Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.8%

Time: 13.7s

Alternatives: 15

Speedup: 1.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0.0005\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0005)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       J
       (*
        l
        (*
         (cos (* K 0.5))
         (+
          2.0
          (*
           (pow l 2.0)
           (+ 0.3333333333333333 (* (pow l 2.0) 0.016666666666666666)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0005)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (l * (cos((K * 0.5)) * (2.0 + (pow(l, 2.0) * (0.3333333333333333 + (pow(l, 2.0) * 0.016666666666666666)))))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0005)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (l * (Math.cos((K * 0.5)) * (2.0 + (Math.pow(l, 2.0) * (0.3333333333333333 + (Math.pow(l, 2.0) * 0.016666666666666666)))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0005):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (J * (l * (math.cos((K * 0.5)) * (2.0 + (math.pow(l, 2.0) * (0.3333333333333333 + (math.pow(l, 2.0) * 0.016666666666666666)))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.0005))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(2.0 + Float64((l ^ 2.0) * Float64(0.3333333333333333 + Float64((l ^ 2.0) * 0.016666666666666666))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.0005)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (J * (l * (cos((K * 0.5)) * (2.0 + ((l ^ 2.0) * (0.3333333333333333 + ((l ^ 2.0) * 0.016666666666666666)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0005]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 5.0000000000000001e-4 < (-.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))) < 5.0000000000000001e-4

   1. Initial program 74.4%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0005\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot 0.016666666666666666\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -0.005 \lor \neg \left(t\_0 \leq 0.0005\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -0.005) (not (<= t_0 0.0005)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       J
       (*
        l
        (* (cos (* K 0.5)) (+ 2.0 (* (pow l 2.0) 0.3333333333333333)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -0.005) || !(t_0 <= 0.0005)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (l * (cos((K * 0.5)) * (2.0 + (pow(l, 2.0) * 0.3333333333333333)))));
	}
	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 <= (-0.005d0)) .or. (.not. (t_0 <= 0.0005d0))) then
        tmp = ((t_0 * j) * cos((k / 2.0d0))) + u
    else
        tmp = u + (j * (l * (cos((k * 0.5d0)) * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0)))))
    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 <= -0.005) || !(t_0 <= 0.0005)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + (J * (l * (Math.cos((K * 0.5)) * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -0.005) or not (t_0 <= 0.0005):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + (J * (l * (math.cos((K * 0.5)) * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -0.005) || !(t_0 <= 0.0005))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333))))));
	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 <= -0.005) || ~((t_0 <= 0.0005)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + (J * (l * (cos((K * 0.5)) * (2.0 + ((l ^ 2.0) * 0.3333333333333333)))));
	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, -0.005], N[Not[LessEqual[t$95$0, 0.0005]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(J * N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.0050000000000000001 or 5.0000000000000001e-4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   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

   if -0.0050000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000001e-4

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

   7. Taylor expanded in l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-rgt-out 99.9%

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

      4. *-commutative 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.005 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.0005\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -0.005:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot t\_0\right)\\ \mathbf{elif}\;t\_1 \leq 0.0005:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(t\_0 \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (- (exp l) (exp (- l)))))
   (if (<= t_1 -0.005)
     (+ U (* t_1 (* J t_0)))
     (if (<= t_1 0.0005)
       (+ U (* J (* l (* t_0 (+ 2.0 (* (pow l 2.0) 0.3333333333333333))))))
       (+ (* (* t_1 J) (cos (/ K 2.0))) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if (t_1 <= -0.005) {
		tmp = U + (t_1 * (J * t_0));
	} else if (t_1 <= 0.0005) {
		tmp = U + (J * (l * (t_0 * (2.0 + (pow(l, 2.0) * 0.3333333333333333)))));
	} else {
		tmp = ((t_1 * J) * cos((K / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = exp(l) - exp(-l)
    if (t_1 <= (-0.005d0)) then
        tmp = u + (t_1 * (j * t_0))
    else if (t_1 <= 0.0005d0) then
        tmp = u + (j * (l * (t_0 * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0)))))
    else
        tmp = ((t_1 * j) * cos((k / 2.0d0))) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if (t_1 <= -0.005) {
		tmp = U + (t_1 * (J * t_0));
	} else if (t_1 <= 0.0005) {
		tmp = U + (J * (l * (t_0 * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333)))));
	} else {
		tmp = ((t_1 * J) * Math.cos((K / 2.0))) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if t_1 <= -0.005:
		tmp = U + (t_1 * (J * t_0))
	elif t_1 <= 0.0005:
		tmp = U + (J * (l * (t_0 * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333)))))
	else:
		tmp = ((t_1 * J) * math.cos((K / 2.0))) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if (t_1 <= -0.005)
		tmp = Float64(U + Float64(t_1 * Float64(J * t_0)));
	elseif (t_1 <= 0.0005)
		tmp = Float64(U + Float64(J * Float64(l * Float64(t_0 * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333))))));
	else
		tmp = Float64(Float64(Float64(t_1 * J) * cos(Float64(K / 2.0))) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if (t_1 <= -0.005)
		tmp = U + (t_1 * (J * t_0));
	elseif (t_1 <= 0.0005)
		tmp = U + (J * (l * (t_0 * (2.0 + ((l ^ 2.0) * 0.3333333333333333)))));
	else
		tmp = ((t_1 * J) * cos((K / 2.0))) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.005], N[(U + N[(t$95$1 * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0005], N[(U + N[(J * N[(l * N[(t$95$0 * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$1 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.0050000000000000001

   1. Initial program 99.7%

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

   3. Taylor expanded in J around 0 99.7%

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

      1. associate-*r* 99.7%

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

      2. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   if -0.0050000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 5.0000000000000001e-4

   1. Initial program 73.7%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

   7. Taylor expanded in l around 0 99.9%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. distribute-rgt-out 99.9%

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

      4. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   if 5.0000000000000001e-4 < (-.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. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.005:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 0.0005:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (<= t_0 (- INFINITY))
     t_0
     (if (<= t_0 2e+282) (fma J (* (cos (/ K 2.0)) (* l 2.0)) U) (+ t_0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_0 <= 2e+282) {
		tmp = fma(J, (cos((K / 2.0)) * (l * 2.0)), U);
	} else {
		tmp = t_0 + U;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_0 <= 2e+282)
		tmp = fma(J, Float64(cos(Float64(K / 2.0)) * Float64(l * 2.0)), U);
	else
		tmp = Float64(t_0 + U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$0, If[LessEqual[t$95$0, 2e+282], N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(t$95$0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 70.2%

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

   4. Taylor expanded in J around inf 70.2%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2.00000000000000007e282

   1. Initial program 74.4%

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

      1. associate-*l* 74.4%

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

      2. fma-define 74.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in l around 0 98.7%

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

   if 2.00000000000000007e282 < (*.f64 J (-.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 80.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq -\infty:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (<= t_0 (- INFINITY))
     t_0
     (if (<= t_0 2e+282)
       (+ U (* J (* l (* 2.0 (cos (* K 0.5))))))
       (+ t_0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_0;
	} else if (t_0 <= 2e+282) {
		tmp = U + (J * (l * (2.0 * cos((K * 0.5)))));
	} else {
		tmp = t_0 + U;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_0;
	} else if (t_0 <= 2e+282) {
		tmp = U + (J * (l * (2.0 * Math.cos((K * 0.5)))));
	} else {
		tmp = t_0 + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_0
	elif t_0 <= 2e+282:
		tmp = U + (J * (l * (2.0 * math.cos((K * 0.5)))))
	else:
		tmp = t_0 + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_0;
	elseif (t_0 <= 2e+282)
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(t_0 + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_0;
	elseif (t_0 <= 2e+282)
		tmp = U + (J * (l * (2.0 * cos((K * 0.5)))));
	else
		tmp = t_0 + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$0, If[LessEqual[t$95$0, 2e+282], N[(U + N[(J * N[(l * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 70.2%

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

   4. Taylor expanded in J around inf 70.2%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2.00000000000000007e282

   1. Initial program 74.4%

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

   3. Taylor expanded in l around 0 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*l* 98.7%

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

      3. associate-*r* 98.7%

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

      4. *-commutative 98.7%

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

   5. Simplified 98.7%

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

   if 2.00000000000000007e282 < (*.f64 J (-.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 80.0%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq -\infty:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq 2 \cdot 10^{+282}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 87.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 2 \cdot 10^{+282}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e+282)))
     t_0
     (+ U (* J (* l (* 2.0 (cos (* K 0.5)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e+282)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (l * (2.0 * cos((K * 0.5)))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e+282)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (l * (2.0 * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e+282):
		tmp = t_0
	else:
		tmp = U + (J * (l * (2.0 * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e+282))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e+282)))
		tmp = t_0;
	else
		tmp = U + (J * (l * (2.0 * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e+282]], $MachinePrecision]], t$95$0, N[(U + N[(J * N[(l * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < -inf.0 or 2.00000000000000007e282 < (*.f64 J (-.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 75.4%

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

   4. Taylor expanded in J around inf 75.4%

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

   if -inf.0 < (*.f64 J (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))) < 2.00000000000000007e282

   1. Initial program 74.4%

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

   3. Taylor expanded in l around 0 98.7%

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

      1. *-commutative 98.7%

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

      2. associate-*l* 98.7%

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

      3. associate-*r* 98.7%

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

      4. *-commutative 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq -\infty \lor \neg \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J \leq 2 \cdot 10^{+282}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+49}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -260:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 0.18:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{+41}:\\ \;\;\;\;t\_1 + U\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (pow l 5.0) (* (cos (* K 0.5)) (* J 0.016666666666666666)))))
        (t_1 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -6.4e+49)
     t_0
     (if (<= l -260.0)
       t_1
       (if (<= l 0.18)
         (fma J (* (cos (/ K 2.0)) (* l 2.0)) U)
         (if (<= l 1.42e+41) (+ t_1 U) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (pow(l, 5.0) * (cos((K * 0.5)) * (J * 0.016666666666666666)));
	double t_1 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -6.4e+49) {
		tmp = t_0;
	} else if (l <= -260.0) {
		tmp = t_1;
	} else if (l <= 0.18) {
		tmp = fma(J, (cos((K / 2.0)) * (l * 2.0)), U);
	} else if (l <= 1.42e+41) {
		tmp = t_1 + U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64((l ^ 5.0) * Float64(cos(Float64(K * 0.5)) * Float64(J * 0.016666666666666666))))
	t_1 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -6.4e+49)
		tmp = t_0;
	elseif (l <= -260.0)
		tmp = t_1;
	elseif (l <= 0.18)
		tmp = fma(J, Float64(cos(Float64(K / 2.0)) * Float64(l * 2.0)), U);
	elseif (l <= 1.42e+41)
		tmp = Float64(t_1 + U);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Power[l, 5.0], $MachinePrecision] * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -6.4e+49], t$95$0, If[LessEqual[l, -260.0], t$95$1, If[LessEqual[l, 0.18], N[(J * N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 1.42e+41], N[(t$95$1 + U), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.40000000000000028e49 or 1.42000000000000007e41 < 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 98.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in l around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*r* 98.1%

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

      3. associate-*l* 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*r* 98.1%

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

      6. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   if -6.40000000000000028e49 < l < -260

   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 85.7%

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

   4. Taylor expanded in J around inf 85.7%

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

   if -260 < l < 0.17999999999999999

   1. Initial program 74.4%

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

      1. associate-*l* 74.4%

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

      2. fma-define 74.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in l around 0 98.7%

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

   if 0.17999999999999999 < l < 1.42000000000000007e41

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 80.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+49}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq -260:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.18:\\ \;\;\;\;\mathsf{fma}\left(J, \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot 2\right), U\right)\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{+41}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{5} \cdot \left(t\_0 \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -26.5:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 0.018:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(t\_0 \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{+41}:\\ \;\;\;\;t\_2 + U\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 5.0) (* t_0 (* J 0.016666666666666666)))))
        (t_2 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -6.4e+49)
     t_1
     (if (<= l -26.5)
       t_2
       (if (<= l 0.018)
         (+ U (* J (* l (* t_0 (+ 2.0 (* (pow l 2.0) 0.3333333333333333))))))
         (if (<= l 1.42e+41) (+ t_2 U) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)));
	double t_2 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -6.4e+49) {
		tmp = t_1;
	} else if (l <= -26.5) {
		tmp = t_2;
	} else if (l <= 0.018) {
		tmp = U + (J * (l * (t_0 * (2.0 + (pow(l, 2.0) * 0.3333333333333333)))));
	} else if (l <= 1.42e+41) {
		tmp = t_2 + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((l ** 5.0d0) * (t_0 * (j * 0.016666666666666666d0)))
    t_2 = (exp(l) - exp(-l)) * j
    if (l <= (-6.4d+49)) then
        tmp = t_1
    else if (l <= (-26.5d0)) then
        tmp = t_2
    else if (l <= 0.018d0) then
        tmp = u + (j * (l * (t_0 * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0)))))
    else if (l <= 1.42d+41) then
        tmp = t_2 + u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (Math.pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)));
	double t_2 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -6.4e+49) {
		tmp = t_1;
	} else if (l <= -26.5) {
		tmp = t_2;
	} else if (l <= 0.018) {
		tmp = U + (J * (l * (t_0 * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333)))));
	} else if (l <= 1.42e+41) {
		tmp = t_2 + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (math.pow(l, 5.0) * (t_0 * (J * 0.016666666666666666)))
	t_2 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -6.4e+49:
		tmp = t_1
	elif l <= -26.5:
		tmp = t_2
	elif l <= 0.018:
		tmp = U + (J * (l * (t_0 * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333)))))
	elif l <= 1.42e+41:
		tmp = t_2 + U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64((l ^ 5.0) * Float64(t_0 * Float64(J * 0.016666666666666666))))
	t_2 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -6.4e+49)
		tmp = t_1;
	elseif (l <= -26.5)
		tmp = t_2;
	elseif (l <= 0.018)
		tmp = Float64(U + Float64(J * Float64(l * Float64(t_0 * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333))))));
	elseif (l <= 1.42e+41)
		tmp = Float64(t_2 + U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((l ^ 5.0) * (t_0 * (J * 0.016666666666666666)));
	t_2 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -6.4e+49)
		tmp = t_1;
	elseif (l <= -26.5)
		tmp = t_2;
	elseif (l <= 0.018)
		tmp = U + (J * (l * (t_0 * (2.0 + ((l ^ 2.0) * 0.3333333333333333)))));
	elseif (l <= 1.42e+41)
		tmp = t_2 + U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[Power[l, 5.0], $MachinePrecision] * N[(t$95$0 * N[(J * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -6.4e+49], t$95$1, If[LessEqual[l, -26.5], t$95$2, If[LessEqual[l, 0.018], N[(U + N[(J * N[(l * N[(t$95$0 * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.42e+41], N[(t$95$2 + U), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -6.40000000000000028e49 or 1.42000000000000007e41 < 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 98.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in l around inf 98.1%

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

      1. *-commutative 98.1%

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

      2. associate-*r* 98.1%

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

      3. associate-*l* 98.1%

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

      4. *-commutative 98.1%

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

      5. associate-*r* 98.1%

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

      6. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   if -6.40000000000000028e49 < l < -26.5

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 85.7%

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

   4. Taylor expanded in J around inf 85.7%

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

   if -26.5 < l < 0.0179999999999999986

   1. Initial program 74.4%

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

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in J around 0 99.9%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)\right)} + U \]

   7. Taylor expanded in l around 0 99.6%

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

      1. associate-*r* 99.6%

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

      2. *-commutative 99.6%

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

      3. distribute-rgt-out 99.6%

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

      4. *-commutative 99.6%

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

   9. Simplified 99.6%

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

   if 0.0179999999999999986 < l < 1.42000000000000007e41

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 80.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.4 \cdot 10^{+49}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \mathbf{elif}\;\ell \leq -26.5:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 0.018:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 1.42 \cdot 10^{+41}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{5} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.016666666666666666\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+16} \lor \neg \left(\ell \leq 1.08 \cdot 10^{+19}\right):\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3.7e+16) (not (<= l 1.08e+19)))
   (+ U (* 0.016666666666666666 (* J (pow l 5.0))))
   (+ U (* J (* l (* 2.0 (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.7e+16) || !(l <= 1.08e+19)) {
		tmp = U + (0.016666666666666666 * (J * pow(l, 5.0)));
	} else {
		tmp = U + (J * (l * (2.0 * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-3.7d+16)) .or. (.not. (l <= 1.08d+19))) then
        tmp = u + (0.016666666666666666d0 * (j * (l ** 5.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.7e+16) || !(l <= 1.08e+19)) {
		tmp = U + (0.016666666666666666 * (J * Math.pow(l, 5.0)));
	} else {
		tmp = U + (J * (l * (2.0 * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -3.7e+16) or not (l <= 1.08e+19):
		tmp = U + (0.016666666666666666 * (J * math.pow(l, 5.0)))
	else:
		tmp = U + (J * (l * (2.0 * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -3.7e+16) || !(l <= 1.08e+19))
		tmp = Float64(U + Float64(0.016666666666666666 * Float64(J * (l ^ 5.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -3.7e+16) || ~((l <= 1.08e+19)))
		tmp = U + (0.016666666666666666 * (J * (l ^ 5.0)));
	else
		tmp = U + (J * (l * (2.0 * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3.7e+16], N[Not[LessEqual[l, 1.08e+19]], $MachinePrecision]], N[(U + N[(0.016666666666666666 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.7e16 or 1.08e19 < 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 90.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 90.2%

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

   5. Simplified 90.2%

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

   6. Taylor expanded in l around inf 90.2%

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

      1. *-commutative 90.2%

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

      2. associate-*r* 90.2%

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

      3. associate-*l* 90.2%

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

      4. *-commutative 90.2%

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

      5. associate-*r* 90.2%

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

      6. *-commutative 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in K around 0 67.1%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)} + U \]

   if -3.7e16 < l < 1.08e19

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0 95.3%

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

      1. *-commutative 95.3%

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

      2. associate-*l* 95.3%

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

      3. associate-*r* 95.3%

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

      4. *-commutative 95.3%

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

   5. Simplified 95.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+16} \lor \neg \left(\ell \leq 1.08 \cdot 10^{+19}\right):\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.15 \cdot 10^{+16} \lor \neg \left(\ell \leq 3.3\right):\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3.15e+16) (not (<= l 3.3)))
   (+ U (* 0.016666666666666666 (* J (pow l 5.0))))
   (fma (* J 2.0) l U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.15e+16) || !(l <= 3.3)) {
		tmp = U + (0.016666666666666666 * (J * pow(l, 5.0)));
	} else {
		tmp = fma((J * 2.0), l, U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -3.15e+16) || !(l <= 3.3))
		tmp = Float64(U + Float64(0.016666666666666666 * Float64(J * (l ^ 5.0))));
	else
		tmp = fma(Float64(J * 2.0), l, U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3.15e+16], N[Not[LessEqual[l, 3.3]], $MachinePrecision]], N[(U + N[(0.016666666666666666 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(J * 2.0), $MachinePrecision] * l + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -3.15e16 or 3.2999999999999998 < 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 88.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 88.9%

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

   5. Simplified 88.9%

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

   6. Taylor expanded in l around inf 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*r* 88.9%

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

      3. associate-*l* 88.9%

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

      4. *-commutative 88.9%

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

      5. associate-*r* 88.9%

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

      6. *-commutative 88.9%

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

   8. Simplified 88.9%

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

   9. Taylor expanded in K around 0 65.5%

      \[\leadsto \color{blue}{0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)} + U \]

   if -3.15e16 < l < 3.2999999999999998

   1. Initial program 74.8%

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

   3. Taylor expanded in l around 0 98.5%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in l around 0 97.3%

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

      1. *-commutative 97.3%

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

      2. associate-*l* 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in K around 0 83.9%

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

       1. +-commutative 83.9%

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

       2. associate-*r* 83.9%

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

       3. fma-define 83.9%

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

   11. Simplified 83.9%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.15 \cdot 10^{+16} \lor \neg \left(\ell \leq 3.3\right):\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J \cdot 2, \ell, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.6%

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

3. Taylor expanded in l around 0 94.0%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 94.0%

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

5. Simplified 94.0%

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

6. Taylor expanded in l around 0 64.2%

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

   1. *-commutative 64.2%

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

   2. associate-*l* 64.2%

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

8. Simplified 64.2%

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

9. Taylor expanded in K around 0 54.1%

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

    1. +-commutative 54.1%

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

    2. associate-*r* 54.1%

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

    3. fma-define 54.1%

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

11. Simplified 54.1%

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

12. Final simplification 54.1%

    \[\leadsto \mathsf{fma}\left(J \cdot 2, \ell, U\right) \]
13. Add Preprocessing

Alternative 12: 41.9% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+49} \lor \neg \left(\ell \leq 1.35 \cdot 10^{+23}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.5e+49) (not (<= l 1.35e+23))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.5e+49) || !(l <= 1.35e+23)) {
		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 <= (-6.5d+49)) .or. (.not. (l <= 1.35d+23))) 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 <= -6.5e+49) || !(l <= 1.35e+23)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.5e+49) or not (l <= 1.35e+23):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.5e+49) || !(l <= 1.35e+23))
		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 <= -6.5e+49) || ~((l <= 1.35e+23)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.5e+49], N[Not[LessEqual[l, 1.35e+23]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -6.5000000000000005e49 or 1.3499999999999999e23 < 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-define 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 14.4%

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

   if -6.5000000000000005e49 < l < 1.3499999999999999e23

   1. Initial program 76.7%

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

      1. associate-*l* 76.7%

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

      2. fma-define 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in J around 0 65.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.5 \cdot 10^{+49} \lor \neg \left(\ell \leq 1.35 \cdot 10^{+23}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 42.2% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -750:\\ \;\;\;\;U \cdot U - U\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+23}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -750.0) (- (* U U) U) (if (<= l 1.4e+23) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -750.0) {
		tmp = (U * U) - U;
	} else if (l <= 1.4e+23) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-750.0d0)) then
        tmp = (u * u) - u
    else if (l <= 1.4d+23) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -750.0) {
		tmp = (U * U) - U;
	} else if (l <= 1.4e+23) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -750.0:
		tmp = (U * U) - U
	elif l <= 1.4e+23:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -750.0)
		tmp = Float64(Float64(U * U) - U);
	elseif (l <= 1.4e+23)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -750.0)
		tmp = (U * U) - U;
	elseif (l <= 1.4e+23)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -750.0], N[(N[(U * U), $MachinePrecision] - U), $MachinePrecision], If[LessEqual[l, 1.4e+23], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -750

   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 24.6%

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

      1. associate-*r* 24.6%

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

   5. Simplified 24.6%

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

   7. Applied egg-rr 16.6%

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

      1. fmm-undef 16.6%

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

   9. Simplified 16.6%

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

   if -750 < l < 1.4e23

   1. Initial program 75.3%

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

      1. associate-*l* 75.3%

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

      2. fma-define 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in J around 0 69.2%

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

   if 1.4e23 < 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-define 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 10.3%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -750:\\ \;\;\;\;U \cdot U - U\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+23}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 54.6% 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 86.6%

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

3. Taylor expanded in l around 0 94.0%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {\ell}^{2}\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 94.0%

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

5. Simplified 94.0%

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

6. Taylor expanded in l around 0 64.2%

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

   1. *-commutative 64.2%

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

   2. associate-*l* 64.2%

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

8. Simplified 64.2%

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

9. Taylor expanded in K around 0 54.1%

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

10. Final simplification 54.1%

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

Alternative 15: 36.9% 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 86.6%

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

   1. associate-*l* 86.6%

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

   2. fma-define 86.6%

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

3. Simplified 86.6%

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

5. Taylor expanded in J around 0 38.6%

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

6. Final simplification 38.6%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))