Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.3% → 99.6%

Time: 11.6s

Alternatives: 16

Speedup: 1.4×

• 50.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -0.02 \lor \neg \left(t_1 \leq 0\right):\\ \;\;\;\;U + \left(t_1 \cdot J\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -0.02) (not (<= t_1 0.0)))
     (+ U (* (* t_1 J) t_0))
     (+
      U
      (* t_0 (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -0.02) || !(t_1 <= 0.0)) {
		tmp = U + ((t_1 * J) * t_0);
	} else {
		tmp = U + (t_0 * ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (l * J))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-0.02d0)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = u + ((t_1 * j) * t_0)
    else
        tmp = u + (t_0 * ((0.3333333333333333d0 * (j * (l ** 3.0d0))) + (2.0d0 * (l * j))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -0.02) || !(t_1 <= 0.0)) {
		tmp = U + ((t_1 * J) * t_0);
	} else {
		tmp = U + (t_0 * ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -0.02) or not (t_1 <= 0.0):
		tmp = U + ((t_1 * J) * t_0)
	else:
		tmp = U + (t_0 * ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -0.02) || !(t_1 <= 0.0))
		tmp = Float64(U + Float64(Float64(t_1 * J) * t_0));
	else
		tmp = Float64(U + Float64(t_0 * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -0.02) || ~((t_1 <= 0.0)))
		tmp = U + ((t_1 * J) * t_0);
	else
		tmp = U + (t_0 * ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.02], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(U + N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.0200000000000000004 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.9%

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

   if -0.0200000000000000004 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 70.8%

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

   2. Taylor expanded in l around 0 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.02 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \]

Alternative 2: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot t_0, \mathsf{fma}\left(0.016666666666666666, t_0 \cdot \left(J \cdot {\ell}^{5}\right), 0.3333333333333333 \cdot \left(J \cdot \left(t_0 \cdot {\ell}^{3}\right)\right) + 2 \cdot \left(t_0 \cdot \left(\ell \cdot J\right)\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(t_1 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K))) (t_1 (- (exp l) (exp (- l)))))
   (if (<= t_1 0.01)
     (+
      (fma
       0.0003968253968253968
       (* (* (pow l 7.0) J) t_0)
       (fma
        0.016666666666666666
        (* t_0 (* J (pow l 5.0)))
        (+
         (* 0.3333333333333333 (* J (* t_0 (pow l 3.0))))
         (* 2.0 (* t_0 (* l J))))))
      U)
     (+ U (* (* t_1 J) (cos (/ K 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 <= 0.01) {
		tmp = fma(0.0003968253968253968, ((pow(l, 7.0) * J) * t_0), fma(0.016666666666666666, (t_0 * (J * pow(l, 5.0))), ((0.3333333333333333 * (J * (t_0 * pow(l, 3.0)))) + (2.0 * (t_0 * (l * J)))))) + U;
	} else {
		tmp = U + ((t_1 * J) * cos((K / 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 <= 0.01)
		tmp = Float64(fma(0.0003968253968253968, Float64(Float64((l ^ 7.0) * J) * t_0), fma(0.016666666666666666, Float64(t_0 * Float64(J * (l ^ 5.0))), Float64(Float64(0.3333333333333333 * Float64(J * Float64(t_0 * (l ^ 3.0)))) + Float64(2.0 * Float64(t_0 * Float64(l * J)))))) + U);
	else
		tmp = Float64(U + Float64(Float64(t_1 * J) * cos(Float64(K / 2.0))));
	end
	return 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[LessEqual[t$95$1, 0.01], N[(N[(0.0003968253968253968 * N[(N[(N[Power[l, 7.0], $MachinePrecision] * J), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.016666666666666666 * N[(t$95$0 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(J * N[(t$95$0 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(t$95$0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(t$95$1 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0100000000000000002

   1. Initial program 81.5%

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

   2. Taylor expanded in l around 0 97.9%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + \left(0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right)\right)} + U \]
   3. Step-by-step derivation

      1. fma-def 97.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.0003968253968253968, J \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right), 0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right)} + U \]

      2. associate-*r* 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \color{blue}{\left(J \cdot {\ell}^{7}\right) \cdot \cos \left(0.5 \cdot K\right)}, 0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right) + U \]

      3. *-commutative 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \color{blue}{\left({\ell}^{7} \cdot J\right)} \cdot \cos \left(0.5 \cdot K\right), 0.016666666666666666 \cdot \left(J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right) + U \]

      4. fma-def 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\mathsf{fma}\left(0.016666666666666666, J \cdot \left({\ell}^{5} \cdot \cos \left(0.5 \cdot K\right)\right), 0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}\right) + U \]

      5. associate-*r* 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \color{blue}{\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(0.5 \cdot K\right)}, 0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right) + U \]

      6. *-commutative 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \color{blue}{\left({\ell}^{5} \cdot J\right)} \cdot \cos \left(0.5 \cdot K\right), 0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right) + U \]

      7. fma-def 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \left({\ell}^{5} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \color{blue}{\mathsf{fma}\left(0.3333333333333333, J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right), 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)}\right)\right) + U \]

      8. associate-*r* 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \left({\ell}^{5} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.3333333333333333, J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right), 2 \cdot \color{blue}{\left(\left(J \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)\right)}\right)\right)\right) + U \]

      9. *-commutative 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \left({\ell}^{5} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.3333333333333333, J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right), 2 \cdot \left(\color{blue}{\left(\ell \cdot J\right)} \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right) + U \]

   4. Simplified 97.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \left({\ell}^{5} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.3333333333333333, J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right), 2 \cdot \left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)\right)\right)\right)} + U \]
   5. Step-by-step derivation

      1. fma-udef 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \left({\ell}^{5} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(\left(\ell \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)\right)}\right)\right) + U \]

      2. *-commutative 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \left({\ell}^{5} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)}\right)\right) + U \]

      3. *-commutative 97.9%

         \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \left({\ell}^{5} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), 0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(J \cdot \ell\right)}\right)\right)\right) + U \]

   6. Applied egg-rr 97.9%

      \[\leadsto \mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \left({\ell}^{5} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \color{blue}{0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{3} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \ell\right)\right)}\right)\right) + U \]

   if 0.0100000000000000002 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(0.0003968253968253968, \left({\ell}^{7} \cdot J\right) \cdot \cos \left(0.5 \cdot K\right), \mathsf{fma}\left(0.016666666666666666, \cos \left(0.5 \cdot K\right) \cdot \left(J \cdot {\ell}^{5}\right), 0.3333333333333333 \cdot \left(J \cdot \left(\cos \left(0.5 \cdot K\right) \cdot {\ell}^{3}\right)\right) + 2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\ell \cdot J\right)\right)\right)\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]

Alternative 3: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.01:\\ \;\;\;\;U + t_1 \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(t_0 \cdot J\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 0.01)
     (+
      U
      (*
       t_1
       (*
        J
        (+
         (* 0.0003968253968253968 (pow l 7.0))
         (+
          (* 0.016666666666666666 (pow l 5.0))
          (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))
     (+ U (* (* t_0 J) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.01) {
		tmp = U + (t_1 * (J * ((0.0003968253968253968 * pow(l, 7.0)) + ((0.016666666666666666 * pow(l, 5.0)) + ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))))));
	} else {
		tmp = U + ((t_0 * J) * 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) :: tmp
    t_0 = exp(l) - exp(-l)
    t_1 = cos((k / 2.0d0))
    if (t_0 <= 0.01d0) then
        tmp = u + (t_1 * (j * ((0.0003968253968253968d0 * (l ** 7.0d0)) + ((0.016666666666666666d0 * (l ** 5.0d0)) + ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))))
    else
        tmp = u + ((t_0 * j) * 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.exp(l) - Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.01) {
		tmp = U + (t_1 * (J * ((0.0003968253968253968 * Math.pow(l, 7.0)) + ((0.016666666666666666 * Math.pow(l, 5.0)) + ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))))));
	} else {
		tmp = U + ((t_0 * J) * t_1);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.01:
		tmp = U + (t_1 * (J * ((0.0003968253968253968 * math.pow(l, 7.0)) + ((0.016666666666666666 * math.pow(l, 5.0)) + ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))))
	else:
		tmp = U + ((t_0 * J) * t_1)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(Float64(0.0003968253968253968 * (l ^ 7.0)) + Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))))));
	else
		tmp = Float64(U + Float64(Float64(t_0 * J) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.01)
		tmp = U + (t_1 * (J * ((0.0003968253968253968 * (l ^ 7.0)) + ((0.016666666666666666 * (l ^ 5.0)) + ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))))));
	else
		tmp = U + ((t_0 * J) * t_1);
	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]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(U + N[(t$95$1 * N[(J * N[(N[(0.0003968253968253968 * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision] + 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(t$95$0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0100000000000000002

   1. Initial program 81.5%

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

   2. Taylor expanded in l around 0 97.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 0.0100000000000000002 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 0.01:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]

Alternative 4: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.01:\\ \;\;\;\;U + t_1 \cdot \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(t_0 \cdot J\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 0.01)
     (+
      U
      (*
       t_1
       (+
        (* 0.016666666666666666 (* J (pow l 5.0)))
        (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J))))))
     (+ U (* (* t_0 J) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.01) {
		tmp = U + (t_1 * ((0.016666666666666666 * (J * pow(l, 5.0))) + ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (l * J)))));
	} else {
		tmp = U + ((t_0 * J) * 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) :: tmp
    t_0 = exp(l) - exp(-l)
    t_1 = cos((k / 2.0d0))
    if (t_0 <= 0.01d0) then
        tmp = u + (t_1 * ((0.016666666666666666d0 * (j * (l ** 5.0d0))) + ((0.3333333333333333d0 * (j * (l ** 3.0d0))) + (2.0d0 * (l * j)))))
    else
        tmp = u + ((t_0 * j) * 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.exp(l) - Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.01) {
		tmp = U + (t_1 * ((0.016666666666666666 * (J * Math.pow(l, 5.0))) + ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J)))));
	} else {
		tmp = U + ((t_0 * J) * t_1);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.01:
		tmp = U + (t_1 * ((0.016666666666666666 * (J * math.pow(l, 5.0))) + ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J)))))
	else:
		tmp = U + ((t_0 * J) * t_1)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(U + Float64(t_1 * Float64(Float64(0.016666666666666666 * Float64(J * (l ^ 5.0))) + Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J))))));
	else
		tmp = Float64(U + Float64(Float64(t_0 * J) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.01)
		tmp = U + (t_1 * ((0.016666666666666666 * (J * (l ^ 5.0))) + ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J)))));
	else
		tmp = U + ((t_0 * J) * t_1);
	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]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(U + N[(t$95$1 * N[(N[(0.016666666666666666 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(t$95$0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0100000000000000002

   1. Initial program 81.5%

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

   2. Taylor expanded in l around 0 97.9%

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

   if 0.0100000000000000002 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 0.01:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]

Alternative 5: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.01:\\ \;\;\;\;U + t_1 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(t_0 \cdot J\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))) (t_1 (cos (/ K 2.0))))
   (if (<= t_0 0.01)
     (+
      U
      (*
       t_1
       (*
        J
        (+
         (* 0.016666666666666666 (pow l 5.0))
         (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))
     (+ U (* (* t_0 J) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.01) {
		tmp = U + (t_1 * (J * ((0.016666666666666666 * pow(l, 5.0)) + ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)))));
	} else {
		tmp = U + ((t_0 * J) * 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) :: tmp
    t_0 = exp(l) - exp(-l)
    t_1 = cos((k / 2.0d0))
    if (t_0 <= 0.01d0) then
        tmp = u + (t_1 * (j * ((0.016666666666666666d0 * (l ** 5.0d0)) + ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))))
    else
        tmp = u + ((t_0 * j) * 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.exp(l) - Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.01) {
		tmp = U + (t_1 * (J * ((0.016666666666666666 * Math.pow(l, 5.0)) + ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)))));
	} else {
		tmp = U + ((t_0 * J) * t_1);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.01:
		tmp = U + (t_1 * (J * ((0.016666666666666666 * math.pow(l, 5.0)) + ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))))
	else:
		tmp = U + ((t_0 * J) * t_1)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.01)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))))));
	else
		tmp = Float64(U + Float64(Float64(t_0 * J) * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= 0.01)
		tmp = U + (t_1 * (J * ((0.016666666666666666 * (l ^ 5.0)) + ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)))));
	else
		tmp = U + ((t_0 * J) * t_1);
	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]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.01], N[(U + N[(t$95$1 * N[(J * 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]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(t$95$0 * J), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0100000000000000002

   1. Initial program 81.5%

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

   2. Taylor expanded in l around 0 97.9%

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

   if 0.0100000000000000002 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq 0.01:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\\ \end{array} \]

Alternative 6: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 0.0002:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 0.0002)
   (+ U (* J (* 2.0 (sinh l))))
   (+
    U
    (*
     (cos (/ K 2.0))
     (+ (* 0.3333333333333333 (* J (pow l 3.0))) (* 2.0 (* l J)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 0.0002) {
		tmp = U + (J * (2.0 * sinh(l)));
	} else {
		tmp = U + (cos((K / 2.0)) * ((0.3333333333333333 * (J * pow(l, 3.0))) + (2.0 * (l * J))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k / 2.0d0) <= 0.0002d0) then
        tmp = u + (j * (2.0d0 * sinh(l)))
    else
        tmp = u + (cos((k / 2.0d0)) * ((0.3333333333333333d0 * (j * (l ** 3.0d0))) + (2.0d0 * (l * j))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 0.0002) {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * ((0.3333333333333333 * (J * Math.pow(l, 3.0))) + (2.0 * (l * J))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 0.0002:
		tmp = U + (J * (2.0 * math.sinh(l)))
	else:
		tmp = U + (math.cos((K / 2.0)) * ((0.3333333333333333 * (J * math.pow(l, 3.0))) + (2.0 * (l * J))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 0.0002)
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))) + Float64(2.0 * Float64(l * J)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K / 2.0) <= 0.0002)
		tmp = U + (J * (2.0 * sinh(l)));
	else
		tmp = U + (cos((K / 2.0)) * ((0.3333333333333333 * (J * (l ^ 3.0))) + (2.0 * (l * J))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 0.0002], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 K 2) < 2.0000000000000001e-4

   1. Initial program 85.3%

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

   2. Taylor expanded in K around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. fma-def 76.6%

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

      3. sinh-undef 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. fma-udef 86.7%

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

      2. *-commutative 86.7%

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

   6. Applied egg-rr 86.7%

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

   if 2.0000000000000001e-4 < (/.f64 K 2)

   1. Initial program 89.5%

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

   2. Taylor expanded in l around 0 88.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 0.0002:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \]

Alternative 7: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 0.0002:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (/ K 2.0) 0.0002)
   (+ U (* J (* 2.0 (sinh l))))
   (+
    U
    (*
     (cos (/ K 2.0))
     (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K / 2.0) <= 0.0002) {
		tmp = U + (J * (2.0 * sinh(l)));
	} else {
		tmp = U + (cos((K / 2.0)) * (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) :: tmp
    if ((k / 2.0d0) <= 0.0002d0) then
        tmp = u + (j * (2.0d0 * sinh(l)))
    else
        tmp = u + (cos((k / 2.0d0)) * (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 tmp;
	if ((K / 2.0) <= 0.0002) {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K / 2.0) <= 0.0002:
		tmp = U + (J * (2.0 * math.sinh(l)))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (Float64(K / 2.0) <= 0.0002)
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * 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)
	tmp = 0.0;
	if ((K / 2.0) <= 0.0002)
		tmp = U + (J * (2.0 * sinh(l)));
	else
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[(K / 2.0), $MachinePrecision], 0.0002], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * 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 K 2) < 2.0000000000000001e-4

   1. Initial program 85.3%

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

   2. Taylor expanded in K around 0 76.6%

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

      1. *-commutative 76.6%

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

      2. fma-def 76.6%

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

      3. sinh-undef 86.7%

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

   4. Applied egg-rr 86.7%

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

      1. fma-udef 86.7%

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

      2. *-commutative 86.7%

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

   6. Applied egg-rr 86.7%

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

   if 2.0000000000000001e-4 < (/.f64 K 2)

   1. Initial program 89.5%

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

   2. Taylor expanded in l around 0 88.0%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{K}{2} \leq 0.0002:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 8: 96.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{7} \cdot \left(0.0003968253968253968 \cdot J\right)\right)\\ \mathbf{if}\;\ell \leq -4.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-28}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 7.0) (* 0.0003968253968253968 J))))))
   (if (<= l -4.2)
     t_1
     (if (<= l 1.45e-28)
       (+ U (* t_0 (* J (* l 2.0))))
       (if (<= l 1.1e+44) (+ U (* J (* 2.0 (sinh l)))) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + (t_0 * (pow(l, 7.0) * (0.0003968253968253968 * J)));
	double tmp;
	if (l <= -4.2) {
		tmp = t_1;
	} else if (l <= 1.45e-28) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else if (l <= 1.1e+44) {
		tmp = U + (J * (2.0 * sinh(l)));
	} 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) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * ((l ** 7.0d0) * (0.0003968253968253968d0 * j)))
    if (l <= (-4.2d0)) then
        tmp = t_1
    else if (l <= 1.45d-28) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else if (l <= 1.1d+44) then
        tmp = u + (j * (2.0d0 * sinh(l)))
    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 / 2.0));
	double t_1 = U + (t_0 * (Math.pow(l, 7.0) * (0.0003968253968253968 * J)));
	double tmp;
	if (l <= -4.2) {
		tmp = t_1;
	} else if (l <= 1.45e-28) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else if (l <= 1.1e+44) {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + (t_0 * (math.pow(l, 7.0) * (0.0003968253968253968 * J)))
	tmp = 0
	if l <= -4.2:
		tmp = t_1
	elif l <= 1.45e-28:
		tmp = U + (t_0 * (J * (l * 2.0)))
	elif l <= 1.1e+44:
		tmp = U + (J * (2.0 * math.sinh(l)))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(t_0 * Float64((l ^ 7.0) * Float64(0.0003968253968253968 * J))))
	tmp = 0.0
	if (l <= -4.2)
		tmp = t_1;
	elseif (l <= 1.45e-28)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	elseif (l <= 1.1e+44)
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + (t_0 * ((l ^ 7.0) * (0.0003968253968253968 * J)));
	tmp = 0.0;
	if (l <= -4.2)
		tmp = t_1;
	elseif (l <= 1.45e-28)
		tmp = U + (t_0 * (J * (l * 2.0)));
	elseif (l <= 1.1e+44)
		tmp = U + (J * (2.0 * sinh(l)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(t$95$0 * N[(N[Power[l, 7.0], $MachinePrecision] * N[(0.0003968253968253968 * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.2], t$95$1, If[LessEqual[l, 1.45e-28], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.1e+44], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.20000000000000018 or 1.09999999999999998e44 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 96.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(0.0003968253968253968 \cdot {\ell}^{7} + \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   3. Taylor expanded in l around inf 96.9%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 96.9%

         \[\leadsto \color{blue}{\left(\left(J \cdot {\ell}^{7}\right) \cdot 0.0003968253968253968\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 96.9%

         \[\leadsto \left(\color{blue}{\left({\ell}^{7} \cdot J\right)} \cdot 0.0003968253968253968\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*l* 96.9%

         \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 96.9%

      \[\leadsto \color{blue}{\left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -4.20000000000000018 < l < 1.45000000000000006e-28

   1. Initial program 71.6%

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

   2. Taylor expanded in l around 0 99.5%

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

   if 1.45000000000000006e-28 < l < 1.09999999999999998e44

   1. Initial program 92.4%

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

   2. Taylor expanded in K around 0 84.7%

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

      1. *-commutative 84.7%

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

      2. fma-def 84.7%

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

      3. sinh-undef 92.3%

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

   4. Applied egg-rr 92.3%

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

      1. fma-udef 92.3%

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

      2. *-commutative 92.3%

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

   6. Applied egg-rr 92.3%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(0.0003968253968253968 \cdot J\right)\right)\\ \mathbf{elif}\;\ell \leq 1.45 \cdot 10^{-28}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(0.0003968253968253968 \cdot J\right)\right)\\ \end{array} \]

Alternative 9: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.22:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.22)
   (+ U (* 2.0 (* J (+ l (* -0.125 (* l (* K K)))))))
   (+ U (* J (* 2.0 (sinh l))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.22) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	} else {
		tmp = U + (J * (2.0 * sinh(l)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.22d0)) then
        tmp = u + (2.0d0 * (j * (l + ((-0.125d0) * (l * (k * k))))))
    else
        tmp = u + (j * (2.0d0 * sinh(l)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.22) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	} else {
		tmp = U + (J * (2.0 * Math.sinh(l)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.22:
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))))
	else:
		tmp = U + (J * (2.0 * math.sinh(l)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.22)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * Float64(K * K)))))));
	else
		tmp = Float64(U + Float64(J * Float64(2.0 * sinh(l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.22)
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	else
		tmp = U + (J * (2.0 * sinh(l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.22], N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(2.0 * N[Sinh[l], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.220000000000000001

   1. Initial program 89.9%

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

   2. Taylor expanded in l around 0 61.7%

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

   3. Taylor expanded in K around 0 64.9%

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

      1. *-commutative 64.9%

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

      2. unpow2 64.9%

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

   5. Simplified 64.9%

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

   if -0.220000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.4%

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

   2. Taylor expanded in K around 0 83.4%

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

      1. *-commutative 83.4%

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

      2. fma-def 83.4%

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

      3. sinh-undef 93.6%

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

   4. Applied egg-rr 93.6%

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

      1. fma-udef 93.6%

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

      2. *-commutative 93.6%

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

   6. Applied egg-rr 93.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.22:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(2 \cdot \sinh \ell\right)\\ \end{array} \]

Alternative 10: 59.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+82}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -880:\\ \;\;\;\;{U}^{-8}\\ \mathbf{elif}\;\ell \leq 0.00335:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+98} \lor \neg \left(\ell \leq 1.26 \cdot 10^{+194}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* J (+ l (* -0.125 (* l (* K K)))))))))
   (if (<= l -2.55e+82)
     t_0
     (if (<= l -880.0)
       (pow U -8.0)
       (if (<= l 0.00335)
         (+ U (* J (* l 2.0)))
         (if (or (<= l 4e+98) (not (<= l 1.26e+194))) t_0 (pow U -8.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	double tmp;
	if (l <= -2.55e+82) {
		tmp = t_0;
	} else if (l <= -880.0) {
		tmp = pow(U, -8.0);
	} else if (l <= 0.00335) {
		tmp = U + (J * (l * 2.0));
	} else if ((l <= 4e+98) || !(l <= 1.26e+194)) {
		tmp = t_0;
	} else {
		tmp = pow(U, -8.0);
	}
	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 + (2.0d0 * (j * (l + ((-0.125d0) * (l * (k * k))))))
    if (l <= (-2.55d+82)) then
        tmp = t_0
    else if (l <= (-880.0d0)) then
        tmp = u ** (-8.0d0)
    else if (l <= 0.00335d0) then
        tmp = u + (j * (l * 2.0d0))
    else if ((l <= 4d+98) .or. (.not. (l <= 1.26d+194))) then
        tmp = t_0
    else
        tmp = u ** (-8.0d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	double tmp;
	if (l <= -2.55e+82) {
		tmp = t_0;
	} else if (l <= -880.0) {
		tmp = Math.pow(U, -8.0);
	} else if (l <= 0.00335) {
		tmp = U + (J * (l * 2.0));
	} else if ((l <= 4e+98) || !(l <= 1.26e+194)) {
		tmp = t_0;
	} else {
		tmp = Math.pow(U, -8.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))))
	tmp = 0
	if l <= -2.55e+82:
		tmp = t_0
	elif l <= -880.0:
		tmp = math.pow(U, -8.0)
	elif l <= 0.00335:
		tmp = U + (J * (l * 2.0))
	elif (l <= 4e+98) or not (l <= 1.26e+194):
		tmp = t_0
	else:
		tmp = math.pow(U, -8.0)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * Float64(K * K)))))))
	tmp = 0.0
	if (l <= -2.55e+82)
		tmp = t_0;
	elseif (l <= -880.0)
		tmp = U ^ -8.0;
	elseif (l <= 0.00335)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	elseif ((l <= 4e+98) || !(l <= 1.26e+194))
		tmp = t_0;
	else
		tmp = U ^ -8.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	tmp = 0.0;
	if (l <= -2.55e+82)
		tmp = t_0;
	elseif (l <= -880.0)
		tmp = U ^ -8.0;
	elseif (l <= 0.00335)
		tmp = U + (J * (l * 2.0));
	elseif ((l <= 4e+98) || ~((l <= 1.26e+194)))
		tmp = t_0;
	else
		tmp = U ^ -8.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.55e+82], t$95$0, If[LessEqual[l, -880.0], N[Power[U, -8.0], $MachinePrecision], If[LessEqual[l, 0.00335], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 4e+98], N[Not[LessEqual[l, 1.26e+194]], $MachinePrecision]], t$95$0, N[Power[U, -8.0], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.5500000000000001e82 or 0.00335000000000000011 < l < 3.99999999999999999e98 or 1.26e194 < 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. Taylor expanded in l around 0 32.6%

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

   3. Taylor expanded in K around 0 42.6%

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

      1. *-commutative 42.6%

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

      2. unpow2 42.6%

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

   5. Simplified 42.6%

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

   if -2.5500000000000001e82 < l < -880 or 3.99999999999999999e98 < l < 1.26e194

   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%

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

   5. Applied egg-rr 48.9%

      \[\leadsto \color{blue}{{U}^{-8}} \]

   if -880 < l < 0.00335000000000000011

   1. Initial program 71.3%

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

   2. Taylor expanded in K around 0 70.5%

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

   3. Taylor expanded in l around 0 87.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.55 \cdot 10^{+82}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{elif}\;\ell \leq -880:\\ \;\;\;\;{U}^{-8}\\ \mathbf{elif}\;\ell \leq 0.00335:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+98} \lor \neg \left(\ell \leq 1.26 \cdot 10^{+194}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{U}^{-8}\\ \end{array} \]

Alternative 11: 59.4% accurate, 16.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+76} \lor \neg \left(\ell \leq 0.00335\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -7e+76) (not (<= l 0.00335)))
   (+ U (* 2.0 (* J (+ l (* -0.125 (* l (* K K)))))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7e+76) || !(l <= 0.00335)) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	} else {
		tmp = U + (J * (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) :: tmp
    if ((l <= (-7d+76)) .or. (.not. (l <= 0.00335d0))) then
        tmp = u + (2.0d0 * (j * (l + ((-0.125d0) * (l * (k * k))))))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -7e+76) || !(l <= 0.00335)) {
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -7e+76) or not (l <= 0.00335):
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -7e+76) || !(l <= 0.00335))
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + Float64(-0.125 * Float64(l * Float64(K * K)))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -7e+76) || ~((l <= 0.00335)))
		tmp = U + (2.0 * (J * (l + (-0.125 * (l * (K * K))))));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -7e+76], N[Not[LessEqual[l, 0.00335]], $MachinePrecision]], N[(U + N[(2.0 * N[(J * N[(l + N[(-0.125 * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -7.00000000000000001e76 or 0.00335000000000000011 < 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. Taylor expanded in l around 0 31.2%

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

   3. Taylor expanded in K around 0 38.4%

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

      1. *-commutative 38.4%

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

      2. unpow2 38.4%

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

   5. Simplified 38.4%

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

   if -7.00000000000000001e76 < l < 0.00335000000000000011

   1. Initial program 73.1%

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

   2. Taylor expanded in K around 0 71.5%

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

   3. Taylor expanded in l around 0 82.1%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7 \cdot 10^{+76} \lor \neg \left(\ell \leq 0.00335\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + -0.125 \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 12: 44.1% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{-29} \lor \neg \left(\ell \leq 1.15 \cdot 10^{-62}\right):\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3.3e-29) (not (<= l 1.15e-62))) (* l (* J 2.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3.3e-29) || !(l <= 1.15e-62)) {
		tmp = l * (J * 2.0);
	} 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 <= (-3.3d-29)) .or. (.not. (l <= 1.15d-62))) then
        tmp = l * (j * 2.0d0)
    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 <= -3.3e-29) || !(l <= 1.15e-62)) {
		tmp = l * (J * 2.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -3.3e-29) or not (l <= 1.15e-62):
		tmp = l * (J * 2.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -3.3e-29) || !(l <= 1.15e-62))
		tmp = Float64(l * Float64(J * 2.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -3.3e-29) || ~((l <= 1.15e-62)))
		tmp = l * (J * 2.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3.3e-29], N[Not[LessEqual[l, 1.15e-62]], $MachinePrecision]], N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -3.30000000000000028e-29 or 1.15e-62 < l

   1. Initial program 94.3%

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

   2. Taylor expanded in K around 0 69.3%

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

   3. Taylor expanded in l around 0 25.7%

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

   4. Taylor expanded in J around inf 23.6%

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

      1. *-commutative 23.6%

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

      2. *-commutative 23.6%

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

      3. associate-*r* 23.6%

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

   6. Simplified 23.6%

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

   if -3.30000000000000028e-29 < l < 1.15e-62

   1. Initial program 75.5%

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

      1. associate-*l* 75.5%

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

      2. fma-def 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in J around 0 75.5%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{-29} \lor \neg \left(\ell \leq 1.15 \cdot 10^{-62}\right):\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 41.5% accurate, 43.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.0115:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 860:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -0.0115) (* U U) (if (<= l 860.0) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.0115) {
		tmp = U * U;
	} else if (l <= 860.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-0.0115d0)) then
        tmp = u * u
    else if (l <= 860.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -0.0115) {
		tmp = U * U;
	} else if (l <= 860.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -0.0115:
		tmp = U * U
	elif l <= 860.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -0.0115)
		tmp = Float64(U * U);
	elseif (l <= 860.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -0.0115)
		tmp = U * U;
	elseif (l <= 860.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -0.0115], N[(U * U), $MachinePrecision], If[LessEqual[l, 860.0], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -0.0115 or 860 < 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%

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

   5. Applied egg-rr 15.0%

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

   if -0.0115 < l < 860

   1. Initial program 71.5%

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

      1. associate-*l* 71.5%

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

      2. fma-def 71.5%

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

   3. Simplified 71.5%

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

   4. Taylor expanded in J around 0 69.7%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.0115:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 860:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]

Alternative 14: 53.8% 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 86.4%

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

2. Taylor expanded in K around 0 71.9%

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

3. Taylor expanded in l around 0 52.3%

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

4. Final simplification 52.3%

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

Alternative 15: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. associate-*l* 86.4%

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

   2. fma-def 86.4%

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

3. Simplified 86.4%

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

5. Applied egg-rr 2.7%

   \[\leadsto \color{blue}{\frac{-8 - U}{-8 - U}} \]
6. Step-by-step derivation

   1. *-inverses 2.7%

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

7. Simplified 2.7%

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

8. Final simplification 2.7%

   \[\leadsto 1 \]

Alternative 16: 36.3% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.4%

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

   1. associate-*l* 86.4%

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

   2. fma-def 86.4%

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

3. Simplified 86.4%

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

4. Taylor expanded in J around 0 34.4%

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

5. Final simplification 34.4%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023271 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))