Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.1% → 99.7%

Time: 14.7s

Alternatives: 21

Speedup: 1.5×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.1% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;t_0 \cdot \left(t_1 \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e-7)))
     (+ (* t_0 (* t_1 J)) U)
     (+ U (* t_0 (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e-7)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 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 <= 5e-7)) {
		tmp = (t_0 * (t_1 * J)) + U;
	} else {
		tmp = U + (t_0 * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 5e-7):
		tmp = (t_0 * (t_1 * J)) + U
	else:
		tmp = U + (t_0 * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e-7))
		tmp = Float64(Float64(t_0 * Float64(t_1 * J)) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 5e-7)))
		tmp = (t_0 * (t_1 * J)) + U;
	else
		tmp = U + (t_0 * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e-7]], $MachinePrecision]], N[(N[(t$95$0 * N[(t$95$1 * J), $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -inf.0 or 4.99999999999999977e-7 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 4.99999999999999977e-7

   1. Initial program 67.6%

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

   2. Taylor expanded in l around 0 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 5 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \left(\frac{K}{2}\right) \cdot \left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 2: 77.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;t_0 \leq -0.985:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.9:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -0.82:\\ \;\;\;\;U + t_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.01:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0)))
        (t_1 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= t_0 -0.985)
     (+ U (* (* J 2.0) (* l (cos (* K 0.5)))))
     (if (<= t_0 -0.9)
       t_1
       (if (<= t_0 -0.82)
         (+ U (* t_0 (* J (* l 2.0))))
         (if (<= t_0 -0.01)
           t_1
           (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (t_0 <= -0.985) {
		tmp = U + ((J * 2.0) * (l * cos((K * 0.5))));
	} else if (t_0 <= -0.9) {
		tmp = t_1;
	} else if (t_0 <= -0.82) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else if (t_0 <= -0.01) {
		tmp = t_1;
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    if (t_0 <= (-0.985d0)) then
        tmp = u + ((j * 2.0d0) * (l * cos((k * 0.5d0))))
    else if (t_0 <= (-0.9d0)) then
        tmp = t_1
    else if (t_0 <= (-0.82d0)) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else if (t_0 <= (-0.01d0)) then
        tmp = t_1
    else
        tmp = u + (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (t_0 <= -0.985) {
		tmp = U + ((J * 2.0) * (l * Math.cos((K * 0.5))));
	} else if (t_0 <= -0.9) {
		tmp = t_1;
	} else if (t_0 <= -0.82) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else if (t_0 <= -0.01) {
		tmp = t_1;
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	tmp = 0
	if t_0 <= -0.985:
		tmp = U + ((J * 2.0) * (l * math.cos((K * 0.5))))
	elif t_0 <= -0.9:
		tmp = t_1
	elif t_0 <= -0.82:
		tmp = U + (t_0 * (J * (l * 2.0)))
	elif t_0 <= -0.01:
		tmp = t_1
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (t_0 <= -0.985)
		tmp = Float64(U + Float64(Float64(J * 2.0) * Float64(l * cos(Float64(K * 0.5)))));
	elseif (t_0 <= -0.9)
		tmp = t_1;
	elseif (t_0 <= -0.82)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	elseif (t_0 <= -0.01)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	tmp = 0.0;
	if (t_0 <= -0.985)
		tmp = U + ((J * 2.0) * (l * cos((K * 0.5))));
	elseif (t_0 <= -0.9)
		tmp = t_1;
	elseif (t_0 <= -0.82)
		tmp = U + (t_0 * (J * (l * 2.0)));
	elseif (t_0 <= -0.01)
		tmp = t_1;
	else
		tmp = U + (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.985], N[(U + N[(N[(J * 2.0), $MachinePrecision] * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.9], t$95$1, If[LessEqual[t$95$0, -0.82], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], t$95$1, N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.984999999999999987

   1. Initial program 67.4%

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

   2. Taylor expanded in l around 0 90.2%

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

   3. Taylor expanded in l around 0 80.2%

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

      1. associate-*r* 80.2%

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

      2. *-commutative 80.2%

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

      3. *-commutative 80.2%

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

      4. associate-*r* 80.2%

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

      5. *-commutative 80.2%

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

      6. *-commutative 80.2%

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

   5. Simplified 80.2%

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

   if -0.984999999999999987 < (cos.f64 (/.f64 K 2)) < -0.900000000000000022 or -0.819999999999999951 < (cos.f64 (/.f64 K 2)) < -0.0100000000000000002

   1. Initial program 90.8%

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

   2. Taylor expanded in l around 0 40.5%

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

      1. *-commutative 40.5%

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

      2. associate-*l* 40.5%

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

      3. *-commutative 40.5%

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

   4. Simplified 40.5%

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

   5. Taylor expanded in K around 0 57.8%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   6. Step-by-step derivation

      1. +-commutative 57.8%

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

      2. associate-*r* 57.8%

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

      3. distribute-rgt-out 65.5%

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

      4. *-commutative 65.5%

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

      5. unpow2 65.5%

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

   7. Simplified 65.5%

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

   if -0.900000000000000022 < (cos.f64 (/.f64 K 2)) < -0.819999999999999951

   1. Initial program 90.4%

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

   2. Taylor expanded in l around 0 80.5%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K 2))

   1. Initial program 84.4%

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

   2. Taylor expanded in l around 0 91.7%

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

   3. Taylor expanded in K around 0 86.6%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J + U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.985:\\ \;\;\;\;U + \left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.9:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 3: 68.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t_0 \leq -0.89:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J + \left(K \cdot K\right) \cdot \left(J \cdot -0.125\right)\right)\right)\\ \mathbf{elif}\;t_0 \leq -0.82:\\ \;\;\;\;U + J \cdot 0.125\\ \mathbf{elif}\;t_0 \leq -0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.89)
     (+ U (* 2.0 (* l (+ J (* (* K K) (* J -0.125))))))
     (if (<= t_0 -0.82)
       (+ U (* J 0.125))
       (if (<= t_0 -0.01)
         (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))
         (+ U (* 0.3333333333333333 (* J (pow l 3.0)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.89) {
		tmp = U + (2.0 * (l * (J + ((K * K) * (J * -0.125)))));
	} else if (t_0 <= -0.82) {
		tmp = U + (J * 0.125);
	} else if (t_0 <= -0.01) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (0.3333333333333333 * (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) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.89d0)) then
        tmp = u + (2.0d0 * (l * (j + ((k * k) * (j * (-0.125d0))))))
    else if (t_0 <= (-0.82d0)) then
        tmp = u + (j * 0.125d0)
    else if (t_0 <= (-0.01d0)) then
        tmp = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    else
        tmp = u + (0.3333333333333333d0 * (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 / 2.0));
	double tmp;
	if (t_0 <= -0.89) {
		tmp = U + (2.0 * (l * (J + ((K * K) * (J * -0.125)))));
	} else if (t_0 <= -0.82) {
		tmp = U + (J * 0.125);
	} else if (t_0 <= -0.01) {
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.89:
		tmp = U + (2.0 * (l * (J + ((K * K) * (J * -0.125)))))
	elif t_0 <= -0.82:
		tmp = U + (J * 0.125)
	elif t_0 <= -0.01:
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.89)
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J + Float64(Float64(K * K) * Float64(J * -0.125))))));
	elseif (t_0 <= -0.82)
		tmp = Float64(U + Float64(J * 0.125));
	elseif (t_0 <= -0.01)
		tmp = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))));
	else
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	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 <= -0.89)
		tmp = U + (2.0 * (l * (J + ((K * K) * (J * -0.125)))));
	elseif (t_0 <= -0.82)
		tmp = U + (J * 0.125);
	elseif (t_0 <= -0.01)
		tmp = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	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, -0.89], N[(U + N[(2.0 * N[(l * N[(J + N[(N[(K * K), $MachinePrecision] * N[(J * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.82], N[(U + N[(J * 0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -0.01], N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.890000000000000013

   1. Initial program 82.8%

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

   2. Taylor expanded in l around 0 48.6%

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

      1. *-commutative 48.6%

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

      2. associate-*l* 48.5%

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

      3. *-commutative 48.5%

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

   4. Simplified 48.5%

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

   5. Taylor expanded in K around 0 0.7%

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

      1. associate-+r+ 0.7%

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

      2. +-commutative 0.7%

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

      3. associate-*r* 0.7%

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

      4. associate-*r* 0.7%

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

      5. distribute-rgt-out 0.7%

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

      6. unpow2 0.7%

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

   7. Simplified 0.7%

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

   8. Taylor expanded in K around 0 44.5%

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

      1. +-commutative 44.5%

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

      2. unpow2 44.5%

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

      3. associate-*r* 44.5%

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

      4. *-commutative 44.5%

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

      5. associate-*l* 44.5%

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

      6. distribute-lft-out 55.1%

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

      7. associate-*r* 55.1%

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

   10. Simplified 55.1%

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

   if -0.890000000000000013 < (cos.f64 (/.f64 K 2)) < -0.819999999999999951

   1. Initial program 89.3%

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

   3. Applied egg-rr 67.3%

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

   4. Taylor expanded in K around 0 67.7%

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

      1. *-commutative 67.7%

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

   6. Simplified 67.7%

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

   if -0.819999999999999951 < (cos.f64 (/.f64 K 2)) < -0.0100000000000000002

   1. Initial program 89.2%

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

   2. Taylor expanded in l around 0 47.3%

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

      1. *-commutative 47.3%

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

      2. associate-*l* 47.3%

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

      3. *-commutative 47.3%

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

   4. Simplified 47.3%

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

   5. Taylor expanded in K around 0 54.5%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   6. Step-by-step derivation

      1. +-commutative 54.5%

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

      2. associate-*r* 54.5%

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

      3. distribute-rgt-out 63.5%

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

      4. *-commutative 63.5%

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

      5. unpow2 63.5%

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

   7. Simplified 63.5%

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

   if -0.0100000000000000002 < (cos.f64 (/.f64 K 2))

   1. Initial program 84.4%

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

   2. Taylor expanded in l around 0 91.7%

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

   3. Taylor expanded in K around 0 86.6%

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

   4. Taylor expanded in l around inf 75.7%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.89:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J + \left(K \cdot K\right) \cdot \left(J \cdot -0.125\right)\right)\right)\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.82:\\ \;\;\;\;U + J \cdot 0.125\\ \mathbf{elif}\;\cos \left(\frac{K}{2}\right) \leq -0.01:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 4: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+90} \lor \neg \left(\ell \leq -52\right) \land \left(\ell \leq 7.4 \lor \neg \left(\ell \leq 5.8 \cdot 10^{+102}\right)\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.9e+90)
         (and (not (<= l -52.0)) (or (<= l 7.4) (not (<= l 5.8e+102)))))
   (+
    U
    (* (cos (/ K 2.0)) (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0)))))
   (+ U (* (- (exp l) (exp (- l))) (+ J (* J (* -0.125 (* K K))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.9e+90) || (!(l <= -52.0) && ((l <= 7.4) || !(l <= 5.8e+102)))) {
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
	} else {
		tmp = U + ((exp(l) - exp(-l)) * (J + (J * (-0.125 * (K * K)))));
	}
	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 <= (-2.9d+90)) .or. (.not. (l <= (-52.0d0))) .and. (l <= 7.4d0) .or. (.not. (l <= 5.8d+102))) then
        tmp = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
    else
        tmp = u + ((exp(l) - exp(-l)) * (j + (j * ((-0.125d0) * (k * k)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.9e+90) || (!(l <= -52.0) && ((l <= 7.4) || !(l <= 5.8e+102)))) {
		tmp = U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
	} else {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * (J + (J * (-0.125 * (K * K)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.9e+90) or (not (l <= -52.0) and ((l <= 7.4) or not (l <= 5.8e+102))):
		tmp = U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))
	else:
		tmp = U + ((math.exp(l) - math.exp(-l)) * (J + (J * (-0.125 * (K * K)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.9e+90) || (!(l <= -52.0) && ((l <= 7.4) || !(l <= 5.8e+102))))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))));
	else
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * Float64(J + Float64(J * Float64(-0.125 * Float64(K * K))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.9e+90) || (~((l <= -52.0)) && ((l <= 7.4) || ~((l <= 5.8e+102)))))
		tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
	else
		tmp = U + ((exp(l) - exp(-l)) * (J + (J * (-0.125 * (K * K)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.9e+90], And[N[Not[LessEqual[l, -52.0]], $MachinePrecision], Or[LessEqual[l, 7.4], N[Not[LessEqual[l, 5.8e+102]], $MachinePrecision]]]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * N[(J + N[(J * N[(-0.125 * N[(K * K), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.9000000000000001e90 or -52 < l < 7.4000000000000004 or 5.8000000000000005e102 < l

   1. Initial program 82.9%

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

   2. Taylor expanded in l around 0 99.1%

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

   if -2.9000000000000001e90 < l < -52 or 7.4000000000000004 < l < 5.8000000000000005e102

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 0.0%

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

      1. *-commutative 0.0%

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

      2. associate-*r* 0.0%

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

      3. *-commutative 0.0%

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

      4. associate-*r* 0.0%

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

      5. distribute-rgt-out 86.1%

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

      6. *-commutative 86.1%

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

      7. unpow2 86.1%

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

   4. Simplified 86.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+90} \lor \neg \left(\ell \leq -52\right) \land \left(\ell \leq 7.4 \lor \neg \left(\ell \leq 5.8 \cdot 10^{+102}\right)\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot \left(J + J \cdot \left(-0.125 \cdot \left(K \cdot K\right)\right)\right)\\ \end{array} \]

Alternative 5: 88.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (* (cos (/ K 2.0)) (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (cos((k / 2.0d0)) * (j * ((0.3333333333333333d0 * (l ** 3.0d0)) + (l * 2.0d0))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0)))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * ((0.3333333333333333 * (l ^ 3.0)) + (l * 2.0))));
end

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in l around 0 89.1%

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

3. Final simplification 89.1%

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

Alternative 6: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -165000 \lor \neg \left(\ell \leq 1800\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -165000.0) (not (<= l 1800.0)))
   (+ U (* (- (exp l) (exp (- l))) J))
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -165000.0) || !(l <= 1800.0)) {
		tmp = U + ((exp(l) - exp(-l)) * J);
	} 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 <= (-165000.0d0)) .or. (.not. (l <= 1800.0d0))) then
        tmp = u + ((exp(l) - exp(-l)) * j)
    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 <= -165000.0) || !(l <= 1800.0)) {
		tmp = U + ((Math.exp(l) - Math.exp(-l)) * J);
	} 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 <= -165000.0) or not (l <= 1800.0):
		tmp = U + ((math.exp(l) - math.exp(-l)) * J)
	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 <= -165000.0) || !(l <= 1800.0))
		tmp = Float64(U + Float64(Float64(exp(l) - exp(Float64(-l))) * J));
	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 <= -165000.0) || ~((l <= 1800.0)))
		tmp = U + ((exp(l) - exp(-l)) * J);
	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, -165000.0], N[Not[LessEqual[l, 1800.0]], $MachinePrecision]], N[(U + N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $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 < -165000 or 1800 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in K around 0 70.4%

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

   if -165000 < l < 1800

   1. Initial program 68.9%

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

   2. Taylor expanded in l around 0 98.2%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -165000 \lor \neg \left(\ell \leq 1800\right):\\ \;\;\;\;U + \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 7: 80.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+92}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -3850000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -4.5e+92)
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (if (<= l -3850000.0)
     (log1p (expm1 (- (/ -8.0 U) U)))
     (if (<= l 2.1e-7)
       (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))
       (+ U (* J (+ (* 0.3333333333333333 (pow l 3.0)) (* l 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.5e+92) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else if (l <= -3850000.0) {
		tmp = log1p(expm1(((-8.0 / U) - U)));
	} else if (l <= 2.1e-7) {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -4.5e+92) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else if (l <= -3850000.0) {
		tmp = Math.log1p(Math.expm1(((-8.0 / U) - U)));
	} else if (l <= 2.1e-7) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((0.3333333333333333 * Math.pow(l, 3.0)) + (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -4.5e+92:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	elif l <= -3850000.0:
		tmp = math.log1p(math.expm1(((-8.0 / U) - U)))
	elif l <= 2.1e-7:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	else:
		tmp = U + (J * ((0.3333333333333333 * math.pow(l, 3.0)) + (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -4.5e+92)
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	elseif (l <= -3850000.0)
		tmp = log1p(expm1(Float64(Float64(-8.0 / U) - U)));
	elseif (l <= 2.1e-7)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(0.3333333333333333 * (l ^ 3.0)) + Float64(l * 2.0))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -4.5e+92], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -3850000.0], N[Log[1 + N[(Exp[N[(N[(-8.0 / U), $MachinePrecision] - U), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 2.1e-7], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision] + N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.4999999999999999e92

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 98.2%

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

   3. Taylor expanded in K around 0 65.5%

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

   4. Taylor expanded in l around inf 65.5%

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

   if -4.4999999999999999e92 < l < -3.85e6

   1. Initial program 100.0%

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

   3. Applied egg-rr 4.4%

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

      1. log1p-expm1-u 80.0%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)} \]

   5. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)} \]

   if -3.85e6 < l < 2.1e-7

   1. Initial program 68.1%

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

   2. Taylor expanded in l around 0 98.3%

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

   if 2.1e-7 < l

   1. Initial program 99.6%

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

   2. Taylor expanded in l around 0 80.1%

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

   3. Taylor expanded in K around 0 64.5%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{3} + 2 \cdot \ell\right) \cdot J + U} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+92}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq -3850000:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{-8}{U} - U\right)\right)\\ \mathbf{elif}\;\ell \leq 2.1 \cdot 10^{-7}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.3333333333333333 \cdot {\ell}^{3} + \ell \cdot 2\right)\\ \end{array} \]

Alternative 8: 78.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+20} \lor \neg \left(\ell \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.16e+20) (not (<= l 3.2e-7)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* 2.0 (* l (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.16e+20) || !(l <= 3.2e-7)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (2.0 * (l * (J * 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 <= (-1.16d+20)) .or. (.not. (l <= 3.2d-7))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (2.0d0 * (l * (j * 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 <= -1.16e+20) || !(l <= 3.2e-7)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (2.0 * (l * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.16e+20) or not (l <= 3.2e-7):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (2.0 * (l * (J * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.16e+20) || !(l <= 3.2e-7))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.16e+20) || ~((l <= 3.2e-7)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (2.0 * (l * (J * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.16e+20], N[Not[LessEqual[l, 3.2e-7]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.16e20 or 3.2000000000000001e-7 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 81.8%

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

   3. Taylor expanded in K around 0 60.5%

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

   4. Taylor expanded in l around inf 60.5%

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

   if -1.16e20 < l < 3.2000000000000001e-7

   1. Initial program 68.4%

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

   2. Taylor expanded in l around 0 97.4%

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

      1. *-commutative 97.4%

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

      2. associate-*l* 97.4%

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

      3. *-commutative 97.4%

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

   4. Simplified 97.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.16 \cdot 10^{+20} \lor \neg \left(\ell \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 9: 78.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+19} \lor \neg \left(\ell \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4.5e+19) (not (<= l 3.2e-7)))
   (+ U (* 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 <= -4.5e+19) || !(l <= 3.2e-7)) {
		tmp = U + (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 <= (-4.5d+19)) .or. (.not. (l <= 3.2d-7))) then
        tmp = u + (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 <= -4.5e+19) || !(l <= 3.2e-7)) {
		tmp = U + (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 <= -4.5e+19) or not (l <= 3.2e-7):
		tmp = U + (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 <= -4.5e+19) || !(l <= 3.2e-7))
		tmp = Float64(U + 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 <= -4.5e+19) || ~((l <= 3.2e-7)))
		tmp = U + (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, -4.5e+19], N[Not[LessEqual[l, 3.2e-7]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $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 < -4.5e19 or 3.2000000000000001e-7 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 81.8%

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

   3. Taylor expanded in K around 0 60.5%

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

   4. Taylor expanded in l around inf 60.5%

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

   if -4.5e19 < l < 3.2000000000000001e-7

   1. Initial program 68.4%

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

   2. Taylor expanded in l around 0 97.4%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+19} \lor \neg \left(\ell \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]

Alternative 10: 58.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ t_1 := \frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{+89}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \mathbf{elif}\;\ell \leq -620:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 660:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+168}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+232}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25)))))
        (t_1 (/ (- (/ 64.0 (* U U)) (* U U)) (+ U (/ -8.0 U)))))
   (if (<= l -1.2e+133)
     t_0
     (if (<= l -7.8e+89)
       (+ U (* J (+ 0.125 (* (* K K) -0.015625))))
       (if (<= l -620.0)
         t_1
         (if (<= l 660.0)
           (fma l (* J 2.0) U)
           (if (<= l 6e+168)
             t_0
             (if (<= l 1.65e+232) t_1 (+ U (* 2.0 (* l J)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double t_1 = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	double tmp;
	if (l <= -1.2e+133) {
		tmp = t_0;
	} else if (l <= -7.8e+89) {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	} else if (l <= -620.0) {
		tmp = t_1;
	} else if (l <= 660.0) {
		tmp = fma(l, (J * 2.0), U);
	} else if (l <= 6e+168) {
		tmp = t_0;
	} else if (l <= 1.65e+232) {
		tmp = t_1;
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	t_1 = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) / Float64(U + Float64(-8.0 / U)))
	tmp = 0.0
	if (l <= -1.2e+133)
		tmp = t_0;
	elseif (l <= -7.8e+89)
		tmp = Float64(U + Float64(J * Float64(0.125 + Float64(Float64(K * K) * -0.015625))));
	elseif (l <= -620.0)
		tmp = t_1;
	elseif (l <= 660.0)
		tmp = fma(l, Float64(J * 2.0), U);
	elseif (l <= 6e+168)
		tmp = t_0;
	elseif (l <= 1.65e+232)
		tmp = t_1;
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.2e+133], t$95$0, If[LessEqual[l, -7.8e+89], N[(U + N[(J * N[(0.125 + N[(N[(K * K), $MachinePrecision] * -0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -620.0], t$95$1, If[LessEqual[l, 660.0], N[(l * N[(J * 2.0), $MachinePrecision] + U), $MachinePrecision], If[LessEqual[l, 6e+168], t$95$0, If[LessEqual[l, 1.65e+232], t$95$1, N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -1.1999999999999999e133 or 660 < l < 5.9999999999999996e168

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 29.1%

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

      1. *-commutative 29.1%

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

      2. associate-*l* 29.1%

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

      3. *-commutative 29.1%

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

   4. Simplified 29.1%

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

   5. Taylor expanded in K around 0 24.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   6. Step-by-step derivation

      1. +-commutative 24.0%

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

      2. associate-*r* 24.0%

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

      3. distribute-rgt-out 43.2%

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

      4. *-commutative 43.2%

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

      5. unpow2 43.2%

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

   7. Simplified 43.2%

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

   if -1.1999999999999999e133 < l < -7.80000000000000021e89

   1. Initial program 100.0%

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

   3. Applied egg-rr 1.5%

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

   4. Taylor expanded in K around 0 56.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot J + -0.015625 \cdot \left({K}^{2} \cdot J\right)\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 56.8%

         \[\leadsto \left(0.125 \cdot J + \color{blue}{\left(-0.015625 \cdot {K}^{2}\right) \cdot J}\right) + U \]

      2. distribute-rgt-out 56.8%

         \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot {K}^{2}\right)} + U \]

      3. unpow2 56.8%

         \[\leadsto J \cdot \left(0.125 + -0.015625 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   6. Simplified 56.8%

      \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot \left(K \cdot K\right)\right)} + U \]

   if -7.80000000000000021e89 < l < -620 or 5.9999999999999996e168 < l < 1.65e232

   1. Initial program 100.0%

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

   3. Applied egg-rr 3.9%

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

      1. flip-- 44.9%

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

      2. frac-times 44.9%

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

      3. metadata-eval 44.9%

         \[\leadsto \frac{\frac{\color{blue}{64}}{U \cdot U} - U \cdot U}{\frac{-8}{U} + U} \]

   5. Applied egg-rr 44.9%

      \[\leadsto \color{blue}{\frac{\frac{64}{U \cdot U} - U \cdot U}{\frac{-8}{U} + U}} \]

   if -620 < l < 660

   1. Initial program 68.4%

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

   2. Taylor expanded in l around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. *-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in K around 0 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*r* 83.9%

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

      3. *-commutative 83.9%

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

      4. fma-def 83.9%

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

   7. Simplified 83.9%

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

   if 1.65e232 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 47.3%

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

      1. *-commutative 47.3%

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

      2. associate-*l* 47.3%

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

      3. *-commutative 47.3%

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

   4. Simplified 47.3%

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

   5. Taylor expanded in K around 0 46.8%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.2 \cdot 10^{+133}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq -7.8 \cdot 10^{+89}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \mathbf{elif}\;\ell \leq -620:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{elif}\;\ell \leq 660:\\ \;\;\;\;\mathsf{fma}\left(\ell, J \cdot 2, U\right)\\ \mathbf{elif}\;\ell \leq 6 \cdot 10^{+168}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+232}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 11: 58.6% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ t_1 := U + 2 \cdot \left(\ell \cdot J\right)\\ t_2 := \frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -4.7 \cdot 10^{+89}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \mathbf{elif}\;\ell \leq -1050:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 270:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+167}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+232}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25)))))
        (t_1 (+ U (* 2.0 (* l J))))
        (t_2 (/ (- (/ 64.0 (* U U)) (* U U)) (+ U (/ -8.0 U)))))
   (if (<= l -3.3e+132)
     t_0
     (if (<= l -4.7e+89)
       (+ U (* J (+ 0.125 (* (* K K) -0.015625))))
       (if (<= l -1050.0)
         t_2
         (if (<= l 270.0)
           t_1
           (if (<= l 5e+167) t_0 (if (<= l 1.55e+232) t_2 t_1))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double t_1 = U + (2.0 * (l * J));
	double t_2 = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	double tmp;
	if (l <= -3.3e+132) {
		tmp = t_0;
	} else if (l <= -4.7e+89) {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	} else if (l <= -1050.0) {
		tmp = t_2;
	} else if (l <= 270.0) {
		tmp = t_1;
	} else if (l <= 5e+167) {
		tmp = t_0;
	} else if (l <= 1.55e+232) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    t_1 = u + (2.0d0 * (l * j))
    t_2 = ((64.0d0 / (u * u)) - (u * u)) / (u + ((-8.0d0) / u))
    if (l <= (-3.3d+132)) then
        tmp = t_0
    else if (l <= (-4.7d+89)) then
        tmp = u + (j * (0.125d0 + ((k * k) * (-0.015625d0))))
    else if (l <= (-1050.0d0)) then
        tmp = t_2
    else if (l <= 270.0d0) then
        tmp = t_1
    else if (l <= 5d+167) then
        tmp = t_0
    else if (l <= 1.55d+232) then
        tmp = t_2
    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 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double t_1 = U + (2.0 * (l * J));
	double t_2 = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	double tmp;
	if (l <= -3.3e+132) {
		tmp = t_0;
	} else if (l <= -4.7e+89) {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	} else if (l <= -1050.0) {
		tmp = t_2;
	} else if (l <= 270.0) {
		tmp = t_1;
	} else if (l <= 5e+167) {
		tmp = t_0;
	} else if (l <= 1.55e+232) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	t_1 = U + (2.0 * (l * J))
	t_2 = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U))
	tmp = 0
	if l <= -3.3e+132:
		tmp = t_0
	elif l <= -4.7e+89:
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)))
	elif l <= -1050.0:
		tmp = t_2
	elif l <= 270.0:
		tmp = t_1
	elif l <= 5e+167:
		tmp = t_0
	elif l <= 1.55e+232:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	t_1 = Float64(U + Float64(2.0 * Float64(l * J)))
	t_2 = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) / Float64(U + Float64(-8.0 / U)))
	tmp = 0.0
	if (l <= -3.3e+132)
		tmp = t_0;
	elseif (l <= -4.7e+89)
		tmp = Float64(U + Float64(J * Float64(0.125 + Float64(Float64(K * K) * -0.015625))));
	elseif (l <= -1050.0)
		tmp = t_2;
	elseif (l <= 270.0)
		tmp = t_1;
	elseif (l <= 5e+167)
		tmp = t_0;
	elseif (l <= 1.55e+232)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	t_1 = U + (2.0 * (l * J));
	t_2 = ((64.0 / (U * U)) - (U * U)) / (U + (-8.0 / U));
	tmp = 0.0;
	if (l <= -3.3e+132)
		tmp = t_0;
	elseif (l <= -4.7e+89)
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	elseif (l <= -1050.0)
		tmp = t_2;
	elseif (l <= 270.0)
		tmp = t_1;
	elseif (l <= 5e+167)
		tmp = t_0;
	elseif (l <= 1.55e+232)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] / N[(U + N[(-8.0 / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.3e+132], t$95$0, If[LessEqual[l, -4.7e+89], N[(U + N[(J * N[(0.125 + N[(N[(K * K), $MachinePrecision] * -0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1050.0], t$95$2, If[LessEqual[l, 270.0], t$95$1, If[LessEqual[l, 5e+167], t$95$0, If[LessEqual[l, 1.55e+232], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -3.3000000000000003e132 or 270 < l < 4.9999999999999997e167

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 29.1%

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

      1. *-commutative 29.1%

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

      2. associate-*l* 29.1%

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

      3. *-commutative 29.1%

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

   4. Simplified 29.1%

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

   5. Taylor expanded in K around 0 24.0%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   6. Step-by-step derivation

      1. +-commutative 24.0%

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

      2. associate-*r* 24.0%

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

      3. distribute-rgt-out 43.2%

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

      4. *-commutative 43.2%

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

      5. unpow2 43.2%

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

   7. Simplified 43.2%

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

   if -3.3000000000000003e132 < l < -4.70000000000000022e89

   1. Initial program 100.0%

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

   3. Applied egg-rr 1.5%

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

   4. Taylor expanded in K around 0 56.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot J + -0.015625 \cdot \left({K}^{2} \cdot J\right)\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 56.8%

         \[\leadsto \left(0.125 \cdot J + \color{blue}{\left(-0.015625 \cdot {K}^{2}\right) \cdot J}\right) + U \]

      2. distribute-rgt-out 56.8%

         \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot {K}^{2}\right)} + U \]

      3. unpow2 56.8%

         \[\leadsto J \cdot \left(0.125 + -0.015625 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   6. Simplified 56.8%

      \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot \left(K \cdot K\right)\right)} + U \]

   if -4.70000000000000022e89 < l < -1050 or 4.9999999999999997e167 < l < 1.54999999999999992e232

   1. Initial program 100.0%

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

   3. Applied egg-rr 3.9%

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

      1. flip-- 44.9%

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

      2. frac-times 44.9%

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

      3. metadata-eval 44.9%

         \[\leadsto \frac{\frac{\color{blue}{64}}{U \cdot U} - U \cdot U}{\frac{-8}{U} + U} \]

   5. Applied egg-rr 44.9%

      \[\leadsto \color{blue}{\frac{\frac{64}{U \cdot U} - U \cdot U}{\frac{-8}{U} + U}} \]

   if -1050 < l < 270 or 1.54999999999999992e232 < l

   1. Initial program 72.6%

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

   2. Taylor expanded in l around 0 92.9%

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

      1. *-commutative 92.9%

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

      2. associate-*l* 92.9%

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

      3. *-commutative 92.9%

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

   4. Simplified 92.9%

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

   5. Taylor expanded in K around 0 79.0%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+132}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq -4.7 \cdot 10^{+89}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \mathbf{elif}\;\ell \leq -1050:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{elif}\;\ell \leq 270:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+167}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq 1.55 \cdot 10^{+232}:\\ \;\;\;\;\frac{\frac{64}{U \cdot U} - U \cdot U}{U + \frac{-8}{U}}\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 12: 56.9% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{64}{U \cdot U} - U \cdot U\\ t_1 := t_0 \cdot \left(U \cdot -0.125\right)\\ t_2 := t_0 \cdot \frac{1}{U}\\ \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+295}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -3.35 \cdot 10^{+193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq -850:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+32} \lor \neg \left(\ell \leq 4.8 \cdot 10^{+132}\right):\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (/ 64.0 (* U U)) (* U U)))
        (t_1 (* t_0 (* U -0.125)))
        (t_2 (* t_0 (/ 1.0 U))))
   (if (<= l -1.7e+295)
     t_2
     (if (<= l -3.35e+193)
       t_1
       (if (<= l -2.05e+73)
         t_2
         (if (<= l -850.0)
           t_1
           (if (or (<= l 2.8e+32) (not (<= l 4.8e+132)))
             (+ U (* 2.0 (* l J)))
             (+ U (* J (+ 0.125 (* (* K K) -0.015625)))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (64.0 / (U * U)) - (U * U);
	double t_1 = t_0 * (U * -0.125);
	double t_2 = t_0 * (1.0 / U);
	double tmp;
	if (l <= -1.7e+295) {
		tmp = t_2;
	} else if (l <= -3.35e+193) {
		tmp = t_1;
	} else if (l <= -2.05e+73) {
		tmp = t_2;
	} else if (l <= -850.0) {
		tmp = t_1;
	} else if ((l <= 2.8e+32) || !(l <= 4.8e+132)) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (64.0d0 / (u * u)) - (u * u)
    t_1 = t_0 * (u * (-0.125d0))
    t_2 = t_0 * (1.0d0 / u)
    if (l <= (-1.7d+295)) then
        tmp = t_2
    else if (l <= (-3.35d+193)) then
        tmp = t_1
    else if (l <= (-2.05d+73)) then
        tmp = t_2
    else if (l <= (-850.0d0)) then
        tmp = t_1
    else if ((l <= 2.8d+32) .or. (.not. (l <= 4.8d+132))) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = u + (j * (0.125d0 + ((k * k) * (-0.015625d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (64.0 / (U * U)) - (U * U);
	double t_1 = t_0 * (U * -0.125);
	double t_2 = t_0 * (1.0 / U);
	double tmp;
	if (l <= -1.7e+295) {
		tmp = t_2;
	} else if (l <= -3.35e+193) {
		tmp = t_1;
	} else if (l <= -2.05e+73) {
		tmp = t_2;
	} else if (l <= -850.0) {
		tmp = t_1;
	} else if ((l <= 2.8e+32) || !(l <= 4.8e+132)) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (64.0 / (U * U)) - (U * U)
	t_1 = t_0 * (U * -0.125)
	t_2 = t_0 * (1.0 / U)
	tmp = 0
	if l <= -1.7e+295:
		tmp = t_2
	elif l <= -3.35e+193:
		tmp = t_1
	elif l <= -2.05e+73:
		tmp = t_2
	elif l <= -850.0:
		tmp = t_1
	elif (l <= 2.8e+32) or not (l <= 4.8e+132):
		tmp = U + (2.0 * (l * J))
	else:
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U))
	t_1 = Float64(t_0 * Float64(U * -0.125))
	t_2 = Float64(t_0 * Float64(1.0 / U))
	tmp = 0.0
	if (l <= -1.7e+295)
		tmp = t_2;
	elseif (l <= -3.35e+193)
		tmp = t_1;
	elseif (l <= -2.05e+73)
		tmp = t_2;
	elseif (l <= -850.0)
		tmp = t_1;
	elseif ((l <= 2.8e+32) || !(l <= 4.8e+132))
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = Float64(U + Float64(J * Float64(0.125 + Float64(Float64(K * K) * -0.015625))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (64.0 / (U * U)) - (U * U);
	t_1 = t_0 * (U * -0.125);
	t_2 = t_0 * (1.0 / U);
	tmp = 0.0;
	if (l <= -1.7e+295)
		tmp = t_2;
	elseif (l <= -3.35e+193)
		tmp = t_1;
	elseif (l <= -2.05e+73)
		tmp = t_2;
	elseif (l <= -850.0)
		tmp = t_1;
	elseif ((l <= 2.8e+32) || ~((l <= 4.8e+132)))
		tmp = U + (2.0 * (l * J));
	else
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(U * -0.125), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[(1.0 / U), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.7e+295], t$95$2, If[LessEqual[l, -3.35e+193], t$95$1, If[LessEqual[l, -2.05e+73], t$95$2, If[LessEqual[l, -850.0], t$95$1, If[Or[LessEqual[l, 2.8e+32], N[Not[LessEqual[l, 4.8e+132]], $MachinePrecision]], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(0.125 + N[(N[(K * K), $MachinePrecision] * -0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.70000000000000001e295 or -3.3500000000000001e193 < l < -2.0499999999999999e73

   1. Initial program 100.0%

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

   3. Applied egg-rr 2.7%

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

      1. flip-- 29.7%

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

      2. div-inv 29.7%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 29.7%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 29.7%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

   5. Applied egg-rr 29.7%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

   6. Taylor expanded in U around inf 41.0%

      \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\frac{1}{U}} \]

   if -1.70000000000000001e295 < l < -3.3500000000000001e193 or -2.0499999999999999e73 < l < -850

   1. Initial program 100.0%

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

   3. Applied egg-rr 3.4%

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

      1. flip-- 36.0%

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

      2. div-inv 36.0%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 36.0%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 36.0%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

   5. Applied egg-rr 36.0%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

   6. Taylor expanded in U around 0 48.7%

      \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(-0.125 \cdot U\right)} \]
   7. Step-by-step derivation

      1. *-commutative 48.7%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(U \cdot -0.125\right)} \]

   8. Simplified 48.7%

      \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(U \cdot -0.125\right)} \]

   if -850 < l < 2.8e32 or 4.8000000000000002e132 < l

   1. Initial program 77.1%

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

   2. Taylor expanded in l around 0 82.9%

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

      1. *-commutative 82.9%

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

      2. associate-*l* 82.9%

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

      3. *-commutative 82.9%

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

   4. Simplified 82.9%

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

   5. Taylor expanded in K around 0 70.6%

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

   if 2.8e32 < l < 4.8000000000000002e132

   1. Initial program 100.0%

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

   3. Applied egg-rr 4.6%

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

   4. Taylor expanded in K around 0 45.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot J + -0.015625 \cdot \left({K}^{2} \cdot J\right)\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 45.0%

         \[\leadsto \left(0.125 \cdot J + \color{blue}{\left(-0.015625 \cdot {K}^{2}\right) \cdot J}\right) + U \]

      2. distribute-rgt-out 45.0%

         \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot {K}^{2}\right)} + U \]

      3. unpow2 45.0%

         \[\leadsto J \cdot \left(0.125 + -0.015625 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   6. Simplified 45.0%

      \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot \left(K \cdot K\right)\right)} + U \]
3. Recombined 4 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.7 \cdot 10^{+295}:\\ \;\;\;\;\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{U}\\ \mathbf{elif}\;\ell \leq -3.35 \cdot 10^{+193}:\\ \;\;\;\;\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \left(U \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq -2.05 \cdot 10^{+73}:\\ \;\;\;\;\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{U}\\ \mathbf{elif}\;\ell \leq -850:\\ \;\;\;\;\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \left(U \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+32} \lor \neg \left(\ell \leq 4.8 \cdot 10^{+132}\right):\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \end{array} \]

Alternative 13: 60.5% accurate, 14.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.65 \cdot 10^{+73}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \mathbf{elif}\;\ell \leq -750:\\ \;\;\;\;\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \left(U \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 270:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* (* l J) (+ 2.0 (* (* K K) -0.25))))))
   (if (<= l -1.4e+129)
     t_0
     (if (<= l -2.65e+73)
       (+ U (* J (+ 0.125 (* (* K K) -0.015625))))
       (if (<= l -750.0)
         (* (- (/ 64.0 (* U U)) (* U U)) (* U -0.125))
         (if (<= l 270.0) (+ U (* 2.0 (* l J))) t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -1.4e+129) {
		tmp = t_0;
	} else if (l <= -2.65e+73) {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	} else if (l <= -750.0) {
		tmp = ((64.0 / (U * U)) - (U * U)) * (U * -0.125);
	} else if (l <= 270.0) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + ((l * j) * (2.0d0 + ((k * k) * (-0.25d0))))
    if (l <= (-1.4d+129)) then
        tmp = t_0
    else if (l <= (-2.65d+73)) then
        tmp = u + (j * (0.125d0 + ((k * k) * (-0.015625d0))))
    else if (l <= (-750.0d0)) then
        tmp = ((64.0d0 / (u * u)) - (u * u)) * (u * (-0.125d0))
    else if (l <= 270.0d0) then
        tmp = u + (2.0d0 * (l * j))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	double tmp;
	if (l <= -1.4e+129) {
		tmp = t_0;
	} else if (l <= -2.65e+73) {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	} else if (l <= -750.0) {
		tmp = ((64.0 / (U * U)) - (U * U)) * (U * -0.125);
	} else if (l <= 270.0) {
		tmp = U + (2.0 * (l * J));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)))
	tmp = 0
	if l <= -1.4e+129:
		tmp = t_0
	elif l <= -2.65e+73:
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)))
	elif l <= -750.0:
		tmp = ((64.0 / (U * U)) - (U * U)) * (U * -0.125)
	elif l <= 270.0:
		tmp = U + (2.0 * (l * J))
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(Float64(l * J) * Float64(2.0 + Float64(Float64(K * K) * -0.25))))
	tmp = 0.0
	if (l <= -1.4e+129)
		tmp = t_0;
	elseif (l <= -2.65e+73)
		tmp = Float64(U + Float64(J * Float64(0.125 + Float64(Float64(K * K) * -0.015625))));
	elseif (l <= -750.0)
		tmp = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) * Float64(U * -0.125));
	elseif (l <= 270.0)
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l * J) * (2.0 + ((K * K) * -0.25)));
	tmp = 0.0;
	if (l <= -1.4e+129)
		tmp = t_0;
	elseif (l <= -2.65e+73)
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	elseif (l <= -750.0)
		tmp = ((64.0 / (U * U)) - (U * U)) * (U * -0.125);
	elseif (l <= 270.0)
		tmp = U + (2.0 * (l * J));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[(l * J), $MachinePrecision] * N[(2.0 + N[(N[(K * K), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.4e+129], t$95$0, If[LessEqual[l, -2.65e+73], N[(U + N[(J * N[(0.125 + N[(N[(K * K), $MachinePrecision] * -0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -750.0], N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] * N[(U * -0.125), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 270.0], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.39999999999999987e129 or 270 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 32.5%

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

      1. *-commutative 32.5%

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

      2. associate-*l* 32.5%

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

      3. *-commutative 32.5%

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

   4. Simplified 32.5%

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

   5. Taylor expanded in K around 0 20.7%

      \[\leadsto \color{blue}{\left(2 \cdot \left(\ell \cdot J\right) + -0.25 \cdot \left({K}^{2} \cdot \left(\ell \cdot J\right)\right)\right)} + U \]
   6. Step-by-step derivation

      1. +-commutative 20.7%

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

      2. associate-*r* 20.7%

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

      3. distribute-rgt-out 40.3%

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

      4. *-commutative 40.3%

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

      5. unpow2 40.3%

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

   7. Simplified 40.3%

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

   if -1.39999999999999987e129 < l < -2.64999999999999998e73

   1. Initial program 100.0%

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

   3. Applied egg-rr 1.3%

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

   4. Taylor expanded in K around 0 36.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot J + -0.015625 \cdot \left({K}^{2} \cdot J\right)\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 36.8%

         \[\leadsto \left(0.125 \cdot J + \color{blue}{\left(-0.015625 \cdot {K}^{2}\right) \cdot J}\right) + U \]

      2. distribute-rgt-out 36.8%

         \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot {K}^{2}\right)} + U \]

      3. unpow2 36.8%

         \[\leadsto J \cdot \left(0.125 + -0.015625 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   6. Simplified 36.8%

      \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot \left(K \cdot K\right)\right)} + U \]

   if -2.64999999999999998e73 < l < -750

   1. Initial program 100.0%

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

   3. Applied egg-rr 4.3%

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

      1. flip-- 46.3%

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

      2. div-inv 46.3%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 46.3%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 46.3%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

   5. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

   6. Taylor expanded in U around 0 48.0%

      \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(-0.125 \cdot U\right)} \]
   7. Step-by-step derivation

      1. *-commutative 48.0%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(U \cdot -0.125\right)} \]

   8. Simplified 48.0%

      \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(U \cdot -0.125\right)} \]

   if -750 < l < 270

   1. Initial program 68.4%

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

   2. Taylor expanded in l around 0 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. *-commutative 99.8%

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

   4. Simplified 99.8%

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

   5. Taylor expanded in K around 0 83.9%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.4 \cdot 10^{+129}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \mathbf{elif}\;\ell \leq -2.65 \cdot 10^{+73}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \mathbf{elif}\;\ell \leq -750:\\ \;\;\;\;\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \left(U \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 270:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot J\right) \cdot \left(2 + \left(K \cdot K\right) \cdot -0.25\right)\\ \end{array} \]

Alternative 14: 52.3% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+134}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-30} \lor \neg \left(\ell \leq 5.8 \cdot 10^{+31}\right) \land \ell \leq 4 \cdot 10^{+132}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -5.7e+134)
   (* U (- U -8.0))
   (if (or (<= l -3.2e-30) (and (not (<= l 5.8e+31)) (<= l 4e+132)))
     (+ U (* J (+ 0.125 (* (* K K) -0.015625))))
     (+ U (* 2.0 (* l J))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.7e+134) {
		tmp = U * (U - -8.0);
	} else if ((l <= -3.2e-30) || (!(l <= 5.8e+31) && (l <= 4e+132))) {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-5.7d+134)) then
        tmp = u * (u - (-8.0d0))
    else if ((l <= (-3.2d-30)) .or. (.not. (l <= 5.8d+31)) .and. (l <= 4d+132)) then
        tmp = u + (j * (0.125d0 + ((k * k) * (-0.015625d0))))
    else
        tmp = u + (2.0d0 * (l * j))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -5.7e+134) {
		tmp = U * (U - -8.0);
	} else if ((l <= -3.2e-30) || (!(l <= 5.8e+31) && (l <= 4e+132))) {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	} else {
		tmp = U + (2.0 * (l * J));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -5.7e+134:
		tmp = U * (U - -8.0)
	elif (l <= -3.2e-30) or (not (l <= 5.8e+31) and (l <= 4e+132)):
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)))
	else:
		tmp = U + (2.0 * (l * J))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -5.7e+134)
		tmp = Float64(U * Float64(U - -8.0));
	elseif ((l <= -3.2e-30) || (!(l <= 5.8e+31) && (l <= 4e+132)))
		tmp = Float64(U + Float64(J * Float64(0.125 + Float64(Float64(K * K) * -0.015625))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(l * J)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -5.7e+134)
		tmp = U * (U - -8.0);
	elseif ((l <= -3.2e-30) || (~((l <= 5.8e+31)) && (l <= 4e+132)))
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	else
		tmp = U + (2.0 * (l * J));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -5.7e+134], N[(U * N[(U - -8.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, -3.2e-30], And[N[Not[LessEqual[l, 5.8e+31]], $MachinePrecision], LessEqual[l, 4e+132]]], N[(U + N[(J * N[(0.125 + N[(N[(K * K), $MachinePrecision] * -0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.70000000000000038e134

   1. Initial program 100.0%

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

   3. Applied egg-rr 24.1%

      \[\leadsto \color{blue}{U \cdot \left(U - -8\right)} \]

   if -5.70000000000000038e134 < l < -3.2e-30 or 5.8000000000000001e31 < l < 3.99999999999999996e132

   1. Initial program 98.7%

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

   3. Applied egg-rr 4.9%

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

   4. Taylor expanded in K around 0 35.4%

      \[\leadsto \color{blue}{\left(0.125 \cdot J + -0.015625 \cdot \left({K}^{2} \cdot J\right)\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 35.4%

         \[\leadsto \left(0.125 \cdot J + \color{blue}{\left(-0.015625 \cdot {K}^{2}\right) \cdot J}\right) + U \]

      2. distribute-rgt-out 35.4%

         \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot {K}^{2}\right)} + U \]

      3. unpow2 35.4%

         \[\leadsto J \cdot \left(0.125 + -0.015625 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   6. Simplified 35.4%

      \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot \left(K \cdot K\right)\right)} + U \]

   if -3.2e-30 < l < 5.8000000000000001e31 or 3.99999999999999996e132 < l

   1. Initial program 77.2%

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

   2. Taylor expanded in l around 0 82.7%

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

      1. *-commutative 82.7%

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

      2. associate-*l* 82.7%

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

      3. *-commutative 82.7%

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

   4. Simplified 82.7%

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

   5. Taylor expanded in K around 0 70.8%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+134}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \mathbf{elif}\;\ell \leq -3.2 \cdot 10^{-30} \lor \neg \left(\ell \leq 5.8 \cdot 10^{+31}\right) \land \ell \leq 4 \cdot 10^{+132}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \end{array} \]

Alternative 15: 57.1% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+293}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1050:\\ \;\;\;\;\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \left(U \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+31} \lor \neg \left(\ell \leq 2.55 \cdot 10^{+132}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 2.0 (* l J)))))
   (if (<= l -3.05e+293)
     t_0
     (if (<= l -1050.0)
       (* (- (/ 64.0 (* U U)) (* U U)) (* U -0.125))
       (if (or (<= l 2.3e+31) (not (<= l 2.55e+132)))
         t_0
         (+ U (* J (+ 0.125 (* (* K K) -0.015625)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (l * J));
	double tmp;
	if (l <= -3.05e+293) {
		tmp = t_0;
	} else if (l <= -1050.0) {
		tmp = ((64.0 / (U * U)) - (U * U)) * (U * -0.125);
	} else if ((l <= 2.3e+31) || !(l <= 2.55e+132)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = u + (2.0d0 * (l * j))
    if (l <= (-3.05d+293)) then
        tmp = t_0
    else if (l <= (-1050.0d0)) then
        tmp = ((64.0d0 / (u * u)) - (u * u)) * (u * (-0.125d0))
    else if ((l <= 2.3d+31) .or. (.not. (l <= 2.55d+132))) then
        tmp = t_0
    else
        tmp = u + (j * (0.125d0 + ((k * k) * (-0.015625d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (2.0 * (l * J));
	double tmp;
	if (l <= -3.05e+293) {
		tmp = t_0;
	} else if (l <= -1050.0) {
		tmp = ((64.0 / (U * U)) - (U * U)) * (U * -0.125);
	} else if ((l <= 2.3e+31) || !(l <= 2.55e+132)) {
		tmp = t_0;
	} else {
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (2.0 * (l * J))
	tmp = 0
	if l <= -3.05e+293:
		tmp = t_0
	elif l <= -1050.0:
		tmp = ((64.0 / (U * U)) - (U * U)) * (U * -0.125)
	elif (l <= 2.3e+31) or not (l <= 2.55e+132):
		tmp = t_0
	else:
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(2.0 * Float64(l * J)))
	tmp = 0.0
	if (l <= -3.05e+293)
		tmp = t_0;
	elseif (l <= -1050.0)
		tmp = Float64(Float64(Float64(64.0 / Float64(U * U)) - Float64(U * U)) * Float64(U * -0.125));
	elseif ((l <= 2.3e+31) || !(l <= 2.55e+132))
		tmp = t_0;
	else
		tmp = Float64(U + Float64(J * Float64(0.125 + Float64(Float64(K * K) * -0.015625))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (2.0 * (l * J));
	tmp = 0.0;
	if (l <= -3.05e+293)
		tmp = t_0;
	elseif (l <= -1050.0)
		tmp = ((64.0 / (U * U)) - (U * U)) * (U * -0.125);
	elseif ((l <= 2.3e+31) || ~((l <= 2.55e+132)))
		tmp = t_0;
	else
		tmp = U + (J * (0.125 + ((K * K) * -0.015625)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(2.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -3.05e+293], t$95$0, If[LessEqual[l, -1050.0], N[(N[(N[(64.0 / N[(U * U), $MachinePrecision]), $MachinePrecision] - N[(U * U), $MachinePrecision]), $MachinePrecision] * N[(U * -0.125), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 2.3e+31], N[Not[LessEqual[l, 2.55e+132]], $MachinePrecision]], t$95$0, N[(U + N[(J * N[(0.125 + N[(N[(K * K), $MachinePrecision] * -0.015625), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.0499999999999998e293 or -1050 < l < 2.3e31 or 2.55e132 < l

   1. Initial program 77.9%

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

   2. Taylor expanded in l around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. associate-*l* 82.4%

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

      3. *-commutative 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in K around 0 70.5%

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

   if -3.0499999999999998e293 < l < -1050

   1. Initial program 100.0%

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

   3. Applied egg-rr 3.1%

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

      1. flip-- 31.1%

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

      2. div-inv 31.1%

         \[\leadsto \color{blue}{\left(\frac{-8}{U} \cdot \frac{-8}{U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

      3. frac-times 31.1%

         \[\leadsto \left(\color{blue}{\frac{-8 \cdot -8}{U \cdot U}} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

      4. metadata-eval 31.1%

         \[\leadsto \left(\frac{\color{blue}{64}}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U} \]

   5. Applied egg-rr 31.1%

      \[\leadsto \color{blue}{\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \frac{1}{\frac{-8}{U} + U}} \]

   6. Taylor expanded in U around 0 33.3%

      \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(-0.125 \cdot U\right)} \]
   7. Step-by-step derivation

      1. *-commutative 33.3%

         \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(U \cdot -0.125\right)} \]

   8. Simplified 33.3%

      \[\leadsto \left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \color{blue}{\left(U \cdot -0.125\right)} \]

   if 2.3e31 < l < 2.55e132

   1. Initial program 100.0%

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

   3. Applied egg-rr 4.6%

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

   4. Taylor expanded in K around 0 45.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot J + -0.015625 \cdot \left({K}^{2} \cdot J\right)\right)} + U \]
   5. Step-by-step derivation

      1. associate-*r* 45.0%

         \[\leadsto \left(0.125 \cdot J + \color{blue}{\left(-0.015625 \cdot {K}^{2}\right) \cdot J}\right) + U \]

      2. distribute-rgt-out 45.0%

         \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot {K}^{2}\right)} + U \]

      3. unpow2 45.0%

         \[\leadsto J \cdot \left(0.125 + -0.015625 \cdot \color{blue}{\left(K \cdot K\right)}\right) + U \]

   6. Simplified 45.0%

      \[\leadsto \color{blue}{J \cdot \left(0.125 + -0.015625 \cdot \left(K \cdot K\right)\right)} + U \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.05 \cdot 10^{+293}:\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{elif}\;\ell \leq -1050:\\ \;\;\;\;\left(\frac{64}{U \cdot U} - U \cdot U\right) \cdot \left(U \cdot -0.125\right)\\ \mathbf{elif}\;\ell \leq 2.3 \cdot 10^{+31} \lor \neg \left(\ell \leq 2.55 \cdot 10^{+132}\right):\\ \;\;\;\;U + 2 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(0.125 + \left(K \cdot K\right) \cdot -0.015625\right)\\ \end{array} \]

Alternative 16: 39.1% accurate, 44.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.28 \cdot 10^{+33}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -1.28e+33) (* U (- U -8.0)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.28e+33) {
		tmp = U * (U - -8.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-1.28d+33)) then
        tmp = u * (u - (-8.0d0))
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.28e+33) {
		tmp = U * (U - -8.0);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.28e+33:
		tmp = U * (U - -8.0)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.28e+33)
		tmp = Float64(U * Float64(U - -8.0));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -1.28e+33)
		tmp = U * (U - -8.0);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.28e+33], N[(U * N[(U - -8.0), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.28e33

   1. Initial program 100.0%

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

   3. Applied egg-rr 20.0%

      \[\leadsto \color{blue}{U \cdot \left(U - -8\right)} \]

   if -1.28e33 < l

   1. Initial program 80.3%

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

   2. Taylor expanded in J around 0 42.6%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.28 \cdot 10^{+33}:\\ \;\;\;\;U \cdot \left(U - -8\right)\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 17: 54.6% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in l around 0 61.4%

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

   1. *-commutative 61.4%

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

   2. associate-*l* 61.4%

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

   3. *-commutative 61.4%

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

4. Simplified 61.4%

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

5. Taylor expanded in K around 0 51.3%

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

6. Final simplification 51.3%

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

Alternative 18: 39.1% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.28 \cdot 10^{+33}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l -1.28e+33) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -1.28e+33) {
		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 <= (-1.28d+33)) 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 <= -1.28e+33) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -1.28e+33:
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -1.28e+33)
		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 <= -1.28e+33)
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -1.28e+33], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.28e33

   1. Initial program 100.0%

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

   3. Applied egg-rr 20.0%

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

   if -1.28e33 < l

   1. Initial program 80.3%

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

   2. Taylor expanded in J around 0 42.6%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.28 \cdot 10^{+33}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 19: 2.4% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 0.0)

C

double code(double J, double l, double K, double U) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return 0.0;
}

Python

def code(J, l, K, U):
	return 0.0

Julia

function code(J, l, K, U)
	return 0.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = 0.0;
end

Wolfram

code[J_, l_, K_, U_] := 0.0

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in K around 0 68.6%

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

4. Applied egg-rr 2.3%

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

   1. +-inverses 2.3%

      \[\leadsto \color{blue}{0} \]

6. Simplified 2.3%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 2.3%

   \[\leadsto 0 \]

Alternative 20: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Applied egg-rr 2.6%

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

   1. *-inverses 2.6%

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

5. Simplified 2.6%

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

6. Final simplification 2.6%

   \[\leadsto 1 \]

Alternative 21: 36.8% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

2. Taylor expanded in J around 0 32.3%

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

3. Final simplification 32.3%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023258 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))