Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.7% → 99.9%

Time: 13.3s

Alternatives: 16

Speedup: 1.5×

• 49.6% 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: 86.7% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -20000 \lor \neg \left(t_1 \leq 0.01\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \end{array} \]

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 -20000.0) (not (<= t_1 0.01)))
     (+ (* (* 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 <= -20000.0) || !(t_1 <= 0.01)) {
		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 <= (-20000.0d0)) .or. (.not. (t_1 <= 0.01d0))) 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 <= -20000.0) || !(t_1 <= 0.01)) {
		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 <= -20000.0) or not (t_1 <= 0.01):
		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 <= -20000.0) || !(t_1 <= 0.01))
		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 <= -20000.0) || ~((t_1 <= 0.01)))
		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, -20000.0], N[Not[LessEqual[t$95$1, 0.01]], $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))) < -2e4 or 0.0100000000000000002 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -2e4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0100000000000000002

   1. Initial program 70.0%

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

   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 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -20000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.01\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.016666666666666666 \cdot {\ell}^{5} + \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -20000 \lor \neg \left(t_1 \leq 0.002\right):\\ \;\;\;\;\left(t_1 \cdot J\right) \cdot t_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 -20000.0) (not (<= t_1 0.002)))
     (+ (* (* 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 <= -20000.0) || !(t_1 <= 0.002)) {
		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 <= (-20000.0d0)) .or. (.not. (t_1 <= 0.002d0))) 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 <= -20000.0) || !(t_1 <= 0.002)) {
		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 <= -20000.0) or not (t_1 <= 0.002):
		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 <= -20000.0) || !(t_1 <= 0.002))
		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 <= -20000.0) || ~((t_1 <= 0.002)))
		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, -20000.0], N[Not[LessEqual[t$95$1, 0.002]], $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))) < -2e4 or 2e-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 -2e4 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 2e-3

   1. Initial program 69.8%

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

   2. Taylor expanded in l around 0 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -20000 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.002\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 3: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq 0.21 \lor \neg \left(t_0 \leq 0.77\right) \land t_0 \leq 0.995:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (or (<= t_0 0.21) (and (not (<= t_0 0.77)) (<= t_0 0.995)))
     (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
     (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((t_0 <= 0.21) || (!(t_0 <= 0.77) && (t_0 <= 0.995))) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if ((t_0 <= 0.21d0) .or. (.not. (t_0 <= 0.77d0)) .and. (t_0 <= 0.995d0)) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if ((t_0 <= 0.21) || (!(t_0 <= 0.77) && (t_0 <= 0.995))) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (t_0 <= 0.21) or (not (t_0 <= 0.77) and (t_0 <= 0.995)):
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if ((t_0 <= 0.21) || (!(t_0 <= 0.77) && (t_0 <= 0.995)))
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((t_0 <= 0.21) || (~((t_0 <= 0.77)) && (t_0 <= 0.995)))
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, 0.21], And[N[Not[LessEqual[t$95$0, 0.77]], $MachinePrecision], LessEqual[t$95$0, 0.995]]], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < 0.209999999999999992 or 0.77000000000000002 < (cos.f64 (/.f64 K 2)) < 0.994999999999999996

   1. Initial program 83.9%

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

   2. Taylor expanded in l around 0 70.2%

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

   if 0.209999999999999992 < (cos.f64 (/.f64 K 2)) < 0.77000000000000002 or 0.994999999999999996 < (cos.f64 (/.f64 K 2))

   1. Initial program 85.8%

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

   2. Taylor expanded in l around 0 86.6%

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

   3. Taylor expanded in K around 0 86.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.21 \lor \neg \left(\cos \left(\frac{K}{2}\right) \leq 0.77\right) \land \cos \left(\frac{K}{2}\right) \leq 0.995:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 4: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ t_1 := U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -8.5:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 0.0031:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* 0.016666666666666666 (* (* J (pow l 5.0)) (cos (* K 0.5))))))
        (t_1 (+ U (* (- (exp l) (exp (- l))) (+ J (* J (* K (* K -0.125))))))))
   (if (<= l -4.5e+61)
     t_0
     (if (<= l -8.5)
       t_1
       (if (<= l 0.0031)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 2.3e+52) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.016666666666666666 * ((J * pow(l, 5.0)) * cos((K * 0.5))));
	double t_1 = U + ((exp(l) - exp(-l)) * (J + (J * (K * (K * -0.125)))));
	double tmp;
	if (l <= -4.5e+61) {
		tmp = t_0;
	} else if (l <= -8.5) {
		tmp = t_1;
	} else if (l <= 0.0031) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2.3e+52) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + (0.016666666666666666d0 * ((j * (l ** 5.0d0)) * cos((k * 0.5d0))))
    t_1 = u + ((exp(l) - exp(-l)) * (j + (j * (k * (k * (-0.125d0))))))
    if (l <= (-4.5d+61)) then
        tmp = t_0
    else if (l <= (-8.5d0)) then
        tmp = t_1
    else if (l <= 0.0031d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else if (l <= 2.3d+52) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.016666666666666666 * ((J * Math.pow(l, 5.0)) * Math.cos((K * 0.5))));
	double t_1 = U + ((Math.exp(l) - Math.exp(-l)) * (J + (J * (K * (K * -0.125)))));
	double tmp;
	if (l <= -4.5e+61) {
		tmp = t_0;
	} else if (l <= -8.5) {
		tmp = t_1;
	} else if (l <= 0.0031) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2.3e+52) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.016666666666666666 * ((J * math.pow(l, 5.0)) * math.cos((K * 0.5))))
	t_1 = U + ((math.exp(l) - math.exp(-l)) * (J + (J * (K * (K * -0.125)))))
	tmp = 0
	if l <= -4.5e+61:
		tmp = t_0
	elif l <= -8.5:
		tmp = t_1
	elif l <= 0.0031:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 2.3e+52:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.016666666666666666 * Float64(Float64(J * (l ^ 5.0)) * cos(Float64(K * 0.5)))))
	t_1 = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * Float64(J + Float64(J * Float64(K * Float64(K * -0.125))))))
	tmp = 0.0
	if (l <= -4.5e+61)
		tmp = t_0;
	elseif (l <= -8.5)
		tmp = t_1;
	elseif (l <= 0.0031)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 2.3e+52)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.016666666666666666 * ((J * (l ^ 5.0)) * cos((K * 0.5))));
	t_1 = U + ((exp(l) - exp(-l)) * (J + (J * (K * (K * -0.125)))));
	tmp = 0.0;
	if (l <= -4.5e+61)
		tmp = t_0;
	elseif (l <= -8.5)
		tmp = t_1;
	elseif (l <= 0.0031)
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	elseif (l <= 2.3e+52)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.016666666666666666 * N[(N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * N[(J + N[(J * N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -4.5e+61], t$95$0, If[LessEqual[l, -8.5], t$95$1, If[LessEqual[l, 0.0031], 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], If[LessEqual[l, 2.3e+52], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -4.5e61 or 2.3e52 < 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 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

   if -4.5e61 < l < -8.5 or 0.00309999999999999989 < l < 2.3e52

   1. Initial program 99.7%

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

   2. Taylor expanded in K around 0 7.7%

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

      1. +-commutative 7.7%

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

      2. associate-*r* 7.7%

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

      3. associate-*r* 7.7%

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

      4. distribute-rgt-out 87.7%

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

      5. associate-*r* 87.7%

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

      6. *-commutative 87.7%

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

      7. associate-*l* 87.7%

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

      8. *-commutative 87.7%

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

      9. unpow2 87.7%

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

      10. associate-*l* 87.7%

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

   4. Simplified 87.7%

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

   if -8.5 < l < 0.00309999999999999989

   1. Initial program 69.8%

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

   2. Taylor expanded in l around 0 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 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+61}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -8.5:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 0.0031:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+52}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 5: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot t_0\right)\\ \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.00016:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 360000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t_0\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* 0.016666666666666666 (* (* J (pow l 5.0)) t_0)))))
   (if (<= l -2.6e+44)
     t_1
     (if (<= l -0.00016)
       (+ (* (- (exp l) (exp (- l))) J) U)
       (if (<= l 360000.0)
         (+ U (* 2.0 (* J (* l t_0))))
         (if (<= l 3.2e+51)
           (+ U (* 2.0 (* J (+ l (log1p (expm1 (* l (* K (* K -0.125)))))))))
           t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (0.016666666666666666 * ((J * pow(l, 5.0)) * t_0));
	double tmp;
	if (l <= -2.6e+44) {
		tmp = t_1;
	} else if (l <= -0.00016) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else if (l <= 360000.0) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 3.2e+51) {
		tmp = U + (2.0 * (J * (l + log1p(expm1((l * (K * (K * -0.125))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (0.016666666666666666 * ((J * Math.pow(l, 5.0)) * t_0));
	double tmp;
	if (l <= -2.6e+44) {
		tmp = t_1;
	} else if (l <= -0.00016) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else if (l <= 360000.0) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (l <= 3.2e+51) {
		tmp = U + (2.0 * (J * (l + Math.log1p(Math.expm1((l * (K * (K * -0.125))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (0.016666666666666666 * ((J * math.pow(l, 5.0)) * t_0))
	tmp = 0
	if l <= -2.6e+44:
		tmp = t_1
	elif l <= -0.00016:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	elif l <= 360000.0:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif l <= 3.2e+51:
		tmp = U + (2.0 * (J * (l + math.log1p(math.expm1((l * (K * (K * -0.125))))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(0.016666666666666666 * Float64(Float64(J * (l ^ 5.0)) * t_0)))
	tmp = 0.0
	if (l <= -2.6e+44)
		tmp = t_1;
	elseif (l <= -0.00016)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	elseif (l <= 360000.0)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif (l <= 3.2e+51)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + log1p(expm1(Float64(l * Float64(K * Float64(K * -0.125)))))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(0.016666666666666666 * N[(N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.6e+44], t$95$1, If[LessEqual[l, -0.00016], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 360000.0], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.2e+51], N[(U + N[(2.0 * N[(J * N[(l + N[Log[1 + N[(Exp[N[(l * N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.5999999999999999e44 or 3.2000000000000002e51 < 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 97.4%

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

   3. Taylor expanded in l around inf 97.4%

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

      1. associate-*r* 97.4%

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

   5. Simplified 97.4%

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

   if -2.5999999999999999e44 < l < -1.60000000000000013e-4

   1. Initial program 98.6%

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

   2. Taylor expanded in K around 0 83.2%

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

   if -1.60000000000000013e-4 < l < 3.6e5

   1. Initial program 69.7%

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

   2. Taylor expanded in l around 0 98.0%

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

   if 3.6e5 < l < 3.2000000000000002e51

   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 2.5%

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

   3. Taylor expanded in K around 0 0.9%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + \left(-0.125 \cdot \left({K}^{2} \cdot \ell\right) + 0.0026041666666666665 \cdot \left({K}^{4} \cdot \ell\right)\right)\right)}\right) + U \]
   4. Step-by-step derivation

      1. associate-*r* 0.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \left(\color{blue}{\left(-0.125 \cdot {K}^{2}\right) \cdot \ell} + 0.0026041666666666665 \cdot \left({K}^{4} \cdot \ell\right)\right)\right)\right) + U \]

      2. associate-*r* 0.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \left(\left(-0.125 \cdot {K}^{2}\right) \cdot \ell + \color{blue}{\left(0.0026041666666666665 \cdot {K}^{4}\right) \cdot \ell}\right)\right)\right) + U \]

      3. distribute-rgt-out 0.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{\ell \cdot \left(-0.125 \cdot {K}^{2} + 0.0026041666666666665 \cdot {K}^{4}\right)}\right)\right) + U \]

      4. unpow2 0.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 0.0026041666666666665 \cdot {K}^{4}\right)\right)\right) + U \]

   5. Simplified 0.9%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 0.0026041666666666665 \cdot {K}^{4}\right)\right)}\right) + U \]

   6. Taylor expanded in K around 0 34.3%

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

      1. unpow2 34.3%

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

      2. *-commutative 34.3%

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

      3. associate-*l* 34.3%

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

   8. Simplified 34.3%

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

      1. log1p-expm1-u 67.5%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\right)}\right)\right) + U \]

   10. Applied egg-rr 67.5%

       \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\right)}\right)\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.00016:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 360000:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;\ell \leq 3.2 \cdot 10^{+51}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 6: 94.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -0.75:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 4800000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* 0.016666666666666666 (* (* J (pow l 5.0)) (cos (* K 0.5)))))))
   (if (<= l -2.7e+40)
     t_0
     (if (<= l -0.75)
       (+ (* (- (exp l) (exp (- l))) J) U)
       (if (<= l 4800000000.0)
         (+
          U
          (*
           (cos (/ K 2.0))
           (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
         (if (<= l 2.4e+51)
           (+ U (* 2.0 (* J (+ l (log1p (expm1 (* l (* K (* K -0.125)))))))))
           t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.016666666666666666 * ((J * pow(l, 5.0)) * cos((K * 0.5))));
	double tmp;
	if (l <= -2.7e+40) {
		tmp = t_0;
	} else if (l <= -0.75) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else if (l <= 4800000000.0) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2.4e+51) {
		tmp = U + (2.0 * (J * (l + log1p(expm1((l * (K * (K * -0.125))))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.016666666666666666 * ((J * Math.pow(l, 5.0)) * Math.cos((K * 0.5))));
	double tmp;
	if (l <= -2.7e+40) {
		tmp = t_0;
	} else if (l <= -0.75) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else if (l <= 4800000000.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else if (l <= 2.4e+51) {
		tmp = U + (2.0 * (J * (l + Math.log1p(Math.expm1((l * (K * (K * -0.125))))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.016666666666666666 * ((J * math.pow(l, 5.0)) * math.cos((K * 0.5))))
	tmp = 0
	if l <= -2.7e+40:
		tmp = t_0
	elif l <= -0.75:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	elif l <= 4800000000.0:
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	elif l <= 2.4e+51:
		tmp = U + (2.0 * (J * (l + math.log1p(math.expm1((l * (K * (K * -0.125))))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.016666666666666666 * Float64(Float64(J * (l ^ 5.0)) * cos(Float64(K * 0.5)))))
	tmp = 0.0
	if (l <= -2.7e+40)
		tmp = t_0;
	elseif (l <= -0.75)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	elseif (l <= 4800000000.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	elseif (l <= 2.4e+51)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l + log1p(expm1(Float64(l * Float64(K * Float64(K * -0.125)))))))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.016666666666666666 * N[(N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.7e+40], t$95$0, If[LessEqual[l, -0.75], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 4800000000.0], 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], If[LessEqual[l, 2.4e+51], N[(U + N[(2.0 * N[(J * N[(l + N[Log[1 + N[(Exp[N[(l * N[(K * N[(K * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.70000000000000009e40 or 2.3999999999999999e51 < 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 97.4%

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

   3. Taylor expanded in l around inf 97.4%

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

      1. associate-*r* 97.4%

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

   5. Simplified 97.4%

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

   if -2.70000000000000009e40 < l < -0.75

   1. Initial program 99.8%

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

   2. Taylor expanded in K around 0 79.8%

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

   if -0.75 < l < 4.8e9

   1. Initial program 70.2%

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

   2. Taylor expanded in l around 0 99.0%

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

   if 4.8e9 < l < 2.3999999999999999e51

   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 2.5%

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

   3. Taylor expanded in K around 0 0.9%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + \left(-0.125 \cdot \left({K}^{2} \cdot \ell\right) + 0.0026041666666666665 \cdot \left({K}^{4} \cdot \ell\right)\right)\right)}\right) + U \]
   4. Step-by-step derivation

      1. associate-*r* 0.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \left(\color{blue}{\left(-0.125 \cdot {K}^{2}\right) \cdot \ell} + 0.0026041666666666665 \cdot \left({K}^{4} \cdot \ell\right)\right)\right)\right) + U \]

      2. associate-*r* 0.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \left(\left(-0.125 \cdot {K}^{2}\right) \cdot \ell + \color{blue}{\left(0.0026041666666666665 \cdot {K}^{4}\right) \cdot \ell}\right)\right)\right) + U \]

      3. distribute-rgt-out 0.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{\ell \cdot \left(-0.125 \cdot {K}^{2} + 0.0026041666666666665 \cdot {K}^{4}\right)}\right)\right) + U \]

      4. unpow2 0.9%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 0.0026041666666666665 \cdot {K}^{4}\right)\right)\right) + U \]

   5. Simplified 0.9%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 0.0026041666666666665 \cdot {K}^{4}\right)\right)}\right) + U \]

   6. Taylor expanded in K around 0 34.3%

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

      1. unpow2 34.3%

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

      2. *-commutative 34.3%

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

      3. associate-*l* 34.3%

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

   8. Simplified 34.3%

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

      1. log1p-expm1-u 67.5%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\right)}\right)\right) + U \]

   10. Applied egg-rr 67.5%

       \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\right)}\right)\right) + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+40}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.75:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 4800000000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 2.4 \cdot 10^{+51}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell + \mathsf{log1p}\left(\mathsf{expm1}\left(\ell \cdot \left(K \cdot \left(K \cdot -0.125\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 7: 94.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot t_0\right)\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -0.000115:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 3.3:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* 0.016666666666666666 (* (* J (pow l 5.0)) t_0)))))
   (if (<= l -2.5e+48)
     t_1
     (if (<= l -0.000115)
       (+ (* (- (exp l) (exp (- l))) J) U)
       (if (<= l 3.3) (+ U (* 2.0 (* J (* l t_0)))) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (0.016666666666666666 * ((J * pow(l, 5.0)) * t_0));
	double tmp;
	if (l <= -2.5e+48) {
		tmp = t_1;
	} else if (l <= -0.000115) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else if (l <= 3.3) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} 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 * 0.5d0))
    t_1 = u + (0.016666666666666666d0 * ((j * (l ** 5.0d0)) * t_0))
    if (l <= (-2.5d+48)) then
        tmp = t_1
    else if (l <= (-0.000115d0)) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else if (l <= 3.3d0) then
        tmp = u + (2.0d0 * (j * (l * t_0)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (0.016666666666666666 * ((J * Math.pow(l, 5.0)) * t_0));
	double tmp;
	if (l <= -2.5e+48) {
		tmp = t_1;
	} else if (l <= -0.000115) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else if (l <= 3.3) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (0.016666666666666666 * ((J * math.pow(l, 5.0)) * t_0))
	tmp = 0
	if l <= -2.5e+48:
		tmp = t_1
	elif l <= -0.000115:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	elif l <= 3.3:
		tmp = U + (2.0 * (J * (l * t_0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64(0.016666666666666666 * Float64(Float64(J * (l ^ 5.0)) * t_0)))
	tmp = 0.0
	if (l <= -2.5e+48)
		tmp = t_1;
	elseif (l <= -0.000115)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	elseif (l <= 3.3)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + (0.016666666666666666 * ((J * (l ^ 5.0)) * t_0));
	tmp = 0.0;
	if (l <= -2.5e+48)
		tmp = t_1;
	elseif (l <= -0.000115)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	elseif (l <= 3.3)
		tmp = U + (2.0 * (J * (l * t_0)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(0.016666666666666666 * N[(N[(J * N[Power[l, 5.0], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.5e+48], t$95$1, If[LessEqual[l, -0.000115], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 3.3], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.49999999999999987e48 or 3.2999999999999998 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 90.2%

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

   3. Taylor expanded in l around inf 90.2%

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

      1. associate-*r* 90.2%

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

   5. Simplified 90.2%

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

   if -2.49999999999999987e48 < l < -1.15e-4

   1. Initial program 98.6%

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

   2. Taylor expanded in K around 0 83.2%

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

   if -1.15e-4 < l < 3.2999999999999998

   1. Initial program 69.4%

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

   2. Taylor expanded in l around 0 98.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+48}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.000115:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 3.3:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.016666666666666666 \cdot \left(\left(J \cdot {\ell}^{5}\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 87.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00016 \lor \neg \left(\ell \leq 0.00065\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.00016) (not (<= l 0.00065)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00016) || !(l <= 0.00065)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.00016d0)) .or. (.not. (l <= 0.00065d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.00016) || !(l <= 0.00065)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.00016) or not (l <= 0.00065):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.00016) || !(l <= 0.00065))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.00016) || ~((l <= 0.00065)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.00016], N[Not[LessEqual[l, 0.00065]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.60000000000000013e-4 or 6.4999999999999997e-4 < l

   1. Initial program 99.8%

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

   2. Taylor expanded in K around 0 73.3%

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

   if -1.60000000000000013e-4 < l < 6.4999999999999997e-4

   1. Initial program 69.2%

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

   2. Taylor expanded in l around 0 99.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00016 \lor \neg \left(\ell \leq 0.00065\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 9: 57.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.58:\\ \;\;\;\;U + 2 \cdot \left(-0.125 \cdot \left(J \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.58)
   (+ U (* 2.0 (* -0.125 (* J (* l (* K K))))))
   (+ U (* 2.0 (* l J)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.58) {
		tmp = U + (2.0 * (-0.125 * (J * (l * (K * K)))));
	} else {
		tmp = U + (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 (cos((k / 2.0d0)) <= (-0.58d0)) then
        tmp = u + (2.0d0 * ((-0.125d0) * (j * (l * (k * k)))))
    else
        tmp = u + (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 (Math.cos((K / 2.0)) <= -0.58) {
		tmp = U + (2.0 * (-0.125 * (J * (l * (K * K)))));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.58:
		tmp = U + (2.0 * (-0.125 * (J * (l * (K * K)))))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.58)
		tmp = Float64(U + Float64(2.0 * Float64(-0.125 * Float64(J * Float64(l * Float64(K * K))))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.58)
		tmp = U + (2.0 * (-0.125 * (J * (l * (K * K)))));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.58], N[(U + N[(2.0 * N[(-0.125 * N[(J * N[(l * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 86.1%

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

   2. Taylor expanded in l around 0 59.1%

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

   3. Taylor expanded in K around 0 7.5%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + \left(-0.125 \cdot \left({K}^{2} \cdot \ell\right) + 0.0026041666666666665 \cdot \left({K}^{4} \cdot \ell\right)\right)\right)}\right) + U \]
   4. Step-by-step derivation

      1. associate-*r* 7.5%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \left(\color{blue}{\left(-0.125 \cdot {K}^{2}\right) \cdot \ell} + 0.0026041666666666665 \cdot \left({K}^{4} \cdot \ell\right)\right)\right)\right) + U \]

      2. associate-*r* 7.5%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \left(\left(-0.125 \cdot {K}^{2}\right) \cdot \ell + \color{blue}{\left(0.0026041666666666665 \cdot {K}^{4}\right) \cdot \ell}\right)\right)\right) + U \]

      3. distribute-rgt-out 7.5%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \color{blue}{\ell \cdot \left(-0.125 \cdot {K}^{2} + 0.0026041666666666665 \cdot {K}^{4}\right)}\right)\right) + U \]

      4. unpow2 7.5%

         \[\leadsto 2 \cdot \left(J \cdot \left(\ell + \ell \cdot \left(-0.125 \cdot \color{blue}{\left(K \cdot K\right)} + 0.0026041666666666665 \cdot {K}^{4}\right)\right)\right) + U \]

   5. Simplified 7.5%

      \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\ell + \ell \cdot \left(-0.125 \cdot \left(K \cdot K\right) + 0.0026041666666666665 \cdot {K}^{4}\right)\right)}\right) + U \]

   6. Taylor expanded in K around 0 59.0%

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

      1. unpow2 59.0%

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

      2. *-commutative 59.0%

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

      3. associate-*l* 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in K around inf 59.0%

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

       1. *-commutative 59.0%

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

       2. unpow2 59.0%

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

   11. Simplified 59.0%

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

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

   1. Initial program 84.8%

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

   2. Taylor expanded in l around 0 67.7%

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

   3. Taylor expanded in K around 0 61.3%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.58:\\ \;\;\;\;U + 2 \cdot \left(-0.125 \cdot \left(J \cdot \left(\ell \cdot \left(K \cdot K\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 10: 64.0% 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 85.0%

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

2. Taylor expanded in l around 0 66.3%

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

3. Final simplification 66.3%

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

Alternative 11: 42.0% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+32}:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+257}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot 0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.2e+18)
   (* U U)
   (if (<= l 8e+32) U (if (<= l 8e+257) (* U U) (+ U (* J 0.125))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.2e+18) {
		tmp = U * U;
	} else if (l <= 8e+32) {
		tmp = U;
	} else if (l <= 8e+257) {
		tmp = U * U;
	} else {
		tmp = U + (J * 0.125);
	}
	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 <= (-2.2d+18)) then
        tmp = u * u
    else if (l <= 8d+32) then
        tmp = u
    else if (l <= 8d+257) then
        tmp = u * u
    else
        tmp = u + (j * 0.125d0)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.2e+18) {
		tmp = U * U;
	} else if (l <= 8e+32) {
		tmp = U;
	} else if (l <= 8e+257) {
		tmp = U * U;
	} else {
		tmp = U + (J * 0.125);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.2e+18:
		tmp = U * U
	elif l <= 8e+32:
		tmp = U
	elif l <= 8e+257:
		tmp = U * U
	else:
		tmp = U + (J * 0.125)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.2e+18)
		tmp = Float64(U * U);
	elseif (l <= 8e+32)
		tmp = U;
	elseif (l <= 8e+257)
		tmp = Float64(U * U);
	else
		tmp = Float64(U + Float64(J * 0.125));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.2e+18)
		tmp = U * U;
	elseif (l <= 8e+32)
		tmp = U;
	elseif (l <= 8e+257)
		tmp = U * U;
	else
		tmp = U + (J * 0.125);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.2e+18], N[(U * U), $MachinePrecision], If[LessEqual[l, 8e+32], U, If[LessEqual[l, 8e+257], N[(U * U), $MachinePrecision], N[(U + N[(J * 0.125), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.2e18 or 8.00000000000000043e32 < l < 8.00000000000000024e257

   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 18.3%

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

   if -2.2e18 < l < 8.00000000000000043e32

   1. Initial program 72.0%

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

      1. associate-*l* 72.0%

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

      2. fma-def 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in J around 0 61.8%

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

   if 8.00000000000000024e257 < 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. Applied egg-rr 4.4%

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

   4. Taylor expanded in K around 0 4.4%

      \[\leadsto \color{blue}{0.125 \cdot J} + U \]
   5. Step-by-step derivation

      1. *-commutative 4.4%

         \[\leadsto \color{blue}{J \cdot 0.125} + U \]

   6. Simplified 4.4%

      \[\leadsto \color{blue}{J \cdot 0.125} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2 \cdot 10^{+18}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+32}:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 8 \cdot 10^{+257}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot 0.125\\ \end{array} \]

Alternative 12: 42.0% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+17}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+31}:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{+259}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -6e+17)
   (* U U)
   (if (<= l 3.8e+31) U (if (<= l 5.1e+259) (* U U) U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6e+17) {
		tmp = U * U;
	} else if (l <= 3.8e+31) {
		tmp = U;
	} else if (l <= 5.1e+259) {
		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 <= (-6d+17)) then
        tmp = u * u
    else if (l <= 3.8d+31) then
        tmp = u
    else if (l <= 5.1d+259) 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 <= -6e+17) {
		tmp = U * U;
	} else if (l <= 3.8e+31) {
		tmp = U;
	} else if (l <= 5.1e+259) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -6e+17:
		tmp = U * U
	elif l <= 3.8e+31:
		tmp = U
	elif l <= 5.1e+259:
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -6e+17)
		tmp = Float64(U * U);
	elseif (l <= 3.8e+31)
		tmp = U;
	elseif (l <= 5.1e+259)
		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 <= -6e+17)
		tmp = U * U;
	elseif (l <= 3.8e+31)
		tmp = U;
	elseif (l <= 5.1e+259)
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -6e+17], N[(U * U), $MachinePrecision], If[LessEqual[l, 3.8e+31], U, If[LessEqual[l, 5.1e+259], N[(U * U), $MachinePrecision], U]]]

Derivation

1. Split input into 2 regimes

2. if l < -6e17 or 3.8000000000000001e31 < l < 5.1000000000000002e259

   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 18.3%

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

   if -6e17 < l < 3.8000000000000001e31 or 5.1000000000000002e259 < l

   1. Initial program 72.9%

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

      1. associate-*l* 72.9%

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

      2. fma-def 72.9%

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

   3. Simplified 72.9%

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

   4. Taylor expanded in J around 0 59.7%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+17}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 3.8 \cdot 10^{+31}:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 5.1 \cdot 10^{+259}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 42.0% accurate, 34.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+17}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+31}:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+253}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U + -8\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -6e+17)
   (* U U)
   (if (<= l 1.4e+31) U (if (<= l 2.5e+253) (* U U) (+ U -8.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -6e+17) {
		tmp = U * U;
	} else if (l <= 1.4e+31) {
		tmp = U;
	} else if (l <= 2.5e+253) {
		tmp = U * U;
	} else {
		tmp = 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) :: tmp
    if (l <= (-6d+17)) then
        tmp = u * u
    else if (l <= 1.4d+31) then
        tmp = u
    else if (l <= 2.5d+253) then
        tmp = u * u
    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 tmp;
	if (l <= -6e+17) {
		tmp = U * U;
	} else if (l <= 1.4e+31) {
		tmp = U;
	} else if (l <= 2.5e+253) {
		tmp = U * U;
	} else {
		tmp = U + -8.0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -6e+17:
		tmp = U * U
	elif l <= 1.4e+31:
		tmp = U
	elif l <= 2.5e+253:
		tmp = U * U
	else:
		tmp = U + -8.0
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -6e+17)
		tmp = Float64(U * U);
	elseif (l <= 1.4e+31)
		tmp = U;
	elseif (l <= 2.5e+253)
		tmp = Float64(U * U);
	else
		tmp = Float64(U + -8.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -6e+17)
		tmp = U * U;
	elseif (l <= 1.4e+31)
		tmp = U;
	elseif (l <= 2.5e+253)
		tmp = U * U;
	else
		tmp = U + -8.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -6e+17], N[(U * U), $MachinePrecision], If[LessEqual[l, 1.4e+31], U, If[LessEqual[l, 2.5e+253], N[(U * U), $MachinePrecision], N[(U + -8.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -6e17 or 1.40000000000000008e31 < l < 2.4999999999999998e253

   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 18.6%

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

   if -6e17 < l < 1.40000000000000008e31

   1. Initial program 72.0%

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

      1. associate-*l* 72.0%

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

      2. fma-def 72.0%

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

   3. Simplified 72.0%

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

   4. Taylor expanded in J around 0 61.8%

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

   if 2.4999999999999998e253 < 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 3.0%

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

      1. +-commutative 3.0%

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

   7. Simplified 3.0%

      \[\leadsto \color{blue}{U + -8} \]
3. Recombined 3 regimes into one program.

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6 \cdot 10^{+17}:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 1.4 \cdot 10^{+31}:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 2.5 \cdot 10^{+253}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U + -8\\ \end{array} \]

Alternative 14: 54.1% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

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 * (l * j))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.0%

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

2. Taylor expanded in l around 0 66.3%

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

3. Taylor expanded in K around 0 56.1%

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

4. Final simplification 56.1%

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

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 85.0%

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

   1. associate-*l* 85.0%

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

   2. fma-def 85.0%

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

3. Simplified 85.0%

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

5. Applied egg-rr 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: 37.2% 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 85.0%

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

   1. associate-*l* 85.0%

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

   2. fma-def 85.0%

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

3. Simplified 85.0%

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

4. Taylor expanded in J around 0 34.1%

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

5. Final simplification 34.1%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023291 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))