Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.8% → 99.5%

Time: 11.2s

Alternatives: 12

Speedup: 1.4×

• 49.7% 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: 86.8% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -\infty \lor \neg \left(t\_0 \leq 0\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

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 (* (* J (cos (* K 0.5))) (* l 2.0))))))

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 + ((J * cos((K * 0.5))) * (l * 2.0));
	}
	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 + ((J * Math.cos((K * 0.5))) * (l * 2.0));
	}
	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 + ((J * math.cos((K * 0.5))) * (l * 2.0))
	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(J * cos(Float64(K * 0.5))) * Float64(l * 2.0)));
	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 + ((J * cos((K * 0.5))) * (l * 2.0));
	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[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $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%

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

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. associate-*r* 99.9%

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

      5. *-commutative 99.9%

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

      6. *-commutative 99.9%

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

      7. distribute-lft-out 99.9%

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

      8. fma-define 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. associate-*l* 99.9%

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

      5. *-commutative 99.9%

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

   8. Simplified 99.9%

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

4. Final simplification 100.0%

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

Alternative 2: 95.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ t_1 := U + {\ell}^{3} \cdot \left(t\_0 \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ t_2 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -8000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-22}:\\ \;\;\;\;U + \left(J \cdot t\_0\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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) U)))
   (if (<= l -2.1e+102)
     t_1
     (if (<= l -8000.0)
       t_2
       (if (<= l 1.3e-22)
         (+ U (* (* J t_0) (* l 2.0)))
         (if (<= l 5.6e+102) t_2 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) + U;
	double tmp;
	if (l <= -2.1e+102) {
		tmp = t_1;
	} else if (l <= -8000.0) {
		tmp = t_2;
	} else if (l <= 1.3e-22) {
		tmp = U + ((J * t_0) * (l * 2.0));
	} else if (l <= 5.6e+102) {
		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 = cos((k * 0.5d0))
    t_1 = u + ((l ** 3.0d0) * (t_0 * (j * 0.3333333333333333d0)))
    t_2 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-2.1d+102)) then
        tmp = t_1
    else if (l <= (-8000.0d0)) then
        tmp = t_2
    else if (l <= 1.3d-22) then
        tmp = u + ((j * t_0) * (l * 2.0d0))
    else if (l <= 5.6d+102) 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 = 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) + U;
	double tmp;
	if (l <= -2.1e+102) {
		tmp = t_1;
	} else if (l <= -8000.0) {
		tmp = t_2;
	} else if (l <= 1.3e-22) {
		tmp = U + ((J * t_0) * (l * 2.0));
	} else if (l <= 5.6e+102) {
		tmp = t_2;
	} 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) + U
	tmp = 0
	if l <= -2.1e+102:
		tmp = t_1
	elif l <= -8000.0:
		tmp = t_2
	elif l <= 1.3e-22:
		tmp = U + ((J * t_0) * (l * 2.0))
	elif l <= 5.6e+102:
		tmp = t_2
	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(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -2.1e+102)
		tmp = t_1;
	elseif (l <= -8000.0)
		tmp = t_2;
	elseif (l <= 1.3e-22)
		tmp = Float64(U + Float64(Float64(J * t_0) * Float64(l * 2.0)));
	elseif (l <= 5.6e+102)
		tmp = t_2;
	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) + U;
	tmp = 0.0;
	if (l <= -2.1e+102)
		tmp = t_1;
	elseif (l <= -8000.0)
		tmp = t_2;
	elseif (l <= 1.3e-22)
		tmp = U + ((J * t_0) * (l * 2.0));
	elseif (l <= 5.6e+102)
		tmp = t_2;
	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[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -2.1e+102], t$95$1, If[LessEqual[l, -8000.0], t$95$2, If[LessEqual[l, 1.3e-22], N[(U + N[(N[(J * t$95$0), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.6e+102], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.10000000000000001e102 or 5.60000000000000037e102 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

   4. Taylor expanded in l around inf 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*r* 100.0%

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

      6. *-commutative 100.0%

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

   6. 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 -2.10000000000000001e102 < l < -8e3 or 1.3e-22 < l < 5.60000000000000037e102

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 77.8%

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

   if -8e3 < l < 1.3e-22

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*r* 99.2%

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

      4. associate-*r* 99.2%

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

      5. *-commutative 99.2%

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

      6. *-commutative 99.2%

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

      7. distribute-lft-out 99.2%

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

      8. fma-define 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*l* 99.2%

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

      4. associate-*l* 99.2%

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

      5. *-commutative 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.1 \cdot 10^{+102}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \mathbf{elif}\;\ell \leq -8000:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 1.3 \cdot 10^{-22}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.5)
   (+ U (* (* J (cos (* K 0.5))) (* l 2.0)))
   (* U (+ 1.0 (/ (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))) U)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.5) {
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))) / 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) :: tmp
    if (cos((k / 2.0d0)) <= 0.5d0) then
        tmp = u + ((j * cos((k * 0.5d0))) * (l * 2.0d0))
    else
        tmp = u * (1.0d0 + ((j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))) / u))
    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)) <= 0.5) {
		tmp = U + ((J * Math.cos((K * 0.5))) * (l * 2.0));
	} else {
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))) / U));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.5:
		tmp = U + ((J * math.cos((K * 0.5))) * (l * 2.0))
	else:
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))) / U))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.5)
		tmp = Float64(U + Float64(Float64(J * cos(Float64(K * 0.5))) * Float64(l * 2.0)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0))))) / U)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.5)
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	else
		tmp = U * (1.0 + ((J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))) / U));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.5], N[(U + N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.5

   1. Initial program 90.6%

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

   3. Taylor expanded in l around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 81.6%

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

      4. associate-*r* 81.6%

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

      5. *-commutative 81.6%

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

      6. *-commutative 81.6%

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

      7. distribute-lft-out 81.6%

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

      8. fma-define 81.6%

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

   5. Simplified 81.6%

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

   6. Taylor expanded in l around 0 58.3%

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

      1. *-commutative 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-*l* 58.3%

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

      4. associate-*l* 58.3%

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

      5. *-commutative 58.3%

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

   8. Simplified 58.3%

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

   if 0.5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.0%

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

   3. Taylor expanded in l around 0 88.1%

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

   4. Taylor expanded in K around 0 85.6%

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

   5. Taylor expanded in U around inf 88.7%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.5:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + \frac{J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}{U}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.5) {
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0)))));
	}
	return tmp;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.5

   1. Initial program 90.6%

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

   3. Taylor expanded in l around 0 81.6%

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

      1. *-commutative 81.6%

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

      2. *-commutative 81.6%

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

      3. associate-*r* 81.6%

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

      4. associate-*r* 81.6%

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

      5. *-commutative 81.6%

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

      6. *-commutative 81.6%

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

      7. distribute-lft-out 81.6%

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

      8. fma-define 81.6%

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

   5. Simplified 81.6%

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

   6. Taylor expanded in l around 0 58.3%

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

      1. *-commutative 58.3%

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

      2. *-commutative 58.3%

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

      3. associate-*l* 58.3%

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

      4. associate-*l* 58.3%

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

      5. *-commutative 58.3%

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

   8. Simplified 58.3%

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

   if 0.5 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.0%

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

   3. Taylor expanded in l around 0 88.1%

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

   4. Taylor expanded in K around 0 85.6%

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

4. Final simplification 77.3%

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

Alternative 5: 86.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -8000 \lor \neg \left(\ell \leq 1.3 \cdot 10^{-22}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -8000.0) (not (<= l 1.3e-22)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* (* J (cos (* K 0.5))) (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -8000.0) || !(l <= 1.3e-22)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + ((J * cos((K * 0.5))) * (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 <= (-8000.0d0)) .or. (.not. (l <= 1.3d-22))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + ((j * cos((k * 0.5d0))) * (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 <= -8000.0) || !(l <= 1.3e-22)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + ((J * Math.cos((K * 0.5))) * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -8000.0) or not (l <= 1.3e-22):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + ((J * math.cos((K * 0.5))) * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -8000.0) || !(l <= 1.3e-22))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(Float64(J * cos(Float64(K * 0.5))) * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -8000.0) || ~((l <= 1.3e-22)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + ((J * cos((K * 0.5))) * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -8000.0], N[Not[LessEqual[l, 1.3e-22]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -8e3 or 1.3e-22 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in K around 0 69.9%

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

   if -8e3 < l < 1.3e-22

   1. Initial program 75.3%

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

   3. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*r* 99.2%

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

      4. associate-*r* 99.2%

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

      5. *-commutative 99.2%

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

      6. *-commutative 99.2%

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

      7. distribute-lft-out 99.2%

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

      8. fma-define 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*l* 99.2%

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

      4. associate-*l* 99.2%

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

      5. *-commutative 99.2%

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

   8. Simplified 99.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -8000 \lor \neg \left(\ell \leq 1.3 \cdot 10^{-22}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq 1050:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))
   (if (<= l -2.5e+39)
     t_0
     (if (<= l 1050.0)
       (+ U (* J (* l 2.0)))
       (if (<= l 2.8e+92) (pow U -3.0) t_0)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	double tmp;
	if (l <= -2.5e+39) {
		tmp = t_0;
	} else if (l <= 1050.0) {
		tmp = U + (J * (l * 2.0));
	} else if (l <= 2.8e+92) {
		tmp = pow(U, -3.0);
	} 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 + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    if (l <= (-2.5d+39)) then
        tmp = t_0
    else if (l <= 1050.0d0) then
        tmp = u + (j * (l * 2.0d0))
    else if (l <= 2.8d+92) then
        tmp = u ** (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	double tmp;
	if (l <= -2.5e+39) {
		tmp = t_0;
	} else if (l <= 1050.0) {
		tmp = U + (J * (l * 2.0));
	} else if (l <= 2.8e+92) {
		tmp = Math.pow(U, -3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	tmp = 0
	if l <= -2.5e+39:
		tmp = t_0
	elif l <= 1050.0:
		tmp = U + (J * (l * 2.0))
	elif l <= 2.8e+92:
		tmp = math.pow(U, -3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))))
	tmp = 0.0
	if (l <= -2.5e+39)
		tmp = t_0;
	elseif (l <= 1050.0)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	elseif (l <= 2.8e+92)
		tmp = U ^ -3.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	tmp = 0.0;
	if (l <= -2.5e+39)
		tmp = t_0;
	elseif (l <= 1050.0)
		tmp = U + (J * (l * 2.0));
	elseif (l <= 2.8e+92)
		tmp = U ^ -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.5e+39], t$95$0, If[LessEqual[l, 1050.0], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 2.8e+92], N[Power[U, -3.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.50000000000000008e39 or 2.80000000000000001e92 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 89.9%

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

   4. Taylor expanded in l around inf 89.9%

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

      1. *-commutative 89.9%

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

      2. associate-*r* 89.9%

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

      3. associate-*l* 89.9%

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

      4. *-commutative 89.9%

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

      5. associate-*r* 89.9%

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

      6. *-commutative 89.9%

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

   6. Simplified 89.9%

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

   7. Taylor expanded in K around 0 59.8%

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

   if -2.50000000000000008e39 < l < 1050

   1. Initial program 77.0%

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

   3. Taylor expanded in l around 0 94.8%

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

   4. Taylor expanded in K around 0 86.2%

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

   5. Taylor expanded in l around 0 86.2%

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

      1. *-commutative 86.2%

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

      2. associate-*r* 86.2%

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

      3. *-commutative 86.2%

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

   7. Simplified 86.2%

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

   if 1050 < l < 2.80000000000000001e92

   1. Initial program 100.0%

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

   4. Applied egg-rr 43.6%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+39}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{elif}\;\ell \leq 1050:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{elif}\;\ell \leq 2.8 \cdot 10^{+92}:\\ \;\;\;\;{U}^{-3}\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;J \leq -3.5 \cdot 10^{+123} \lor \neg \left(J \leq 1.05 \cdot 10^{+61}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= J -3.5e+123) (not (<= J 1.05e+61)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))
   (+ U (* 0.3333333333333333 (* J (pow l 3.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((J <= -3.5e+123) || !(J <= 1.05e+61)) {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	} 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 ((j <= (-3.5d+123)) .or. (.not. (j <= 1.05d+61))) then
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    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 ((J <= -3.5e+123) || !(J <= 1.05e+61)) {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (J <= -3.5e+123) or not (J <= 1.05e+61):
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	else:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((J <= -3.5e+123) || !(J <= 1.05e+61))
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	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 ((J <= -3.5e+123) || ~((J <= 1.05e+61)))
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	else
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[J, -3.5e+123], N[Not[LessEqual[J, 1.05e+61]], $MachinePrecision]], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $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 2 regimes

2. if J < -3.5e123 or 1.0500000000000001e61 < J

   1. Initial program 71.9%

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

   3. Taylor expanded in l around 0 88.2%

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

   if -3.5e123 < J < 1.0500000000000001e61

   1. Initial program 96.6%

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

   3. Taylor expanded in l around 0 83.3%

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

   4. Taylor expanded in l around inf 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*r* 80.0%

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

      3. associate-*l* 80.0%

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

      4. *-commutative 80.0%

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

      5. associate-*r* 80.0%

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

      6. *-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in K around 0 68.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.5 \cdot 10^{+123} \lor \neg \left(J \leq 1.05 \cdot 10^{+61}\right):\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot 0.5\right)\\ \mathbf{if}\;J \leq -3.4 \cdot 10^{+123}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot t\_0\right)\right)\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot t\_0\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (* K 0.5))))
   (if (<= J -3.4e+123)
     (+ U (* 2.0 (* J (* l t_0))))
     (if (<= J 4.2e+59)
       (+ U (* 0.3333333333333333 (* J (pow l 3.0))))
       (+ U (* (* J t_0) (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K * 0.5));
	double tmp;
	if (J <= -3.4e+123) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (J <= 4.2e+59) {
		tmp = U + (0.3333333333333333 * (J * pow(l, 3.0)));
	} else {
		tmp = U + ((J * t_0) * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k * 0.5d0))
    if (j <= (-3.4d+123)) then
        tmp = u + (2.0d0 * (j * (l * t_0)))
    else if (j <= 4.2d+59) then
        tmp = u + (0.3333333333333333d0 * (j * (l ** 3.0d0)))
    else
        tmp = u + ((j * t_0) * (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 * 0.5));
	double tmp;
	if (J <= -3.4e+123) {
		tmp = U + (2.0 * (J * (l * t_0)));
	} else if (J <= 4.2e+59) {
		tmp = U + (0.3333333333333333 * (J * Math.pow(l, 3.0)));
	} else {
		tmp = U + ((J * t_0) * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K * 0.5))
	tmp = 0
	if J <= -3.4e+123:
		tmp = U + (2.0 * (J * (l * t_0)))
	elif J <= 4.2e+59:
		tmp = U + (0.3333333333333333 * (J * math.pow(l, 3.0)))
	else:
		tmp = U + ((J * t_0) * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K * 0.5))
	tmp = 0.0
	if (J <= -3.4e+123)
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * t_0))));
	elseif (J <= 4.2e+59)
		tmp = Float64(U + Float64(0.3333333333333333 * Float64(J * (l ^ 3.0))));
	else
		tmp = Float64(U + Float64(Float64(J * t_0) * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K * 0.5));
	tmp = 0.0;
	if (J <= -3.4e+123)
		tmp = U + (2.0 * (J * (l * t_0)));
	elseif (J <= 4.2e+59)
		tmp = U + (0.3333333333333333 * (J * (l ^ 3.0)));
	else
		tmp = U + ((J * t_0) * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[J, -3.4e+123], N[(U + N[(2.0 * N[(J * N[(l * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[J, 4.2e+59], N[(U + N[(0.3333333333333333 * N[(J * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * t$95$0), $MachinePrecision] * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if J < -3.40000000000000001e123

   1. Initial program 68.1%

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

   3. Taylor expanded in l around 0 93.4%

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

   if -3.40000000000000001e123 < J < 4.19999999999999968e59

   1. Initial program 96.6%

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

   3. Taylor expanded in l around 0 83.3%

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

   4. Taylor expanded in l around inf 80.0%

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

      1. *-commutative 80.0%

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

      2. associate-*r* 80.0%

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

      3. associate-*l* 80.0%

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

      4. *-commutative 80.0%

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

      5. associate-*r* 80.0%

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

      6. *-commutative 80.0%

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

   6. Simplified 80.0%

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

   7. Taylor expanded in K around 0 68.7%

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

   if 4.19999999999999968e59 < J

   1. Initial program 75.2%

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

   3. Taylor expanded in l around 0 94.0%

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

      1. *-commutative 94.0%

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

      2. *-commutative 94.0%

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

      3. associate-*r* 94.0%

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

      4. associate-*r* 94.0%

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

      5. *-commutative 94.0%

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

      6. *-commutative 94.0%

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

      7. distribute-lft-out 94.0%

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

      8. fma-define 94.0%

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

   5. Simplified 94.0%

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

   6. Taylor expanded in l around 0 83.8%

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

      1. *-commutative 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*l* 83.8%

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

      4. associate-*l* 83.8%

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

      5. *-commutative 83.8%

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

   8. Simplified 83.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;J \leq -3.4 \cdot 10^{+123}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{elif}\;J \leq 4.2 \cdot 10^{+59}:\\ \;\;\;\;U + 0.3333333333333333 \cdot \left(J \cdot {\ell}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \left(\ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.7% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+41} \lor \neg \left(\ell \leq 12000000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.35e+41) (not (<= l 12000000000.0))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.35e+41) || !(l <= 12000000000.0)) {
		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 <= (-1.35d+41)) .or. (.not. (l <= 12000000000.0d0))) 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 <= -1.35e+41) || !(l <= 12000000000.0)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.35e+41) or not (l <= 12000000000.0):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.35e+41) || !(l <= 12000000000.0))
		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 <= -1.35e+41) || ~((l <= 12000000000.0)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.35e+41], N[Not[LessEqual[l, 12000000000.0]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -1.35e41 or 1.2e10 < l

   1. Initial program 100.0%

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

   4. Applied egg-rr 17.9%

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

   if -1.35e41 < l < 1.2e10

   1. Initial program 77.5%

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

   3. Taylor expanded in J around 0 70.3%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.35 \cdot 10^{+41} \lor \neg \left(\ell \leq 12000000000\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.4% 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 88.1%

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

3. Taylor expanded in l around 0 88.0%

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

4. Taylor expanded in K around 0 69.8%

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

5. Taylor expanded in l around 0 54.9%

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

   1. *-commutative 54.9%

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

   2. associate-*r* 54.9%

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

   3. *-commutative 54.9%

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

7. Simplified 54.9%

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

8. Final simplification 54.9%

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

Alternative 11: 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 88.1%

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

4. Applied egg-rr 2.7%

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

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 12: 37.2% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.1%

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

3. Taylor expanded in J around 0 38.1%

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

4. Final simplification 38.1%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024073 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))