Simplified 35.8%

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

   6. Taylor expanded in K around 0 37.4%

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

      1. +-commutative 37.4%

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

      2. metadata-eval 37.4%

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

      3. metadata-eval 37.4%

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

      4. distribute-lft-neg-in 37.4%

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

      5. associate-*l* 37.4%

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

      6. *-commutative 37.4%

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

      7. associate-*r* 37.4%

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

      8. distribute-lft-neg-in 37.4%

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

      9. *-commutative 37.4%

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

      10. associate-*l* 37.4%

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

      11. *-commutative 37.4%

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

      12. distribute-neg-in 37.4%

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

      13. distribute-lft-in 47.7%

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

      14. distribute-rgt-neg-in 47.7%

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

      15. associate-*r* 47.7%

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

      16. distribute-rgt-out 47.7%

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

   8. Simplified 47.7%

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

   if -1.90000000000000011e164 < l < -6e13

   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 1.6%

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

   5. Taylor expanded in K around 0 32.8%

      \[\leadsto \color{blue}{\left(-0.03125 \cdot \left(J \cdot {K}^{2}\right) + 0.25 \cdot J\right)} + U \]

   6. Taylor expanded in K around inf 33.0%

      \[\leadsto \color{blue}{-0.03125 \cdot \left(J \cdot {K}^{2}\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 33.0%

         \[\leadsto \color{blue}{\left(J \cdot {K}^{2}\right) \cdot -0.03125} + U \]

      2. *-commutative 33.0%

         \[\leadsto \color{blue}{\left({K}^{2} \cdot J\right)} \cdot -0.03125 + U \]

      3. associate-*l* 33.0%

         \[\leadsto \color{blue}{{K}^{2} \cdot \left(J \cdot -0.03125\right)} + U \]

   8. Simplified 33.0%

      \[\leadsto \color{blue}{{K}^{2} \cdot \left(J \cdot -0.03125\right)} + U \]

   if -2.7e9 < l < 3.5e15

   1. Initial program 74.3%

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

   3. Taylor expanded in l around 0 94.4%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+164}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{elif}\;\ell \leq -60000000000000:\\ \;\;\;\;U + {K}^{2} \cdot \left(J \cdot -0.03125\right)\\ \mathbf{elif}\;\ell \leq -2700000000 \lor \neg \left(\ell \leq 3.5 \cdot 10^{+15}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {K}^{2} \cdot -0.25\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -19500000000000:\\ \;\;\;\;U + {K}^{2} \cdot \left(J \cdot -0.03125\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+15} \lor \neg \left(\ell \leq 2.7 \cdot 10^{+177}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{U}^{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* J (* l 2.0)))))
   (if (<= l -4e+166)
     t_0
     (if (<= l -19500000000000.0)
       (+ U (* (pow K 2.0) (* J -0.03125)))
       (if (or (<= l 1.8e+15) (not (<= l 2.7e+177))) t_0 (pow U -3.0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * 2.0));
	double tmp;
	if (l <= -4e+166) {
		tmp = t_0;
	} else if (l <= -19500000000000.0) {
		tmp = U + (pow(K, 2.0) * (J * -0.03125));
	} else if ((l <= 1.8e+15) || !(l <= 2.7e+177)) {
		tmp = t_0;
	} else {
		tmp = pow(U, -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) :: tmp
    t_0 = u + (j * (l * 2.0d0))
    if (l <= (-4d+166)) then
        tmp = t_0
    else if (l <= (-19500000000000.0d0)) then
        tmp = u + ((k ** 2.0d0) * (j * (-0.03125d0)))
    else if ((l <= 1.8d+15) .or. (.not. (l <= 2.7d+177))) then
        tmp = t_0
    else
        tmp = u ** (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (J * (l * 2.0));
	double tmp;
	if (l <= -4e+166) {
		tmp = t_0;
	} else if (l <= -19500000000000.0) {
		tmp = U + (Math.pow(K, 2.0) * (J * -0.03125));
	} else if ((l <= 1.8e+15) || !(l <= 2.7e+177)) {
		tmp = t_0;
	} else {
		tmp = Math.pow(U, -3.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (J * (l * 2.0))
	tmp = 0
	if l <= -4e+166:
		tmp = t_0
	elif l <= -19500000000000.0:
		tmp = U + (math.pow(K, 2.0) * (J * -0.03125))
	elif (l <= 1.8e+15) or not (l <= 2.7e+177):
		tmp = t_0
	else:
		tmp = math.pow(U, -3.0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(J * Float64(l * 2.0)))
	tmp = 0.0
	if (l <= -4e+166)
		tmp = t_0;
	elseif (l <= -19500000000000.0)
		tmp = Float64(U + Float64((K ^ 2.0) * Float64(J * -0.03125)));
	elseif ((l <= 1.8e+15) || !(l <= 2.7e+177))
		tmp = t_0;
	else
		tmp = U ^ -3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (J * (l * 2.0));
	tmp = 0.0;
	if (l <= -4e+166)
		tmp = t_0;
	elseif (l <= -19500000000000.0)
		tmp = U + ((K ^ 2.0) * (J * -0.03125));
	elseif ((l <= 1.8e+15) || ~((l <= 2.7e+177)))
		tmp = t_0;
	else
		tmp = U ^ -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4e+166], t$95$0, If[LessEqual[l, -19500000000000.0], N[(U + N[(N[Power[K, 2.0], $MachinePrecision] * N[(J * -0.03125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 1.8e+15], N[Not[LessEqual[l, 2.7e+177]], $MachinePrecision]], t$95$0, N[Power[U, -3.0], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.99999999999999976e166 or -1.95e13 < l < 1.8e15 or 2.69999999999999991e177 < l

   1. Initial program 82.5%

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

   3. Taylor expanded in l around 0 79.3%

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

   4. Taylor expanded in K around 0 68.0%

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

      1. *-commutative 68.0%

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

      2. associate-*r* 68.0%

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

   6. Simplified 68.0%

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

   if -3.99999999999999976e166 < l < -1.95e13

   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 1.6%

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

   5. Taylor expanded in K around 0 32.8%

      \[\leadsto \color{blue}{\left(-0.03125 \cdot \left(J \cdot {K}^{2}\right) + 0.25 \cdot J\right)} + U \]

   6. Taylor expanded in K around inf 33.0%

      \[\leadsto \color{blue}{-0.03125 \cdot \left(J \cdot {K}^{2}\right)} + U \]
   7. Step-by-step derivation

      1. *-commutative 33.0%

         \[\leadsto \color{blue}{\left(J \cdot {K}^{2}\right) \cdot -0.03125} + U \]

      2. *-commutative 33.0%

         \[\leadsto \color{blue}{\left({K}^{2} \cdot J\right)} \cdot -0.03125 + U \]

      3. associate-*l* 33.0%

         \[\leadsto \color{blue}{{K}^{2} \cdot \left(J \cdot -0.03125\right)} + U \]

   8. Simplified 33.0%

      \[\leadsto \color{blue}{{K}^{2} \cdot \left(J \cdot -0.03125\right)} + U \]

   if 1.8e15 < l < 2.69999999999999991e177

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 28.0%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4 \cdot 10^{+166}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -19500000000000:\\ \;\;\;\;U + {K}^{2} \cdot \left(J \cdot -0.03125\right)\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+15} \lor \neg \left(\ell \leq 2.7 \cdot 10^{+177}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+15} \lor \neg \left(\ell \leq 2.7 \cdot 10^{+177}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l 1.8e+15) (not (<= l 2.7e+177)))
   (+ U (* J (* l 2.0)))
   (pow U -3.0)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 1.8e+15) || !(l <= 2.7e+177)) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = pow(U, -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= 1.8d+15) .or. (.not. (l <= 2.7d+177))) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = u ** (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= 1.8e+15) || !(l <= 2.7e+177)) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = Math.pow(U, -3.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= 1.8e+15) or not (l <= 2.7e+177):
		tmp = U + (J * (l * 2.0))
	else:
		tmp = math.pow(U, -3.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= 1.8e+15) || !(l <= 2.7e+177))
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = U ^ -3.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= 1.8e+15) || ~((l <= 2.7e+177)))
		tmp = U + (J * (l * 2.0));
	else
		tmp = U ^ -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, 1.8e+15], N[Not[LessEqual[l, 2.7e+177]], $MachinePrecision]], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[U, -3.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.8e15 or 2.69999999999999991e177 < l

   1. Initial program 85.1%

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

   3. Taylor expanded in l around 0 69.6%

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

   4. Taylor expanded in K around 0 59.5%

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

      1. *-commutative 59.5%

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

      2. associate-*r* 59.5%

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

   6. Simplified 59.5%

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

   if 1.8e15 < l < 2.69999999999999991e177

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 28.0%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.8 \cdot 10^{+15} \lor \neg \left(\ell \leq 2.7 \cdot 10^{+177}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (2.0d0 * (j * (l * cos((0.5d0 * k)))))
end function

Java

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

Python

def code(J, l, K, U):
	return U + (2.0 * (J * (l * math.cos((0.5 * K)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in l around 0 62.0%

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

4. Final simplification 62.0%

   \[\leadsto U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right) \]
5. Add Preprocessing

Alternative 11: 42.9% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -200000000000 \lor \neg \left(\ell \leq 1000\right):\\ \;\;\;\;U \cdot \left(U + -1\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -200000000000.0) (not (<= l 1000.0))) (* U (+ U -1.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -200000000000.0) || !(l <= 1000.0)) {
		tmp = U * (U + -1.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-200000000000.0d0)) .or. (.not. (l <= 1000.0d0))) then
        tmp = u * (u + (-1.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -200000000000.0) || !(l <= 1000.0)) {
		tmp = U * (U + -1.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -200000000000.0) or not (l <= 1000.0):
		tmp = U * (U + -1.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -200000000000.0) || !(l <= 1000.0))
		tmp = Float64(U * Float64(U + -1.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -200000000000.0) || ~((l <= 1000.0)))
		tmp = U * (U + -1.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -200000000000.0], N[Not[LessEqual[l, 1000.0]], $MachinePrecision]], N[(U * N[(U + -1.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -2e11 or 1e3 < 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 29.6%

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

      1. associate-*r* 29.6%

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

      2. *-commutative 29.6%

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

      3. associate-*l* 29.6%

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

   5. Simplified 29.6%

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

   7. Applied egg-rr 12.4%

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

      1. fma-udef 12.4%

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

      2. neg-mul-1 12.4%

         \[\leadsto U \cdot U + \color{blue}{-1 \cdot U} \]

      3. distribute-rgt-out 12.4%

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

   9. Simplified 12.4%

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

   if -2e11 < l < 1e3

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. fma-def 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in J around 0 68.0%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -200000000000 \lor \neg \left(\ell \leq 1000\right):\\ \;\;\;\;U \cdot \left(U + -1\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.9% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -41000000000 \lor \neg \left(\ell \leq 620\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -41000000000.0) (not (<= l 620.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -41000000000.0) || !(l <= 620.0)) {
		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 <= (-41000000000.0d0)) .or. (.not. (l <= 620.0d0))) 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 <= -41000000000.0) || !(l <= 620.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -41000000000.0) or not (l <= 620.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -41000000000.0) || !(l <= 620.0))
		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 <= -41000000000.0) || ~((l <= 620.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -41000000000.0], N[Not[LessEqual[l, 620.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -4.1e10 or 620 < l

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-def 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 12.4%

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

   if -4.1e10 < l < 620

   1. Initial program 73.7%

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

      1. associate-*l* 73.7%

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

      2. fma-def 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in J around 0 68.0%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -41000000000 \lor \neg \left(\ell \leq 620\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 54.2% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

3. Taylor expanded in l around 0 62.0%

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

4. Taylor expanded in K around 0 51.6%

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

   1. *-commutative 51.6%

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

   2. associate-*r* 51.6%

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

6. Simplified 51.6%

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

7. Final simplification 51.6%

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

Alternative 14: 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 87.2%

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

   1. associate-*l* 87.3%

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

   2. fma-def 87.3%

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

3. Simplified 87.3%

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

6. Applied egg-rr 2.6%

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

   1. *-inverses 2.6%

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

8. Simplified 2.6%

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

9. Final simplification 2.6%

   \[\leadsto 1 \]
10. Add Preprocessing

Alternative 15: 36.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.2%

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

   1. associate-*l* 87.3%

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

   2. fma-def 87.3%

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

3. Simplified 87.3%

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

5. Taylor expanded in J around 0 34.1%

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

6. Final simplification 34.1%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))