Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.6% → 99.5%

Time: 10.1s

Alternatives: 16

Speedup: 1.5×

• 48.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0\right):\\ \;\;\;\;\left(t_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.0)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+ U (* (* l 2.0) (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.0)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.0)) {
		tmp = ((t_0 * J) * Math.cos((K / 2.0))) + U;
	} else {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.0):
		tmp = ((t_0 * J) * math.cos((K / 2.0))) + U
	else:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.0))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.0)))
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	else
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $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 0.0 < (-.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))) < 0.0

   1. Initial program 76.3%

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

   2. Taylor expanded in l around 0 100.0%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 100.0%

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

      4. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 2: 93.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{3} \cdot \left(t_0 \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+142}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -15500000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-30}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot t_0\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;t_2 + U\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5)))
        (t_1 (+ U (* (pow l 3.0) (* t_0 (* J 0.3333333333333333)))))
        (t_2 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -4.5e+142)
     t_1
     (if (<= l -15500000000.0)
       t_2
       (if (<= l 7e-30)
         (+ U (* (* l 2.0) (* J t_0)))
         (if (<= l 1.1e+64) (+ t_2 U) t_1))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double t_1 = U + (pow(l, 3.0) * (t_0 * (J * 0.3333333333333333)));
	double t_2 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -4.5e+142) {
		tmp = t_1;
	} else if (l <= -15500000000.0) {
		tmp = t_2;
	} else if (l <= 7e-30) {
		tmp = U + ((l * 2.0) * (J * t_0));
	} else if (l <= 1.1e+64) {
		tmp = t_2 + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    t_1 = u + ((l ** 3.0d0) * (t_0 * (j * 0.3333333333333333d0)))
    t_2 = (exp(l) - exp(-l)) * j
    if (l <= (-4.5d+142)) then
        tmp = t_1
    else if (l <= (-15500000000.0d0)) then
        tmp = t_2
    else if (l <= 7d-30) then
        tmp = u + ((l * 2.0d0) * (j * t_0))
    else if (l <= 1.1d+64) then
        tmp = t_2 + u
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K * 0.5));
	double t_1 = U + (Math.pow(l, 3.0) * (t_0 * (J * 0.3333333333333333)));
	double t_2 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -4.5e+142) {
		tmp = t_1;
	} else if (l <= -15500000000.0) {
		tmp = t_2;
	} else if (l <= 7e-30) {
		tmp = U + ((l * 2.0) * (J * t_0));
	} else if (l <= 1.1e+64) {
		tmp = t_2 + U;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	t_1 = U + (math.pow(l, 3.0) * (t_0 * (J * 0.3333333333333333)))
	t_2 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -4.5e+142:
		tmp = t_1
	elif l <= -15500000000.0:
		tmp = t_2
	elif l <= 7e-30:
		tmp = U + ((l * 2.0) * (J * t_0))
	elif l <= 1.1e+64:
		tmp = t_2 + U
	else:
		tmp = t_1
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	t_1 = Float64(U + Float64((l ^ 3.0) * Float64(t_0 * Float64(J * 0.3333333333333333))))
	t_2 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -4.5e+142)
		tmp = t_1;
	elseif (l <= -15500000000.0)
		tmp = t_2;
	elseif (l <= 7e-30)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * t_0)));
	elseif (l <= 1.1e+64)
		tmp = Float64(t_2 + U);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	t_1 = U + ((l ^ 3.0) * (t_0 * (J * 0.3333333333333333)));
	t_2 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -4.5e+142)
		tmp = t_1;
	elseif (l <= -15500000000.0)
		tmp = t_2;
	elseif (l <= 7e-30)
		tmp = U + ((l * 2.0) * (J * t_0));
	elseif (l <= 1.1e+64)
		tmp = t_2 + U;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(t$95$0 * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -4.5e+142], t$95$1, If[LessEqual[l, -15500000000.0], t$95$2, If[LessEqual[l, 7e-30], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.1e+64], N[(t$95$2 + U), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.4999999999999999e142 or 1.10000000000000001e64 < l

   1. Initial program 100.0%

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

   2. Taylor expanded in l around 0 100.0%

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

   3. Taylor expanded in l around inf 100.0%

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

      1. associate-*r* 100.0%

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

      2. *-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*l* 100.0%

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

      7. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   if -4.4999999999999999e142 < l < -1.55e10

   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 85.7%

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

   3. Taylor expanded in J around inf 85.7%

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

   if -1.55e10 < l < 7.0000000000000006e-30

   1. Initial program 75.9%

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

   2. Taylor expanded in l around 0 100.0%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 100.0%

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

      4. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

   if 7.0000000000000006e-30 < l < 1.10000000000000001e64

   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 81.8%

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

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+142}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -15500000000:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-30}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{+64}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 3: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))
   (+ U (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * 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) :: tmp
    if (cos((k / 2.0d0)) <= (-0.1d0)) then
        tmp = u + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
    else
        tmp = u + (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.1) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.1:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	else:
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.1)
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	else
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.10000000000000001

   1. Initial program 81.8%

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

   2. Taylor expanded in l around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*l* 71.0%

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

      4. associate-*r* 71.0%

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

   4. Simplified 71.0%

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

   if -0.10000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.4%

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

   2. Taylor expanded in l around 0 92.0%

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

   3. Taylor expanded in K around 0 89.3%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 4: 88.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

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 * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

2. Taylor expanded in l around 0 91.8%

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

3. Final simplification 91.8%

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

Alternative 5: 68.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + \left(\ell \cdot \left(J \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ U (* (* l (* J (pow K 2.0))) -0.25))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = U + ((l * (J * pow(K, 2.0))) * -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) :: tmp
    if (cos((k / 2.0d0)) <= (-0.1d0)) then
        tmp = u + ((l * (j * (k ** 2.0d0))) * (-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 tmp;
	if (Math.cos((K / 2.0)) <= -0.1) {
		tmp = U + ((l * (J * Math.pow(K, 2.0))) * -0.25);
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.1:
		tmp = U + ((l * (J * math.pow(K, 2.0))) * -0.25)
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(U + Float64(Float64(l * Float64(J * (K ^ 2.0))) * -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)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.1)
		tmp = U + ((l * (J * (K ^ 2.0))) * -0.25);
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(U + N[(N[(l * N[(J * N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K 2)) < -0.10000000000000001

   1. Initial program 81.8%

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

   2. Taylor expanded in l around 0 71.0%

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

      1. associate-*r* 71.0%

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

      2. *-commutative 71.0%

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

      3. associate-*l* 71.0%

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

      4. associate-*r* 71.0%

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

   4. Simplified 71.0%

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

   5. Taylor expanded in K around 0 60.1%

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

   6. Taylor expanded in K around inf 60.1%

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

      1. *-commutative 60.1%

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

      2. associate-*r* 60.1%

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

   8. Simplified 60.1%

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

   if -0.10000000000000001 < (cos.f64 (/.f64 K 2))

   1. Initial program 89.4%

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

   2. Taylor expanded in l around 0 92.0%

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

   3. Taylor expanded in l around inf 81.3%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

      3. *-commutative 81.3%

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

      4. associate-*l* 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-*l* 81.3%

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

      7. *-commutative 81.3%

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

   5. Simplified 81.3%

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

   6. Taylor expanded in K around 0 81.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + \left(\ell \cdot \left(J \cdot {K}^{2}\right)\right) \cdot -0.25\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]

Alternative 6: 86.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{if}\;\ell \leq -15500000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-30}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* (- (exp l) (exp (- l))) J)))
   (if (<= l -15500000000.0)
     t_0
     (if (<= l 7e-30) (+ U (* (* l 2.0) (* J (cos (* K 0.5))))) (+ t_0 U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = (exp(l) - exp(-l)) * J;
	double tmp;
	if (l <= -15500000000.0) {
		tmp = t_0;
	} else if (l <= 7e-30) {
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	} else {
		tmp = t_0 + 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) :: t_0
    real(8) :: tmp
    t_0 = (exp(l) - exp(-l)) * j
    if (l <= (-15500000000.0d0)) then
        tmp = t_0
    else if (l <= 7d-30) then
        tmp = u + ((l * 2.0d0) * (j * cos((k * 0.5d0))))
    else
        tmp = t_0 + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = (Math.exp(l) - Math.exp(-l)) * J;
	double tmp;
	if (l <= -15500000000.0) {
		tmp = t_0;
	} else if (l <= 7e-30) {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	} else {
		tmp = t_0 + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = (math.exp(l) - math.exp(-l)) * J
	tmp = 0
	if l <= -15500000000.0:
		tmp = t_0
	elif l <= 7e-30:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	else:
		tmp = t_0 + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(Float64(exp(l) - exp(Float64(-l))) * J)
	tmp = 0.0
	if (l <= -15500000000.0)
		tmp = t_0;
	elseif (l <= 7e-30)
		tmp = Float64(U + Float64(Float64(l * 2.0) * Float64(J * cos(Float64(K * 0.5)))));
	else
		tmp = Float64(t_0 + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = (exp(l) - exp(-l)) * J;
	tmp = 0.0;
	if (l <= -15500000000.0)
		tmp = t_0;
	elseif (l <= 7e-30)
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	else
		tmp = t_0 + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision]}, If[LessEqual[l, -15500000000.0], t$95$0, If[LessEqual[l, 7e-30], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + U), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.55e10

   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 80.4%

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

   3. Taylor expanded in J around inf 80.4%

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

   if -1.55e10 < l < 7.0000000000000006e-30

   1. Initial program 75.9%

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

   2. Taylor expanded in l around 0 100.0%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 100.0%

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

      4. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

   if 7.0000000000000006e-30 < 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 78.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -15500000000:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-30}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]

Alternative 7: 87.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -15500000000 \lor \neg \left(\ell \leq 310\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -15500000000.0) (not (<= l 310.0)))
   (* (- (exp l) (exp (- l))) J)
   (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -15500000000.0) || !(l <= 310.0)) {
		tmp = (exp(l) - exp(-l)) * J;
	} else {
		tmp = U + ((l * 2.0) * (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 <= (-15500000000.0d0)) .or. (.not. (l <= 310.0d0))) then
        tmp = (exp(l) - exp(-l)) * j
    else
        tmp = u + ((l * 2.0d0) * (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 <= -15500000000.0) || !(l <= 310.0)) {
		tmp = (Math.exp(l) - Math.exp(-l)) * J;
	} else {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -15500000000.0) or not (l <= 310.0):
		tmp = (math.exp(l) - math.exp(-l)) * J
	else:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -15500000000.0) || !(l <= 310.0))
		tmp = Float64(Float64(exp(l) - exp(Float64(-l))) * J);
	else
		tmp = Float64(U + Float64(Float64(l * 2.0) * 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 <= -15500000000.0) || ~((l <= 310.0)))
		tmp = (exp(l) - exp(-l)) * J;
	else
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -15500000000.0], N[Not[LessEqual[l, 310.0]], $MachinePrecision]], N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.55e10 or 310 < 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 78.7%

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

   3. Taylor expanded in J around inf 78.7%

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

   if -1.55e10 < l < 310

   1. Initial program 76.8%

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

   2. Taylor expanded in l around 0 100.0%

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

      1. associate-*r* 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 100.0%

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

      4. associate-*r* 100.0%

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

   4. Simplified 100.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -15500000000 \lor \neg \left(\ell \leq 310\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 8: 79.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -21000000000000 \lor \neg \left(\ell \leq 5.7 \cdot 10^{+16}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -21000000000000.0) (not (<= l 5.7e+16)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -21000000000000.0) || !(l <= 5.7e+16)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-21000000000000.0d0)) .or. (.not. (l <= 5.7d+16))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -21000000000000.0) || !(l <= 5.7e+16)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -21000000000000.0) or not (l <= 5.7e+16):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -21000000000000.0) || !(l <= 5.7e+16))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -21000000000000.0) || ~((l <= 5.7e+16)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -21000000000000.0], N[Not[LessEqual[l, 5.7e+16]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.1e13 or 5.7e16 < 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 84.8%

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

   3. Taylor expanded in l around inf 84.8%

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

      1. associate-*r* 84.8%

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

      2. *-commutative 84.8%

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

      3. *-commutative 84.8%

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

      4. associate-*l* 84.8%

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

      5. *-commutative 84.8%

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

      6. associate-*l* 84.8%

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

      7. *-commutative 84.8%

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

   5. Simplified 84.8%

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

   6. Taylor expanded in K around 0 67.0%

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

   if -2.1e13 < l < 5.7e16

   1. Initial program 77.3%

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

   2. Taylor expanded in l around 0 97.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -21000000000000 \lor \neg \left(\ell \leq 5.7 \cdot 10^{+16}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 9: 79.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -270000000000 \lor \neg \left(\ell \leq 5.7 \cdot 10^{+16}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -270000000000.0) (not (<= l 5.7e+16)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* (* l 2.0) (* J (cos (* K 0.5)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -270000000000.0) || !(l <= 5.7e+16)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + ((l * 2.0) * (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 <= (-270000000000.0d0)) .or. (.not. (l <= 5.7d+16))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + ((l * 2.0d0) * (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 <= -270000000000.0) || !(l <= 5.7e+16)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + ((l * 2.0) * (J * Math.cos((K * 0.5))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -270000000000.0) or not (l <= 5.7e+16):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + ((l * 2.0) * (J * math.cos((K * 0.5))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -270000000000.0) || !(l <= 5.7e+16))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(Float64(l * 2.0) * 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 <= -270000000000.0) || ~((l <= 5.7e+16)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + ((l * 2.0) * (J * cos((K * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -270000000000.0], N[Not[LessEqual[l, 5.7e+16]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(l * 2.0), $MachinePrecision] * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.7e11 or 5.7e16 < 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 84.8%

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

   3. Taylor expanded in l around inf 84.8%

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

      1. associate-*r* 84.8%

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

      2. *-commutative 84.8%

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

      3. *-commutative 84.8%

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

      4. associate-*l* 84.8%

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

      5. *-commutative 84.8%

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

      6. associate-*l* 84.8%

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

      7. *-commutative 84.8%

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

   5. Simplified 84.8%

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

   6. Taylor expanded in K around 0 67.0%

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

   if -2.7e11 < l < 5.7e16

   1. Initial program 77.3%

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

   2. Taylor expanded in l around 0 97.9%

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

      1. associate-*r* 97.9%

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

      2. *-commutative 97.9%

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

      3. associate-*l* 97.9%

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

      4. associate-*r* 97.9%

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

   4. Simplified 97.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -270000000000 \lor \neg \left(\ell \leq 5.7 \cdot 10^{+16}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(\ell \cdot 2\right) \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 10: 72.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+22} \lor \neg \left(\ell \leq 3 \cdot 10^{-46}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.15e+22) (not (<= l 3e-46)))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.15e+22) || !(l <= 3e-46)) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.15d+22)) .or. (.not. (l <= 3d-46))) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.15e+22) || !(l <= 3e-46)) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.15e+22) or not (l <= 3e-46):
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.15e+22) || !(l <= 3e-46))
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.15e+22) || ~((l <= 3e-46)))
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.15e+22], N[Not[LessEqual[l, 3e-46]], $MachinePrecision]], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.1500000000000001e22 or 2.99999999999999987e-46 < l

   1. Initial program 99.3%

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

   2. Taylor expanded in l around 0 84.8%

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

   3. Taylor expanded in l around inf 84.1%

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

      1. associate-*r* 84.1%

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

      2. *-commutative 84.1%

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

      3. *-commutative 84.1%

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

      4. associate-*l* 84.1%

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

      5. *-commutative 84.1%

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

      6. associate-*l* 84.1%

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

      7. *-commutative 84.1%

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

   5. Simplified 84.1%

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

   6. Taylor expanded in K around 0 68.0%

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

   if -1.1500000000000001e22 < l < 2.99999999999999987e-46

   1. Initial program 75.7%

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

   2. Taylor expanded in l around 0 99.2%

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

      1. associate-*r* 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*l* 99.2%

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

      4. associate-*r* 99.2%

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

   4. Simplified 99.2%

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

   5. Taylor expanded in K around 0 86.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+22} \lor \neg \left(\ell \leq 3 \cdot 10^{-46}\right):\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]

Alternative 11: 47.1% accurate, 27.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot \left(J \cdot 2\right)\\ \mathbf{if}\;\ell \leq -1.12 \cdot 10^{+195}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -16000000000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+16}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (* l (* J 2.0))))
   (if (<= l -1.12e+195)
     t_0
     (if (<= l -16000000000.0) (* U U) (if (<= l 5.7e+16) U t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = l * (J * 2.0);
	double tmp;
	if (l <= -1.12e+195) {
		tmp = t_0;
	} else if (l <= -16000000000.0) {
		tmp = U * U;
	} else if (l <= 5.7e+16) {
		tmp = U;
	} 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 = l * (j * 2.0d0)
    if (l <= (-1.12d+195)) then
        tmp = t_0
    else if (l <= (-16000000000.0d0)) then
        tmp = u * u
    else if (l <= 5.7d+16) then
        tmp = u
    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 = l * (J * 2.0);
	double tmp;
	if (l <= -1.12e+195) {
		tmp = t_0;
	} else if (l <= -16000000000.0) {
		tmp = U * U;
	} else if (l <= 5.7e+16) {
		tmp = U;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = l * (J * 2.0)
	tmp = 0
	if l <= -1.12e+195:
		tmp = t_0
	elif l <= -16000000000.0:
		tmp = U * U
	elif l <= 5.7e+16:
		tmp = U
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(l * Float64(J * 2.0))
	tmp = 0.0
	if (l <= -1.12e+195)
		tmp = t_0;
	elseif (l <= -16000000000.0)
		tmp = Float64(U * U);
	elseif (l <= 5.7e+16)
		tmp = U;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = l * (J * 2.0);
	tmp = 0.0;
	if (l <= -1.12e+195)
		tmp = t_0;
	elseif (l <= -16000000000.0)
		tmp = U * U;
	elseif (l <= 5.7e+16)
		tmp = U;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.12e+195], t$95$0, If[LessEqual[l, -16000000000.0], N[(U * U), $MachinePrecision], If[LessEqual[l, 5.7e+16], U, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -1.12000000000000004e195 or 5.7e16 < 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 34.9%

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

   3. Taylor expanded in J around inf 34.8%

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

   4. Taylor expanded in K around 0 25.3%

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

      1. associate-*r* 25.3%

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

      2. *-commutative 25.3%

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

      3. *-commutative 25.3%

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

   6. Simplified 25.3%

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

   if -1.12000000000000004e195 < l < -1.6e10

   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 73.3%

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

   4. Applied egg-rr 21.1%

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

   if -1.6e10 < l < 5.7e16

   1. Initial program 77.3%

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

   3. Applied egg-rr 54.0%

      \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in J around 0 74.3%

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

4. Final simplification 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.12 \cdot 10^{+195}:\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \mathbf{elif}\;\ell \leq -16000000000:\\ \;\;\;\;U \cdot U\\ \mathbf{elif}\;\ell \leq 5.7 \cdot 10^{+16}:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(J \cdot 2\right)\\ \end{array} \]

Alternative 12: 43.6% accurate, 43.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -6.7 \cdot 10^{+19} \lor \neg \left(\ell \leq 680\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -6.7e+19) (not (<= l 680.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.7e+19) || !(l <= 680.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-6.7d+19)) .or. (.not. (l <= 680.0d0))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -6.7e+19) || !(l <= 680.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -6.7e+19) or not (l <= 680.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -6.7e+19) || !(l <= 680.0))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -6.7e+19) || ~((l <= 680.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -6.7e+19], N[Not[LessEqual[l, 680.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -6.7e19 or 680 < 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 78.7%

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

   4. Applied egg-rr 16.2%

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

   if -6.7e19 < l < 680

   1. Initial program 76.8%

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

   3. Applied egg-rr 55.1%

      \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   4. Taylor expanded in J around 0 75.8%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -6.7 \cdot 10^{+19} \lor \neg \left(\ell \leq 680\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]

Alternative 13: 55.1% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.9%

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

2. Taylor expanded in l around 0 66.1%

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

   1. associate-*r* 66.1%

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

   2. *-commutative 66.1%

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

   3. associate-*l* 66.1%

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

   4. associate-*r* 66.1%

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

4. Simplified 66.1%

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

5. Taylor expanded in K around 0 55.7%

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

6. Final simplification 55.7%

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

Alternative 14: 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 87.9%

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

2. Taylor expanded in K around 0 77.3%

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

4. Applied egg-rr 2.5%

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

   1. +-inverses 2.5%

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

6. Simplified 2.5%

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

7. Final simplification 2.5%

   \[\leadsto 0 \]

Alternative 15: 2.8% 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 87.9%

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

2. Taylor expanded in K around 0 77.3%

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

4. Applied egg-rr 2.7%

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

   1. *-inverses 2.7%

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

6. Simplified 2.7%

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

7. Final simplification 2.7%

   \[\leadsto 1 \]

Alternative 16: 38.0% 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 87.9%

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

3. Applied egg-rr 30.1%

   \[\leadsto \left(J \cdot \color{blue}{-4}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

4. Taylor expanded in J around 0 41.0%

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

5. Final simplification 41.0%

   \[\leadsto U \]

Reproduce

herbie shell --seed 2023333 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))