Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.9%

Time: 14.4s

Alternatives: 15

Speedup: 1.5×

• 49.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+20} \lor \neg \left(t_1 \leq 0.001\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -1e+20) (not (<= t_1 0.001)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (+
        (* 0.016666666666666666 (* J (pow l 5.0)))
        (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -1e+20) || !(t_1 <= 0.001)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * ((0.016666666666666666 * (J * pow(l, 5.0))) + (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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-1d+20)) .or. (.not. (t_1 <= 0.001d0))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * ((0.016666666666666666d0 * (j * (l ** 5.0d0))) + (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 t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -1e+20) || !(t_1 <= 0.001)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * ((0.016666666666666666 * (J * Math.pow(l, 5.0))) + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -1e+20) or not (t_1 <= 0.001):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * ((0.016666666666666666 * (J * math.pow(l, 5.0))) + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -1e+20) || !(t_1 <= 0.001))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(Float64(0.016666666666666666 * Float64(J * (l ^ 5.0))) + 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)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -1e+20) || ~((t_1 <= 0.001)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * ((0.016666666666666666 * (J * (l ^ 5.0))) + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1e+20], N[Not[LessEqual[t$95$1, 0.001]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(N[(0.016666666666666666 * N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1e20 or 1e-3 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -1e20 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1e-3

   1. Initial program 79.0%

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

   2. Taylor expanded in l around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-lft-in 99.9%

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

      3. fma-def 99.9%

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

      4. *-commutative 99.9%

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

      5. *-commutative 99.9%

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

      6. associate-*l* 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. *-commutative 99.9%

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

      3. +-commutative 99.9%

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

      4. *-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -1 \cdot 10^{+20} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.001\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.016666666666666666 \cdot \left(J \cdot {\ell}^{5}\right) + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+20} \lor \neg \left(t_1 \leq 0.001\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

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 -1e+20) (not (<= t_1 0.001)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (*
       t_0
       (*
        J
        (+
         (* 0.016666666666666666 (pow l 5.0))
         (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -1e+20) || !(t_1 <= 0.001)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.016666666666666666 * pow(l, 5.0)) + ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-1d+20)) .or. (.not. (t_1 <= 0.001d0))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (j * ((0.016666666666666666d0 * (l ** 5.0d0)) + ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -1e+20) || !(t_1 <= 0.001)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.016666666666666666 * Math.pow(l, 5.0)) + ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -1e+20) or not (t_1 <= 0.001):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.016666666666666666 * math.pow(l, 5.0)) + ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -1e+20) || !(t_1 <= 0.001))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.016666666666666666 * (l ^ 5.0)) + Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -1e+20) || ~((t_1 <= 0.001)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.016666666666666666 * (l ^ 5.0)) + ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(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, -1e+20], N[Not[LessEqual[t$95$1, 0.001]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * 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]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1e20 or 1e-3 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -1e20 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1e-3

   1. Initial program 79.0%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -1 \cdot 10^{+20} \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.001\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -0.01 \lor \neg \left(t_1 \leq 0.001\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -0.01) (not (<= t_1 0.001)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -0.01) || !(t_1 <= 0.001)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-0.01d0)) .or. (.not. (t_1 <= 0.001d0))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -0.01) || !(t_1 <= 0.001)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -0.01) or not (t_1 <= 0.001):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -0.01) || !(t_1 <= 0.001))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_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)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -0.01) || ~((t_1 <= 0.001)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.01], N[Not[LessEqual[t$95$1, 0.001]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * 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 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.0100000000000000002 or 1e-3 < (-.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 -0.0100000000000000002 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1e-3

   1. Initial program 78.7%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.01 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.001\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 4: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{if}\;\ell \leq -5.6:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.00195:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 7.0) (* J 0.0003968253968253968))))))
   (if (<= l -5.6)
     t_1
     (if (<= l 0.00195)
       (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
       (if (<= l 5.6e+43) (+ (* (- (exp l) (exp (- l))) J) U) 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) * (J * 0.0003968253968253968)));
	double tmp;
	if (l <= -5.6) {
		tmp = t_1;
	} else if (l <= 0.00195) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 5.6e+43) {
		tmp = ((exp(l) - exp(-l)) * J) + 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) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * ((l ** 7.0d0) * (j * 0.0003968253968253968d0)))
    if (l <= (-5.6d0)) then
        tmp = t_1
    else if (l <= 0.00195d0) then
        tmp = u + (t_0 * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 5.6d+43) then
        tmp = ((exp(l) - exp(-l)) * j) + 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 / 2.0));
	double t_1 = U + (t_0 * (Math.pow(l, 7.0) * (J * 0.0003968253968253968)));
	double tmp;
	if (l <= -5.6) {
		tmp = t_1;
	} else if (l <= 0.00195) {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 5.6e+43) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} 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) * (J * 0.0003968253968253968)))
	tmp = 0
	if l <= -5.6:
		tmp = t_1
	elif l <= 0.00195:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 5.6e+43:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	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(J * 0.0003968253968253968))))
	tmp = 0.0
	if (l <= -5.6)
		tmp = t_1;
	elseif (l <= 0.00195)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 5.6e+43)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	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) * (J * 0.0003968253968253968)));
	tmp = 0.0;
	if (l <= -5.6)
		tmp = t_1;
	elseif (l <= 0.00195)
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 5.6e+43)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	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[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -5.6], t$95$1, If[LessEqual[l, 0.00195], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+43], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.5999999999999996 or 5.60000000000000038e43 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 95.3%

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

   3. Taylor expanded in l around inf 95.3%

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

      1. associate-*r* 95.3%

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

      2. *-commutative 95.3%

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

      3. *-commutative 95.3%

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

   5. Simplified 95.3%

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

   if -5.5999999999999996 < l < 0.0019499999999999999

   1. Initial program 79.0%

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

   2. Taylor expanded in l around 0 99.7%

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

   if 0.0019499999999999999 < l < 5.60000000000000038e43

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 92.3%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{elif}\;\ell \leq 0.00195:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \end{array} \]

Alternative 5: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + t_0 \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{if}\;\ell \leq -4.2:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;U + t_0 \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* t_0 (* (pow l 7.0) (* J 0.0003968253968253968))))))
   (if (<= l -4.2)
     t_1
     (if (<= l 2.2e-9)
       (+ U (* t_0 (* l (* J 2.0))))
       (if (<= l 5.6e+43) (+ (* (- (exp l) (exp (- l))) J) U) 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) * (J * 0.0003968253968253968)));
	double tmp;
	if (l <= -4.2) {
		tmp = t_1;
	} else if (l <= 2.2e-9) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else if (l <= 5.6e+43) {
		tmp = ((exp(l) - exp(-l)) * J) + 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) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + (t_0 * ((l ** 7.0d0) * (j * 0.0003968253968253968d0)))
    if (l <= (-4.2d0)) then
        tmp = t_1
    else if (l <= 2.2d-9) then
        tmp = u + (t_0 * (l * (j * 2.0d0)))
    else if (l <= 5.6d+43) then
        tmp = ((exp(l) - exp(-l)) * j) + 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 / 2.0));
	double t_1 = U + (t_0 * (Math.pow(l, 7.0) * (J * 0.0003968253968253968)));
	double tmp;
	if (l <= -4.2) {
		tmp = t_1;
	} else if (l <= 2.2e-9) {
		tmp = U + (t_0 * (l * (J * 2.0)));
	} else if (l <= 5.6e+43) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} 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) * (J * 0.0003968253968253968)))
	tmp = 0
	if l <= -4.2:
		tmp = t_1
	elif l <= 2.2e-9:
		tmp = U + (t_0 * (l * (J * 2.0)))
	elif l <= 5.6e+43:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	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(J * 0.0003968253968253968))))
	tmp = 0.0
	if (l <= -4.2)
		tmp = t_1;
	elseif (l <= 2.2e-9)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	elseif (l <= 5.6e+43)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	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) * (J * 0.0003968253968253968)));
	tmp = 0.0;
	if (l <= -4.2)
		tmp = t_1;
	elseif (l <= 2.2e-9)
		tmp = U + (t_0 * (l * (J * 2.0)));
	elseif (l <= 5.6e+43)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	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[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.2], t$95$1, If[LessEqual[l, 2.2e-9], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+43], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.20000000000000018 or 5.60000000000000038e43 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 95.3%

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

   3. Taylor expanded in l around inf 95.3%

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

      1. associate-*r* 95.3%

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

      2. *-commutative 95.3%

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

      3. *-commutative 95.3%

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

   5. Simplified 95.3%

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

   if -4.20000000000000018 < l < 2.1999999999999998e-9

   1. Initial program 78.7%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

   4. Simplified 99.2%

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

   if 2.1999999999999998e-9 < l < 5.60000000000000038e43

   1. Initial program 99.4%

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

   2. Taylor expanded in K around 0 92.7%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.2:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \mathbf{elif}\;\ell \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+43}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left({\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\right)\\ \end{array} \]

Alternative 6: 87.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -47 \lor \neg \left(\ell \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -47.0) (not (<= l 2.2e-9)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -47.0) || !(l <= 2.2e-9)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-47.0d0)) .or. (.not. (l <= 2.2d-9))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -47.0) || !(l <= 2.2e-9)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -47.0) or not (l <= 2.2e-9):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -47.0) || !(l <= 2.2e-9))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -47.0) || ~((l <= 2.2e-9)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -47.0], N[Not[LessEqual[l, 2.2e-9]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -47 or 2.1999999999999998e-9 < 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 K around 0 72.5%

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

   if -47 < l < 2.1999999999999998e-9

   1. Initial program 78.9%

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

   2. Taylor expanded in l around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. *-commutative 98.4%

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

   4. Simplified 98.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -47 \lor \neg \left(\ell \leq 2.2 \cdot 10^{-9}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]

Alternative 7: 64.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \ell \cdot \left(J \cdot 2 + -0.25 \cdot \left(J \cdot {K}^{2}\right)\right)\\ t_1 := \mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1300000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+191}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+229}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+245}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* l (+ (* J 2.0) (* -0.25 (* J (pow K 2.0)))))))
        (t_1 (log1p (expm1 U))))
   (if (<= l -9.5e+31)
     t_0
     (if (<= l 1300000.0)
       (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))
       (if (<= l 6.5e+191)
         t_1
         (if (<= l 1.75e+229) (pow U -4.0) (if (<= l 3.3e+245) t_1 t_0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (l * ((J * 2.0) + (-0.25 * (J * pow(K, 2.0)))));
	double t_1 = log1p(expm1(U));
	double tmp;
	if (l <= -9.5e+31) {
		tmp = t_0;
	} else if (l <= 1300000.0) {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 6.5e+191) {
		tmp = t_1;
	} else if (l <= 1.75e+229) {
		tmp = pow(U, -4.0);
	} else if (l <= 3.3e+245) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (l * ((J * 2.0) + (-0.25 * (J * Math.pow(K, 2.0)))));
	double t_1 = Math.log1p(Math.expm1(U));
	double tmp;
	if (l <= -9.5e+31) {
		tmp = t_0;
	} else if (l <= 1300000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 6.5e+191) {
		tmp = t_1;
	} else if (l <= 1.75e+229) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 3.3e+245) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (l * ((J * 2.0) + (-0.25 * (J * math.pow(K, 2.0)))))
	t_1 = math.log1p(math.expm1(U))
	tmp = 0
	if l <= -9.5e+31:
		tmp = t_0
	elif l <= 1300000.0:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	elif l <= 6.5e+191:
		tmp = t_1
	elif l <= 1.75e+229:
		tmp = math.pow(U, -4.0)
	elif l <= 3.3e+245:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(l * Float64(Float64(J * 2.0) + Float64(-0.25 * Float64(J * (K ^ 2.0))))))
	t_1 = log1p(expm1(U))
	tmp = 0.0
	if (l <= -9.5e+31)
		tmp = t_0;
	elseif (l <= 1300000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	elseif (l <= 6.5e+191)
		tmp = t_1;
	elseif (l <= 1.75e+229)
		tmp = U ^ -4.0;
	elseif (l <= 3.3e+245)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(l * N[(N[(J * 2.0), $MachinePrecision] + N[(-0.25 * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[1 + N[(Exp[U] - 1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -9.5e+31], t$95$0, If[LessEqual[l, 1300000.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 6.5e+191], t$95$1, If[LessEqual[l, 1.75e+229], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 3.3e+245], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -9.5000000000000008e31 or 3.30000000000000011e245 < 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 29.5%

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

      1. associate-*r* 29.5%

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

      2. *-commutative 29.5%

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

      3. associate-*l* 29.5%

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

   4. Simplified 29.5%

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

   5. Taylor expanded in K around 0 43.8%

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

   if -9.5000000000000008e31 < l < 1.3e6

   1. Initial program 80.6%

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

   2. Taylor expanded in l around 0 93.8%

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

      1. associate-*r* 93.8%

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

      2. *-commutative 93.8%

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

   4. Simplified 93.8%

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

   if 1.3e6 < l < 6.50000000000000008e191 or 1.7500000000000001e229 < l < 3.30000000000000011e245

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 30.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)} \]

   if 6.50000000000000008e191 < l < 1.7500000000000001e229

   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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{{U}^{-4}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.5 \cdot 10^{+31}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2 + -0.25 \cdot \left(J \cdot {K}^{2}\right)\right)\\ \mathbf{elif}\;\ell \leq 1300000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+229}:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 3.3 \cdot 10^{+245}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(U\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2 + -0.25 \cdot \left(J \cdot {K}^{2}\right)\right)\\ \end{array} \]

Alternative 8: 65.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2 + -0.25 \cdot \left(J \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.35e+32)
   (+ U (* l (+ (* J 2.0) (* -0.25 (* J (pow K 2.0))))))
   (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.35e+32) {
		tmp = U + (l * ((J * 2.0) + (-0.25 * (J * pow(K, 2.0)))));
	} else {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.35d+32)) then
        tmp = u + (l * ((j * 2.0d0) + ((-0.25d0) * (j * (k ** 2.0d0)))))
    else
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.35e+32) {
		tmp = U + (l * ((J * 2.0) + (-0.25 * (J * Math.pow(K, 2.0)))));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.35e+32:
		tmp = U + (l * ((J * 2.0) + (-0.25 * (J * math.pow(K, 2.0)))))
	else:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.35e+32)
		tmp = Float64(U + Float64(l * Float64(Float64(J * 2.0) + Float64(-0.25 * Float64(J * (K ^ 2.0))))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.35e+32)
		tmp = U + (l * ((J * 2.0) + (-0.25 * (J * (K ^ 2.0)))));
	else
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.35e+32], N[(U + N[(l * N[(N[(J * 2.0), $MachinePrecision] + N[(-0.25 * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.35000000000000006e32

   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 29.6%

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

      1. associate-*r* 29.6%

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

      2. *-commutative 29.6%

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

      3. associate-*l* 29.6%

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

   4. Simplified 29.6%

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

   5. Taylor expanded in K around 0 46.1%

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

   if -1.35000000000000006e32 < l

   1. Initial program 86.8%

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

   2. Taylor expanded in l around 0 69.2%

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

      1. associate-*r* 69.2%

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

      2. *-commutative 69.2%

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

   4. Simplified 69.2%

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

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+32}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2 + -0.25 \cdot \left(J \cdot {K}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]

Alternative 9: 64.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

2. Taylor expanded in l around 0 59.8%

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

3. Final simplification 59.8%

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

Alternative 10: 64.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

2. Taylor expanded in l around 0 59.8%

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

   1. associate-*r* 59.8%

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

   2. *-commutative 59.8%

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

   3. associate-*l* 59.8%

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

4. Simplified 59.8%

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

5. Final simplification 59.8%

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

Alternative 11: 64.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

2. Taylor expanded in l around 0 59.8%

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

   1. associate-*r* 59.8%

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

   2. *-commutative 59.8%

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

4. Simplified 59.8%

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

5. Final simplification 59.8%

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

Alternative 12: 42.7% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -360 \lor \neg \left(\ell \leq 4200000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -360.0) (not (<= l 4200000000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -360.0) || !(l <= 4200000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-360.0d0)) .or. (.not. (l <= 4200000000.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -360.0) || !(l <= 4200000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -360.0) or not (l <= 4200000000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -360.0) || !(l <= 4200000000.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -360.0) || ~((l <= 4200000000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -360.0], N[Not[LessEqual[l, 4200000000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -360 or 4.2e9 < 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. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Applied egg-rr 11.6%

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

   if -360 < l < 4.2e9

   1. Initial program 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*l* 80.0%

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

      3. fma-def 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in l around 0 74.2%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -360 \lor \neg \left(\ell \leq 4200000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 54.6% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

2. Taylor expanded in l around 0 59.8%

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

   1. associate-*r* 59.8%

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

   2. *-commutative 59.8%

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

4. Simplified 59.8%

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

5. Taylor expanded in K around 0 50.2%

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

   1. +-commutative 50.2%

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

   2. associate-*r* 50.2%

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

7. Simplified 50.2%

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

8. Final simplification 50.2%

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

Alternative 14: 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 90.0%

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

   1. *-commutative 90.0%

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

   2. associate-*l* 90.0%

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

   3. fma-def 90.0%

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

3. Simplified 90.0%

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

5. Applied egg-rr 2.6%

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

   1. *-inverses 2.6%

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

7. Simplified 2.6%

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

8. Final simplification 2.6%

   \[\leadsto 1 \]

Alternative 15: 37.3% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.0%

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

   1. *-commutative 90.0%

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

   2. associate-*l* 90.0%

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

   3. fma-def 90.0%

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

3. Simplified 90.0%

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

4. Taylor expanded in l around 0 38.2%

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

5. Final simplification 38.2%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023310 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))