Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.7% → 99.7%

Time: 10.7s

Alternatives: 19

Speedup: 1.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.7% 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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-7}:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, t\_1 \cdot t\_0, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (<= t_1 (- INFINITY))
     (+ (* (* t_1 J) t_0) U)
     (if (<= t_1 2e-7)
       (+
        U
        (* t_0 (* l (+ (* 0.3333333333333333 (* J (pow l 2.0))) (* J 2.0)))))
       (fma J (* t_1 t_0) U)))))

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 <= -((double) INFINITY)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else if (t_1 <= 2e-7) {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * pow(l, 2.0))) + (J * 2.0))));
	} else {
		tmp = fma(J, (t_1 * t_0), U);
	}
	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 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	elseif (t_1 <= 2e-7)
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 2.0))) + Float64(J * 2.0)))));
	else
		tmp = fma(J, Float64(t_1 * t_0), U);
	end
	return 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[LessEqual[t$95$1, (-Infinity)], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[t$95$1, 2e-7], N[(U + N[(t$95$0 * N[(l * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(t$95$1 * t$95$0), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.9999999999999999e-7

   1. Initial program 66.6%

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

   3. Taylor expanded in l around 0 99.9%

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

   if 1.9999999999999999e-7 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 98.3%

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

      1. associate-*l* 98.3%

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

      2. fma-define 98.3%

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

   3. Simplified 98.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)} \]
   4. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty:\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{elif}\;e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + J \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \left(e^{\ell} - e^{-\ell}\right) \cdot \cos \left(\frac{K}{2}\right), U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + J \cdot 2\right)\right)\\ \end{array} \end{array} \]

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 (- INFINITY)) (not (<= t_1 2e-7)))
     (+ (* (* t_1 J) t_0) U)
     (+
      U
      (* t_0 (* l (+ (* 0.3333333333333333 (* J (pow l 2.0))) (* J 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 <= -((double) INFINITY)) || !(t_1 <= 2e-7)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * pow(l, 2.0))) + (J * 2.0))));
	}
	return tmp;
}

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 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 2e-7)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * Math.pow(l, 2.0))) + (J * 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 <= -math.inf) or not (t_1 <= 2e-7):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * math.pow(l, 2.0))) + (J * 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 <= Float64(-Inf)) || !(t_1 <= 2e-7))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(Float64(0.3333333333333333 * Float64(J * (l ^ 2.0))) + Float64(J * 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 <= -Inf) || ~((t_1 <= 2e-7)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (l * ((0.3333333333333333 * (J * (l ^ 2.0))) + (J * 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, (-Infinity)], N[Not[LessEqual[t$95$1, 2e-7]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(l * N[(N[(0.3333333333333333 * N[(J * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * 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))) < -inf.0 or 1.9999999999999999e-7 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.2%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 1.9999999999999999e-7

   1. Initial program 66.6%

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

   3. Taylor expanded in l around 0 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 2 \cdot 10^{-7}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+81}:\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot t\_0\right)\\ \mathbf{elif}\;\ell \leq -23:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 840:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(t\_0 \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))) (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -2.7e+81)
     (+ U (* (* J 0.3333333333333333) (* (pow l 3.0) t_0)))
     (if (<= l -23.0)
       t_1
       (if (<= l 840.0)
         (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
         (if (<= l 3.4e+89)
           t_1
           (* 0.3333333333333333 (* t_0 (* J (pow l 3.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.7e+81) {
		tmp = U + ((J * 0.3333333333333333) * (pow(l, 3.0) * t_0));
	} else if (l <= -23.0) {
		tmp = t_1;
	} else if (l <= 840.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 3.4e+89) {
		tmp = t_1;
	} else {
		tmp = 0.3333333333333333 * (t_0 * (J * pow(l, 3.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-2.7d+81)) then
        tmp = u + ((j * 0.3333333333333333d0) * ((l ** 3.0d0) * t_0))
    else if (l <= (-23.0d0)) then
        tmp = t_1
    else if (l <= 840.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else if (l <= 3.4d+89) then
        tmp = t_1
    else
        tmp = 0.3333333333333333d0 * (t_0 * (j * (l ** 3.0d0)))
    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 * 0.5));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -2.7e+81) {
		tmp = U + ((J * 0.3333333333333333) * (Math.pow(l, 3.0) * t_0));
	} else if (l <= -23.0) {
		tmp = t_1;
	} else if (l <= 840.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 3.4e+89) {
		tmp = t_1;
	} else {
		tmp = 0.3333333333333333 * (t_0 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -2.7e+81:
		tmp = U + ((J * 0.3333333333333333) * (math.pow(l, 3.0) * t_0))
	elif l <= -23.0:
		tmp = t_1
	elif l <= 840.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	elif l <= 3.4e+89:
		tmp = t_1
	else:
		tmp = 0.3333333333333333 * (t_0 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -2.7e+81)
		tmp = Float64(U + Float64(Float64(J * 0.3333333333333333) * Float64((l ^ 3.0) * t_0)));
	elseif (l <= -23.0)
		tmp = t_1;
	elseif (l <= 840.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif (l <= 3.4e+89)
		tmp = t_1;
	else
		tmp = Float64(0.3333333333333333 * Float64(t_0 * Float64(J * (l ^ 3.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -2.7e+81)
		tmp = U + ((J * 0.3333333333333333) * ((l ^ 3.0) * t_0));
	elseif (l <= -23.0)
		tmp = t_1;
	elseif (l <= 840.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	elseif (l <= 3.4e+89)
		tmp = t_1;
	else
		tmp = 0.3333333333333333 * (t_0 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -2.7e+81], N[(U + N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -23.0], t$95$1, If[LessEqual[l, 840.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.4e+89], t$95$1, N[(0.3333333333333333 * N[(t$95$0 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.6999999999999999e81

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.2%

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

   4. Taylor expanded in l around inf 98.2%

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

      1. associate-*r* 98.2%

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

      2. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   if -2.6999999999999999e81 < l < -23 or 840 < l < 3.4000000000000002e89

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 70.6%

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

      1. +-commutative 70.6%

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

   5. Simplified 70.6%

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

   if -23 < l < 840

   1. Initial program 66.9%

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

   3. Taylor expanded in l around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

   5. Simplified 98.3%

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

   if 3.4000000000000002e89 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. distribute-lft-in 100.0%

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

      2. distribute-rgt-in 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. unpow2 100.0%

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

      6. pow3 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in U around inf 100.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. *-rgt-identity 100.0%

         \[\leadsto U \cdot \color{blue}{\left(\left(1 + \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}\right) \cdot 1\right)} \]

      2. metadata-eval 100.0%

         \[\leadsto U \cdot \left(\left(1 + \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}\right) \cdot \color{blue}{\left(--1\right)}\right) \]

      3. distribute-rgt-neg-in 100.0%

         \[\leadsto U \cdot \color{blue}{\left(-\left(1 + \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}\right) \cdot -1\right)} \]

      4. +-commutative 100.0%

         \[\leadsto U \cdot \left(-\color{blue}{\left(\frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U} + 1\right)} \cdot -1\right) \]

      5. distribute-lft1-in 100.0%

         \[\leadsto U \cdot \left(-\color{blue}{\left(\frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U} \cdot -1 + -1\right)}\right) \]

      6. *-commutative 100.0%

         \[\leadsto U \cdot \left(-\left(\color{blue}{-1 \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}} + -1\right)\right) \]

      7. metadata-eval 100.0%

         \[\leadsto U \cdot \left(-\left(-1 \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U} + \color{blue}{\left(-1\right)}\right)\right) \]

      8. sub-neg 100.0%

         \[\leadsto U \cdot \left(-\color{blue}{\left(-1 \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U} - 1\right)}\right) \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{U \cdot \left(1 + \cos \left(0.5 \cdot K\right) \cdot \frac{\mathsf{fma}\left(0.3333333333333333, J \cdot {\ell}^{3}, J \cdot \left(\ell \cdot 2\right)\right)}{U}\right)} \]

   9. Taylor expanded in l around inf 100.0%

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

       1. associate-*r* 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\left(J \cdot {\ell}^{3}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+81}:\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -23:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 840:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 3.4 \cdot 10^{+89}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ t_1 := 0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -23:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 750:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+88}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ (* (- (exp l) (exp (- l))) J) U))
        (t_1 (* 0.3333333333333333 (* (cos (* K 0.5)) (* J (pow l 3.0))))))
   (if (<= l -2.7e+81)
     t_1
     (if (<= l -23.0)
       t_0
       (if (<= l 750.0)
         (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
         (if (<= l 1.55e+88) t_0 t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = ((exp(l) - exp(-l)) * J) + U;
	double t_1 = 0.3333333333333333 * (cos((K * 0.5)) * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -2.7e+81) {
		tmp = t_1;
	} else if (l <= -23.0) {
		tmp = t_0;
	} else if (l <= 750.0) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.55e+88) {
		tmp = 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 = ((exp(l) - exp(-l)) * j) + u
    t_1 = 0.3333333333333333d0 * (cos((k * 0.5d0)) * (j * (l ** 3.0d0)))
    if (l <= (-2.7d+81)) then
        tmp = t_1
    else if (l <= (-23.0d0)) then
        tmp = t_0
    else if (l <= 750.0d0) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    else if (l <= 1.55d+88) then
        tmp = 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.exp(l) - Math.exp(-l)) * J) + U;
	double t_1 = 0.3333333333333333 * (Math.cos((K * 0.5)) * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -2.7e+81) {
		tmp = t_1;
	} else if (l <= -23.0) {
		tmp = t_0;
	} else if (l <= 750.0) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else if (l <= 1.55e+88) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = ((math.exp(l) - math.exp(-l)) * J) + U
	t_1 = 0.3333333333333333 * (math.cos((K * 0.5)) * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -2.7e+81:
		tmp = t_1
	elif l <= -23.0:
		tmp = t_0
	elif l <= 750.0:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	elif l <= 1.55e+88:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	t_1 = Float64(0.3333333333333333 * Float64(cos(Float64(K * 0.5)) * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -2.7e+81)
		tmp = t_1;
	elseif (l <= -23.0)
		tmp = t_0;
	elseif (l <= 750.0)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	elseif (l <= 1.55e+88)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = ((exp(l) - exp(-l)) * J) + U;
	t_1 = 0.3333333333333333 * (cos((K * 0.5)) * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -2.7e+81)
		tmp = t_1;
	elseif (l <= -23.0)
		tmp = t_0;
	elseif (l <= 750.0)
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	elseif (l <= 1.55e+88)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.7e+81], t$95$1, If[LessEqual[l, -23.0], t$95$0, If[LessEqual[l, 750.0], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.55e+88], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.6999999999999999e81 or 1.5500000000000001e88 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.9%

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

      1. distribute-lft-in 98.9%

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

      2. distribute-rgt-in 98.9%

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

      3. *-commutative 98.9%

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

      4. associate-*l* 98.9%

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

      5. unpow2 98.9%

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

      6. pow3 98.9%

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

   5. Applied egg-rr 98.9%

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

   6. Taylor expanded in U around inf 98.9%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. *-rgt-identity 98.9%

         \[\leadsto U \cdot \color{blue}{\left(\left(1 + \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}\right) \cdot 1\right)} \]

      2. metadata-eval 98.9%

         \[\leadsto U \cdot \left(\left(1 + \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}\right) \cdot \color{blue}{\left(--1\right)}\right) \]

      3. distribute-rgt-neg-in 98.9%

         \[\leadsto U \cdot \color{blue}{\left(-\left(1 + \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}\right) \cdot -1\right)} \]

      4. +-commutative 98.9%

         \[\leadsto U \cdot \left(-\color{blue}{\left(\frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U} + 1\right)} \cdot -1\right) \]

      5. distribute-lft1-in 98.9%

         \[\leadsto U \cdot \left(-\color{blue}{\left(\frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U} \cdot -1 + -1\right)}\right) \]

      6. *-commutative 98.9%

         \[\leadsto U \cdot \left(-\left(\color{blue}{-1 \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U}} + -1\right)\right) \]

      7. metadata-eval 98.9%

         \[\leadsto U \cdot \left(-\left(-1 \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U} + \color{blue}{\left(-1\right)}\right)\right) \]

      8. sub-neg 98.9%

         \[\leadsto U \cdot \left(-\color{blue}{\left(-1 \cdot \frac{\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(J \cdot \ell\right)\right)}{U} - 1\right)}\right) \]

   8. Simplified 98.9%

      \[\leadsto \color{blue}{U \cdot \left(1 + \cos \left(0.5 \cdot K\right) \cdot \frac{\mathsf{fma}\left(0.3333333333333333, J \cdot {\ell}^{3}, J \cdot \left(\ell \cdot 2\right)\right)}{U}\right)} \]

   9. Taylor expanded in l around inf 98.9%

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

       1. associate-*r* 98.9%

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

   11. Simplified 98.9%

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

   if -2.6999999999999999e81 < l < -23 or 750 < l < 1.5500000000000001e88

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 70.6%

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

      1. +-commutative 70.6%

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

   5. Simplified 70.6%

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

   if -23 < l < 750

   1. Initial program 66.9%

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

   3. Taylor expanded in l around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+81}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq -23:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 750:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+88}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+81}:\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -23:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.9e+81)
   (+ U (* (* J 0.3333333333333333) (* (pow l 3.0) (cos (* K 0.5)))))
   (if (<= l -23.0)
     (+ (* (- (exp l) (exp (- l))) J) U)
     (+
      U
      (*
       (cos (/ K 2.0))
       (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.9e+81) {
		tmp = U + ((J * 0.3333333333333333) * (pow(l, 3.0) * cos((K * 0.5))));
	} else if (l <= -23.0) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.9d+81)) then
        tmp = u + ((j * 0.3333333333333333d0) * ((l ** 3.0d0) * cos((k * 0.5d0))))
    else if (l <= (-23.0d0)) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.9e+81) {
		tmp = U + ((J * 0.3333333333333333) * (Math.pow(l, 3.0) * Math.cos((K * 0.5))));
	} else if (l <= -23.0) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.9e+81:
		tmp = U + ((J * 0.3333333333333333) * (math.pow(l, 3.0) * math.cos((K * 0.5))))
	elif l <= -23.0:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.9e+81)
		tmp = Float64(U + Float64(Float64(J * 0.3333333333333333) * Float64((l ^ 3.0) * cos(Float64(K * 0.5)))));
	elseif (l <= -23.0)
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.9e+81)
		tmp = U + ((J * 0.3333333333333333) * ((l ^ 3.0) * cos((K * 0.5))));
	elseif (l <= -23.0)
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.9e+81], N[(U + N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[(N[Power[l, 3.0], $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -23.0], 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[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.9e81

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 98.2%

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

   4. Taylor expanded in l around inf 98.2%

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

      1. associate-*r* 98.2%

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

      2. *-commutative 98.2%

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

   6. Simplified 98.2%

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

   if -1.9e81 < l < -23

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 68.4%

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

      1. +-commutative 68.4%

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

   5. Simplified 68.4%

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

   if -23 < l

   1. Initial program 76.4%

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

   3. Taylor expanded in l around 0 92.7%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.9 \cdot 10^{+81}:\\ \;\;\;\;U + \left(J \cdot 0.3333333333333333\right) \cdot \left({\ell}^{3} \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -23:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.58:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right) + 2 \cdot \left(\ell \cdot J\right)\right)\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.57999999999999996

   1. Initial program 78.8%

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

   3. Taylor expanded in l around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. associate-*r* 67.9%

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

   5. Simplified 67.9%

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

   if 0.57999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.7%

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

   3. Taylor expanded in l around 0 89.2%

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

      1. distribute-lft-in 89.2%

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

      2. distribute-rgt-in 89.2%

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

      3. *-commutative 89.2%

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

      4. associate-*l* 89.2%

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

      5. unpow2 89.2%

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

      6. pow3 89.2%

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

   5. Applied egg-rr 89.2%

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

   6. Taylor expanded in K around 0 84.9%

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

4. Final simplification 79.7%

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

Alternative 7: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.58:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.58)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (pow 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 <= 0.58) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(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 <= 0.58d0) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (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 <= 0.58) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.58:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * math.pow(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 <= 0.58)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (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 <= 0.58)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (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[LessEqual[t$95$0, 0.58], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.57999999999999996

   1. Initial program 78.8%

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

   3. Taylor expanded in l around 0 67.9%

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

      1. *-commutative 67.9%

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

      2. associate-*r* 67.9%

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

   5. Simplified 67.9%

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

   if 0.57999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 84.7%

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

   3. Taylor expanded in l around 0 89.2%

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

   4. Taylor expanded in K around 0 84.9%

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

4. Final simplification 79.7%

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

Alternative 8: 86.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -23 \lor \neg \left(\ell \leq 750\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -23.0) (not (<= l 750.0)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -23.0) || !(l <= 750.0)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-23.0d0)) .or. (.not. (l <= 750.0d0))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -23.0) || !(l <= 750.0)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -23.0) or not (l <= 750.0):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -23.0) || !(l <= 750.0))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -23.0) || ~((l <= 750.0)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -23.0], N[Not[LessEqual[l, 750.0]], $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[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -23 or 750 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 77.4%

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

      1. +-commutative 77.4%

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

   5. Simplified 77.4%

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

   if -23 < l < 750

   1. Initial program 66.9%

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

   3. Taylor expanded in l around 0 98.3%

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

      1. *-commutative 98.3%

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

      2. associate-*r* 98.3%

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

   5. Simplified 98.3%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -23 \lor \neg \left(\ell \leq 750\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+54} \lor \neg \left(\ell \leq 3.5 \cdot 10^{+63}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.05e+54) (not (<= l 3.5e+63)))
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.05e+54) || !(l <= 3.5e+63)) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.05d+54)) .or. (.not. (l <= 3.5d+63))) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.05e+54) || !(l <= 3.5e+63)) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.05e+54) or not (l <= 3.5e+63):
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.05e+54) || !(l <= 3.5e+63))
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.05e+54) || ~((l <= 3.5e+63)))
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.05e+54], N[Not[LessEqual[l, 3.5e+63]], $MachinePrecision]], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.04999999999999984e54 or 3.50000000000000029e63 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 88.3%

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

   4. Taylor expanded in K around 0 70.3%

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

   5. Taylor expanded in l around inf 70.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]

   if -2.04999999999999984e54 < l < 3.50000000000000029e63

   1. Initial program 70.8%

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

   3. Taylor expanded in l around 0 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-*r* 88.9%

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

   5. Simplified 88.9%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.05 \cdot 10^{+54} \lor \neg \left(\ell \leq 3.5 \cdot 10^{+63}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+53} \lor \neg \left(\ell \leq 1.9 \cdot 10^{+63}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.15e+53) (not (<= l 1.9e+63)))
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.15e+53) || !(l <= 1.9e+63)) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} 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 <= (-2.15d+53)) .or. (.not. (l <= 1.9d+63))) then
        tmp = 0.3333333333333333d0 * (j * (l ** 3.0d0))
    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 <= -2.15e+53) || !(l <= 1.9e+63)) {
		tmp = 0.3333333333333333 * (J * Math.pow(l, 3.0));
	} 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 <= -2.15e+53) or not (l <= 1.9e+63):
		tmp = 0.3333333333333333 * (J * math.pow(l, 3.0))
	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 <= -2.15e+53) || !(l <= 1.9e+63))
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	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 <= -2.15e+53) || ~((l <= 1.9e+63)))
		tmp = 0.3333333333333333 * (J * (l ^ 3.0));
	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, -2.15e+53], N[Not[LessEqual[l, 1.9e+63]], $MachinePrecision]], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $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 < -2.1499999999999999e53 or 1.9000000000000001e63 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 88.3%

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

   4. Taylor expanded in K around 0 70.3%

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

   5. Taylor expanded in l around inf 70.3%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]

   if -2.1499999999999999e53 < l < 1.9000000000000001e63

   1. Initial program 70.8%

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

   3. Taylor expanded in l around 0 88.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.15 \cdot 10^{+53} \lor \neg \left(\ell \leq 1.9 \cdot 10^{+63}\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 71.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+53} \lor \neg \left(\ell \leq 740\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.1e+53) (not (<= l 740.0)))
   (* 0.3333333333333333 (* J (pow l 3.0)))
   (fma J (* l 2.0) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.1e+53) || !(l <= 740.0)) {
		tmp = 0.3333333333333333 * (J * pow(l, 3.0));
	} else {
		tmp = fma(J, (l * 2.0), U);
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.1e+53) || !(l <= 740.0))
		tmp = Float64(0.3333333333333333 * Float64(J * (l ^ 3.0)));
	else
		tmp = fma(J, Float64(l * 2.0), U);
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.1e+53], N[Not[LessEqual[l, 740.0]], $MachinePrecision]], N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.1000000000000002e53 or 740 < l

   1. Initial program 99.2%

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

   3. Taylor expanded in l around 0 82.0%

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

   4. Taylor expanded in K around 0 64.7%

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

   5. Taylor expanded in l around inf 64.7%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)} \]

   if -2.1000000000000002e53 < l < 740

   1. Initial program 69.2%

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

   3. Taylor expanded in l around 0 94.4%

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

   4. Taylor expanded in K around 0 77.6%

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

   5. Taylor expanded in l around 0 77.6%

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

      1. +-commutative 77.6%

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

      2. *-commutative 77.6%

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

      3. associate-*r* 77.6%

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

      4. *-commutative 77.6%

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

      5. fma-define 77.6%

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

      6. *-commutative 77.6%

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

   7. Simplified 77.6%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+53} \lor \neg \left(\ell \leq 740\right):\\ \;\;\;\;0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 1.06 \cdot 10^{+189} \lor \neg \left(K \leq 1.6 \cdot 10^{+256}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{K}^{2} \cdot 32\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= K 1.06e+189) (not (<= K 1.6e+256)))
   (+ U (* l (* J 2.0)))
   (* (pow K 2.0) 32.0)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((K <= 1.06e+189) || !(K <= 1.6e+256)) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = pow(K, 2.0) * 32.0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((k <= 1.06d+189) .or. (.not. (k <= 1.6d+256))) then
        tmp = u + (l * (j * 2.0d0))
    else
        tmp = (k ** 2.0d0) * 32.0d0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((K <= 1.06e+189) || !(K <= 1.6e+256)) {
		tmp = U + (l * (J * 2.0));
	} else {
		tmp = Math.pow(K, 2.0) * 32.0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (K <= 1.06e+189) or not (K <= 1.6e+256):
		tmp = U + (l * (J * 2.0))
	else:
		tmp = math.pow(K, 2.0) * 32.0
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((K <= 1.06e+189) || !(K <= 1.6e+256))
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	else
		tmp = Float64((K ^ 2.0) * 32.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((K <= 1.06e+189) || ~((K <= 1.6e+256)))
		tmp = U + (l * (J * 2.0));
	else
		tmp = (K ^ 2.0) * 32.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[K, 1.06e+189], N[Not[LessEqual[K, 1.6e+256]], $MachinePrecision]], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[K, 2.0], $MachinePrecision] * 32.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 1.05999999999999998e189 or 1.59999999999999998e256 < K

   1. Initial program 82.8%

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

   3. Taylor expanded in l around 0 88.9%

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

   4. Taylor expanded in K around 0 73.5%

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

   5. Taylor expanded in l around 0 54.3%

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

      1. associate-*r* 54.3%

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

      2. *-commutative 54.3%

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

   7. Simplified 54.3%

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

   if 1.05999999999999998e189 < K < 1.59999999999999998e256

   1. Initial program 85.7%

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

   4. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \cos \left(-4 \cdot K\right), -U\right)} \]
   5. Step-by-step derivation

      1. fma-neg 2.3%

         \[\leadsto \color{blue}{-4 \cdot \cos \left(-4 \cdot K\right) - U} \]

      2. *-commutative 2.3%

         \[\leadsto -4 \cdot \cos \color{blue}{\left(K \cdot -4\right)} - U \]

   6. Simplified 2.3%

      \[\leadsto \color{blue}{-4 \cdot \cos \left(K \cdot -4\right) - U} \]

   7. Taylor expanded in K around 0 46.5%

      \[\leadsto \color{blue}{32 \cdot {K}^{2} - \left(4 + U\right)} \]

   8. Taylor expanded in K around inf 46.5%

      \[\leadsto \color{blue}{32 \cdot {K}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative 46.5%

         \[\leadsto \color{blue}{{K}^{2} \cdot 32} \]

   10. Simplified 46.5%

       \[\leadsto \color{blue}{{K}^{2} \cdot 32} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 1.06 \cdot 10^{+189} \lor \neg \left(K \leq 1.6 \cdot 10^{+256}\right):\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;{K}^{2} \cdot 32\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 52.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 8.5 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{elif}\;K \leq 4.6 \cdot 10^{+255}:\\ \;\;\;\;{K}^{2} \cdot 32\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 8.5e+188)
   (fma J (* l 2.0) U)
   (if (<= K 4.6e+255) (* (pow K 2.0) 32.0) (+ U (* l (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 8.5e+188) {
		tmp = fma(J, (l * 2.0), U);
	} else if (K <= 4.6e+255) {
		tmp = pow(K, 2.0) * 32.0;
	} else {
		tmp = U + (l * (J * 2.0));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 8.5e+188)
		tmp = fma(J, Float64(l * 2.0), U);
	elseif (K <= 4.6e+255)
		tmp = Float64((K ^ 2.0) * 32.0);
	else
		tmp = Float64(U + Float64(l * Float64(J * 2.0)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 8.5e+188], N[(J * N[(l * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[K, 4.6e+255], N[(N[Power[K, 2.0], $MachinePrecision] * 32.0), $MachinePrecision], N[(U + N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if K < 8.49999999999999958e188

   1. Initial program 83.3%

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

   3. Taylor expanded in l around 0 88.4%

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

   4. Taylor expanded in K around 0 74.0%

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

   5. Taylor expanded in l around 0 54.2%

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

      1. +-commutative 54.2%

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

      2. *-commutative 54.2%

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

      3. associate-*r* 54.2%

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

      4. *-commutative 54.2%

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

      5. fma-define 54.3%

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

      6. *-commutative 54.3%

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

   7. Simplified 54.3%

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

   if 8.49999999999999958e188 < K < 4.6000000000000001e255

   1. Initial program 85.7%

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

   4. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \cos \left(-4 \cdot K\right), -U\right)} \]
   5. Step-by-step derivation

      1. fma-neg 2.3%

         \[\leadsto \color{blue}{-4 \cdot \cos \left(-4 \cdot K\right) - U} \]

      2. *-commutative 2.3%

         \[\leadsto -4 \cdot \cos \color{blue}{\left(K \cdot -4\right)} - U \]

   6. Simplified 2.3%

      \[\leadsto \color{blue}{-4 \cdot \cos \left(K \cdot -4\right) - U} \]

   7. Taylor expanded in K around 0 46.5%

      \[\leadsto \color{blue}{32 \cdot {K}^{2} - \left(4 + U\right)} \]

   8. Taylor expanded in K around inf 46.5%

      \[\leadsto \color{blue}{32 \cdot {K}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative 46.5%

         \[\leadsto \color{blue}{{K}^{2} \cdot 32} \]

   10. Simplified 46.5%

       \[\leadsto \color{blue}{{K}^{2} \cdot 32} \]

   if 4.6000000000000001e255 < K

   1. Initial program 72.8%

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

   3. Taylor expanded in l around 0 99.7%

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

   4. Taylor expanded in K around 0 63.6%

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

   5. Taylor expanded in l around 0 55.7%

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

      1. associate-*r* 55.7%

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

      2. *-commutative 55.7%

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

   7. Simplified 55.7%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 8.5 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(J, \ell \cdot 2, U\right)\\ \mathbf{elif}\;K \leq 4.6 \cdot 10^{+255}:\\ \;\;\;\;{K}^{2} \cdot 32\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(J \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.6% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -560000000000 \lor \neg \left(\ell \leq 260\right):\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -560000000000.0) (not (<= l 260.0))) (- -4.0 (* U U)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -560000000000.0) || !(l <= 260.0)) {
		tmp = -4.0 - (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 <= (-560000000000.0d0)) .or. (.not. (l <= 260.0d0))) then
        tmp = (-4.0d0) - (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 <= -560000000000.0) || !(l <= 260.0)) {
		tmp = -4.0 - (U * U);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -560000000000.0) or not (l <= 260.0):
		tmp = -4.0 - (U * U)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -560000000000.0) || !(l <= 260.0))
		tmp = Float64(-4.0 - Float64(U * U));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -560000000000.0) || ~((l <= 260.0)))
		tmp = -4.0 - (U * U);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -560000000000.0], N[Not[LessEqual[l, 260.0]], $MachinePrecision]], N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -5.6e11 or 260 < l

   1. Initial program 99.2%

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

      1. associate-*l* 99.2%

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

      2. fma-define 99.2%

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

   3. Simplified 99.2%

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

   6. Applied egg-rr 14.8%

      \[\leadsto \color{blue}{-4 + \left(-U\right) \cdot U} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 14.8%

         \[\leadsto \color{blue}{-4 - U \cdot U} \]

   8. Simplified 14.8%

      \[\leadsto \color{blue}{-4 - U \cdot U} \]

   if -5.6e11 < l < 260

   1. Initial program 67.6%

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

   3. Taylor expanded in J around 0 64.6%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -560000000000 \lor \neg \left(\ell \leq 260\right):\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.4% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+34}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{elif}\;\ell \leq 260:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-4 - U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.4e+34) (* U (- 2.0 U)) (if (<= l 260.0) U (- -4.0 (* U U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.4e+34) {
		tmp = U * (2.0 - U);
	} else if (l <= 260.0) {
		tmp = U;
	} else {
		tmp = -4.0 - (U * U);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-5.4d+34)) then
        tmp = u * (2.0d0 - u)
    else if (l <= 260.0d0) then
        tmp = u
    else
        tmp = (-4.0d0) - (u * u)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.4e+34) {
		tmp = U * (2.0 - U);
	} else if (l <= 260.0) {
		tmp = U;
	} else {
		tmp = -4.0 - (U * U);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.4e+34:
		tmp = U * (2.0 - U)
	elif l <= 260.0:
		tmp = U
	else:
		tmp = -4.0 - (U * U)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.4e+34)
		tmp = Float64(U * Float64(2.0 - U));
	elseif (l <= 260.0)
		tmp = U;
	else
		tmp = Float64(-4.0 - Float64(U * U));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.4e+34)
		tmp = U * (2.0 - U);
	elseif (l <= 260.0)
		tmp = U;
	else
		tmp = -4.0 - (U * U);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.4e+34], N[(U * N[(2.0 - U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 260.0], U, N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.4000000000000001e34

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 16.6%

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

      1. +-commutative 16.6%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-U, U, U\right) + U} \]

      2. fma-undefine 16.6%

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

      3. associate-+l+ 16.6%

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

      4. count-2 16.6%

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

      5. distribute-rgt-out 16.6%

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

   8. Simplified 16.6%

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

   if -5.4000000000000001e34 < l < 260

   1. Initial program 67.8%

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

   3. Taylor expanded in J around 0 64.2%

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

   if 260 < l

   1. Initial program 98.2%

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

      1. associate-*l* 98.2%

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

      2. fma-define 98.2%

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

   3. Simplified 98.2%

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

   6. Applied egg-rr 12.9%

      \[\leadsto \color{blue}{-4 + \left(-U\right) \cdot U} \]
   7. Step-by-step derivation

      1. cancel-sign-sub-inv 12.9%

         \[\leadsto \color{blue}{-4 - U \cdot U} \]

   8. Simplified 12.9%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.4 \cdot 10^{+34}:\\ \;\;\;\;U \cdot \left(2 - U\right)\\ \mathbf{elif}\;\ell \leq 260:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;-4 - U \cdot U\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 41.4% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+69} \lor \neg \left(\ell \leq 4.2 \cdot 10^{+89}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.2e+69) (not (<= l 4.2e+89))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.2e+69) || !(l <= 4.2e+89)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-6.2d+69)) .or. (.not. (l <= 4.2d+89))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.2e+69) || !(l <= 4.2e+89)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.2e+69) or not (l <= 4.2e+89):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.2e+69) || !(l <= 4.2e+89))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -6.2e+69) || ~((l <= 4.2e+89)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.2e+69], N[Not[LessEqual[l, 4.2e+89]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -6.1999999999999997e69 or 4.19999999999999972e89 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 14.3%

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

   if -6.1999999999999997e69 < l < 4.19999999999999972e89

   1. Initial program 73.2%

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

   3. Taylor expanded in J around 0 52.8%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+69} \lor \neg \left(\ell \leq 4.2 \cdot 10^{+89}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 41.3% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.75 \cdot 10^{+72}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \mathbf{elif}\;\ell \leq 4.2 \cdot 10^{+89}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -1.75e+72) (* U (- U -4.0)) (if (<= l 4.2e+89) U (* U U))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.75e+72) {
		tmp = U * (U - -4.0);
	} else if (l <= 4.2e+89) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.75d+72)) then
        tmp = u * (u - (-4.0d0))
    else if (l <= 4.2d+89) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.75e+72) {
		tmp = U * (U - -4.0);
	} else if (l <= 4.2e+89) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.75e+72:
		tmp = U * (U - -4.0)
	elif l <= 4.2e+89:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.75e+72)
		tmp = Float64(U * Float64(U - -4.0));
	elseif (l <= 4.2e+89)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.75e+72)
		tmp = U * (U - -4.0);
	elseif (l <= 4.2e+89)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.75e+72], N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.2e+89], U, N[(U * U), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.75000000000000005e72

   1. Initial program 100.0%

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

   4. Applied egg-rr 14.5%

      \[\leadsto \color{blue}{U \cdot \left(U - -4\right)} \]

   if -1.75000000000000005e72 < l < 4.19999999999999972e89

   1. Initial program 73.5%

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

   3. Taylor expanded in J around 0 52.1%

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

   if 4.19999999999999972e89 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 14.8%

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

Alternative 18: 53.1% 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 82.9%

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

3. Taylor expanded in l around 0 88.7%

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

4. Taylor expanded in K around 0 71.7%

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

5. Taylor expanded in l around 0 52.5%

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

   1. associate-*r* 52.5%

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

   2. *-commutative 52.5%

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

7. Simplified 52.5%

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

8. Final simplification 52.5%

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

Alternative 19: 36.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.9%

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

3. Taylor expanded in J around 0 34.5%

   \[\leadsto \color{blue}{U} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))