Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 87.2% → 99.6%

Time: 10.6s

Alternatives: 20

Speedup: 1.4×

• 50.2% 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: 87.2% 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: 99.6% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 1e-7)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* l (* J 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 1e-7)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * (J * 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 1e-7)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 1e-7):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 1e-7))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 1e-7)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (l * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 1e-7]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(l * N[(J * 2.0), $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 9.9999999999999995e-8 < (-.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))) < 9.9999999999999995e-8

   1. Initial program 71.2%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 94.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := U + {\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ t_1 := \left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+75}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\ell \leq -0.118:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-6}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0
         (+ U (* (pow l 3.0) (* (* J 0.3333333333333333) (cos (* K 0.5))))))
        (t_1 (+ (* (- (exp l) (exp (- l))) J) U)))
   (if (<= l -5.6e+75)
     t_0
     (if (<= l -0.118)
       t_1
       (if (<= l 7.8e-6)
         (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))
         (if (<= l 5e+102) t_1 t_0))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = U + (pow(l, 3.0) * ((J * 0.3333333333333333) * cos((K * 0.5))));
	double t_1 = ((exp(l) - exp(-l)) * J) + U;
	double tmp;
	if (l <= -5.6e+75) {
		tmp = t_0;
	} else if (l <= -0.118) {
		tmp = t_1;
	} else if (l <= 7.8e-6) {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 5e+102) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = u + ((l ** 3.0d0) * ((j * 0.3333333333333333d0) * cos((k * 0.5d0))))
    t_1 = ((exp(l) - exp(-l)) * j) + u
    if (l <= (-5.6d+75)) then
        tmp = t_0
    else if (l <= (-0.118d0)) then
        tmp = t_1
    else if (l <= 7.8d-6) then
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    else if (l <= 5d+102) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = U + (Math.pow(l, 3.0) * ((J * 0.3333333333333333) * Math.cos((K * 0.5))));
	double t_1 = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	double tmp;
	if (l <= -5.6e+75) {
		tmp = t_0;
	} else if (l <= -0.118) {
		tmp = t_1;
	} else if (l <= 7.8e-6) {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	} else if (l <= 5e+102) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = U + (math.pow(l, 3.0) * ((J * 0.3333333333333333) * math.cos((K * 0.5))))
	t_1 = ((math.exp(l) - math.exp(-l)) * J) + U
	tmp = 0
	if l <= -5.6e+75:
		tmp = t_0
	elif l <= -0.118:
		tmp = t_1
	elif l <= 7.8e-6:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	elif l <= 5e+102:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(J, l, K, U)
	t_0 = Float64(U + Float64((l ^ 3.0) * Float64(Float64(J * 0.3333333333333333) * cos(Float64(K * 0.5)))))
	t_1 = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U)
	tmp = 0.0
	if (l <= -5.6e+75)
		tmp = t_0;
	elseif (l <= -0.118)
		tmp = t_1;
	elseif (l <= 7.8e-6)
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	elseif (l <= 5e+102)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = U + ((l ^ 3.0) * ((J * 0.3333333333333333) * cos((K * 0.5))));
	t_1 = ((exp(l) - exp(-l)) * J) + U;
	tmp = 0.0;
	if (l <= -5.6e+75)
		tmp = t_0;
	elseif (l <= -0.118)
		tmp = t_1;
	elseif (l <= 7.8e-6)
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	elseif (l <= 5e+102)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(N[(J * 0.3333333333333333), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]}, If[LessEqual[l, -5.6e+75], t$95$0, If[LessEqual[l, -0.118], t$95$1, If[LessEqual[l, 7.8e-6], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5e+102], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.60000000000000023e75 or 5e102 < 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 96.9%

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

   4. Taylor expanded in l around inf 96.9%

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

      1. *-commutative 96.9%

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

      2. associate-*r* 96.9%

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

      3. associate-*l* 96.9%

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

      4. *-commutative 96.9%

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

      5. associate-*r* 96.9%

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

      6. *-commutative 96.9%

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

   6. Simplified 96.9%

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

   if -5.60000000000000023e75 < l < -0.11799999999999999 or 7.7999999999999999e-6 < l < 5e102

   1. Initial program 99.9%

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

   3. Taylor expanded in K around 0 79.3%

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

   if -0.11799999999999999 < l < 7.7999999999999999e-6

   1. Initial program 71.2%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \cdot 10^{+75}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \mathbf{elif}\;\ell \leq -0.118:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{elif}\;\ell \leq 7.8 \cdot 10^{-6}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + {\ell}^{3} \cdot \left(\left(J \cdot 0.3333333333333333\right) \cdot \cos \left(K \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.7:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.7)
   (+ U (* l (* 2.0 (* J (cos (* K 0.5))))))
   (+ 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)) <= 0.7) {
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	} 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)) <= 0.7d0) then
        tmp = u + (l * (2.0d0 * (j * cos((k * 0.5d0)))))
    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)) <= 0.7) {
		tmp = U + (l * (2.0 * (J * Math.cos((K * 0.5)))));
	} 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)) <= 0.7:
		tmp = U + (l * (2.0 * (J * math.cos((K * 0.5)))))
	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)) <= 0.7)
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(K * 0.5))))));
	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)) <= 0.7)
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	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], 0.7], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.69999999999999996

   1. Initial program 84.1%

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

   3. Taylor expanded in l around 0 83.1%

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

      1. associate-*r* 83.1%

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

      2. *-commutative 83.1%

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

      3. associate-*r* 83.1%

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

      4. associate-*r* 83.1%

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

      5. *-commutative 83.1%

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

      6. associate-*l* 83.1%

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

      7. *-commutative 83.1%

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

      8. distribute-lft-out 83.1%

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

      9. fma-define 83.1%

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

   5. Simplified 83.1%

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

   6. Taylor expanded in l around 0 71.0%

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

   if 0.69999999999999996 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.2%

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

   3. Taylor expanded in l around 0 92.2%

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

   4. Taylor expanded in K around 0 90.8%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.7:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.06 \lor \neg \left(\ell \leq 7.8 \cdot 10^{-6}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.06) (not (<= l 7.8e-6)))
   (+ (* (- (exp l) (exp (- l))) J) U)
   (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.06) || !(l <= 7.8e-6)) {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	} else {
		tmp = U + (cos((K / 2.0)) * (l * (J * 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 <= (-0.06d0)) .or. (.not. (l <= 7.8d-6))) then
        tmp = ((exp(l) - exp(-l)) * j) + u
    else
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 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 <= -0.06) || !(l <= 7.8e-6)) {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.06) or not (l <= 7.8e-6):
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	else:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.06) || !(l <= 7.8e-6))
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.06) || ~((l <= 7.8e-6)))
		tmp = ((exp(l) - exp(-l)) * J) + U;
	else
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.06], N[Not[LessEqual[l, 7.8e-6]], $MachinePrecision]], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -0.059999999999999998 or 7.7999999999999999e-6 < 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 76.4%

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

   if -0.059999999999999998 < l < 7.7999999999999999e-6

   1. Initial program 71.2%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. associate-*r* 100.0%

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

   5. Simplified 100.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.06 \lor \neg \left(\ell \leq 7.8 \cdot 10^{-6}\right):\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in l around 0 90.8%

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

4. Final simplification 90.8%

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

Alternative 6: 44.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+197}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{+62}:\\ \;\;\;\;U + J \cdot \left(8 - {K}^{2}\right)\\ \mathbf{elif}\;\ell \leq -72000000000000:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 12200:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 6.7 \cdot 10^{+140} \lor \neg \left(\ell \leq 4.1 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{{U}^{3}}{U + -4}\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(0.4444444444444444 + U \cdot \left(U \cdot 0.7901234567901234 - 0.5925925925925926\right)\right) - 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -3e+197)
   (- -4.0 (* U U))
   (if (<= l -1.05e+62)
     (+ U (* J (- 8.0 (pow K 2.0))))
     (if (<= l -72000000000000.0)
       (pow U -4.0)
       (if (<= l 12200.0)
         U
         (if (or (<= l 6.7e+140) (not (<= l 4.1e+273)))
           (/ (pow U 3.0) (+ U -4.0))
           (-
            (*
             U
             (+
              0.4444444444444444
              (* U (- (* U 0.7901234567901234) 0.5925925925925926))))
            0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3e+197) {
		tmp = -4.0 - (U * U);
	} else if (l <= -1.05e+62) {
		tmp = U + (J * (8.0 - pow(K, 2.0)));
	} else if (l <= -72000000000000.0) {
		tmp = pow(U, -4.0);
	} else if (l <= 12200.0) {
		tmp = U;
	} else if ((l <= 6.7e+140) || !(l <= 4.1e+273)) {
		tmp = pow(U, 3.0) / (U + -4.0);
	} else {
		tmp = (U * (0.4444444444444444 + (U * ((U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333;
	}
	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 <= (-3d+197)) then
        tmp = (-4.0d0) - (u * u)
    else if (l <= (-1.05d+62)) then
        tmp = u + (j * (8.0d0 - (k ** 2.0d0)))
    else if (l <= (-72000000000000.0d0)) then
        tmp = u ** (-4.0d0)
    else if (l <= 12200.0d0) then
        tmp = u
    else if ((l <= 6.7d+140) .or. (.not. (l <= 4.1d+273))) then
        tmp = (u ** 3.0d0) / (u + (-4.0d0))
    else
        tmp = (u * (0.4444444444444444d0 + (u * ((u * 0.7901234567901234d0) - 0.5925925925925926d0)))) - 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -3e+197) {
		tmp = -4.0 - (U * U);
	} else if (l <= -1.05e+62) {
		tmp = U + (J * (8.0 - Math.pow(K, 2.0)));
	} else if (l <= -72000000000000.0) {
		tmp = Math.pow(U, -4.0);
	} else if (l <= 12200.0) {
		tmp = U;
	} else if ((l <= 6.7e+140) || !(l <= 4.1e+273)) {
		tmp = Math.pow(U, 3.0) / (U + -4.0);
	} else {
		tmp = (U * (0.4444444444444444 + (U * ((U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -3e+197:
		tmp = -4.0 - (U * U)
	elif l <= -1.05e+62:
		tmp = U + (J * (8.0 - math.pow(K, 2.0)))
	elif l <= -72000000000000.0:
		tmp = math.pow(U, -4.0)
	elif l <= 12200.0:
		tmp = U
	elif (l <= 6.7e+140) or not (l <= 4.1e+273):
		tmp = math.pow(U, 3.0) / (U + -4.0)
	else:
		tmp = (U * (0.4444444444444444 + (U * ((U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -3e+197)
		tmp = Float64(-4.0 - Float64(U * U));
	elseif (l <= -1.05e+62)
		tmp = Float64(U + Float64(J * Float64(8.0 - (K ^ 2.0))));
	elseif (l <= -72000000000000.0)
		tmp = U ^ -4.0;
	elseif (l <= 12200.0)
		tmp = U;
	elseif ((l <= 6.7e+140) || !(l <= 4.1e+273))
		tmp = Float64((U ^ 3.0) / Float64(U + -4.0));
	else
		tmp = Float64(Float64(U * Float64(0.4444444444444444 + Float64(U * Float64(Float64(U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -3e+197)
		tmp = -4.0 - (U * U);
	elseif (l <= -1.05e+62)
		tmp = U + (J * (8.0 - (K ^ 2.0)));
	elseif (l <= -72000000000000.0)
		tmp = U ^ -4.0;
	elseif (l <= 12200.0)
		tmp = U;
	elseif ((l <= 6.7e+140) || ~((l <= 4.1e+273)))
		tmp = (U ^ 3.0) / (U + -4.0);
	else
		tmp = (U * (0.4444444444444444 + (U * ((U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -3e+197], N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -1.05e+62], N[(U + N[(J * N[(8.0 - N[Power[K, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -72000000000000.0], N[Power[U, -4.0], $MachinePrecision], If[LessEqual[l, 12200.0], U, If[Or[LessEqual[l, 6.7e+140], N[Not[LessEqual[l, 4.1e+273]], $MachinePrecision]], N[(N[Power[U, 3.0], $MachinePrecision] / N[(U + -4.0), $MachinePrecision]), $MachinePrecision], N[(N[(U * N[(0.4444444444444444 + N[(U * N[(N[(U * 0.7901234567901234), $MachinePrecision] - 0.5925925925925926), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if l < -3.0000000000000002e197

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 27.7%

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

      1. cancel-sign-sub-inv 27.7%

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

   8. Simplified 27.7%

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

   if -3.0000000000000002e197 < l < -1.05e62

   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 1.1%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in K around 0 23.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \left(J \cdot {K}^{2}\right) + 8 \cdot J\right)} + U \]
   6. Step-by-step derivation

      1. associate-*r* 23.3%

         \[\leadsto \left(\color{blue}{\left(-1 \cdot J\right) \cdot {K}^{2}} + 8 \cdot J\right) + U \]

      2. +-commutative 23.3%

         \[\leadsto \color{blue}{\left(8 \cdot J + \left(-1 \cdot J\right) \cdot {K}^{2}\right)} + U \]

      3. mul-1-neg 23.3%

         \[\leadsto \left(8 \cdot J + \color{blue}{\left(-J\right)} \cdot {K}^{2}\right) + U \]

      4. cancel-sign-sub-inv 23.3%

         \[\leadsto \color{blue}{\left(8 \cdot J - J \cdot {K}^{2}\right)} + U \]

      5. *-commutative 23.3%

         \[\leadsto \left(\color{blue}{J \cdot 8} - J \cdot {K}^{2}\right) + U \]

      6. distribute-lft-out-- 26.5%

         \[\leadsto \color{blue}{J \cdot \left(8 - {K}^{2}\right)} + U \]

   7. Simplified 26.5%

      \[\leadsto \color{blue}{J \cdot \left(8 - {K}^{2}\right)} + U \]

   if -1.05e62 < l < -7.2e13

   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 30.7%

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

   if -7.2e13 < l < 12200

   1. Initial program 72.4%

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

   3. Taylor expanded in J around 0 69.8%

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

   if 12200 < l < 6.7e140 or 4.09999999999999991e273 < 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 31.6%

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

      1. *-commutative 31.6%

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

      2. flip-- 31.6%

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

      3. associate-*l/ 34.3%

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

      4. fmm-def 34.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(U, U, --4 \cdot -4\right)} \cdot U}{U + -4} \]

      5. metadata-eval 34.3%

         \[\leadsto \frac{\mathsf{fma}\left(U, U, -\color{blue}{16}\right) \cdot U}{U + -4} \]

      6. metadata-eval 34.3%

         \[\leadsto \frac{\mathsf{fma}\left(U, U, \color{blue}{-16}\right) \cdot U}{U + -4} \]

   6. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(U, U, -16\right) \cdot U}{U + -4}} \]

   7. Taylor expanded in U around inf 34.3%

      \[\leadsto \frac{\color{blue}{{U}^{3}}}{U + -4} \]

   if 6.7e140 < l < 4.09999999999999991e273

   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 1.8%

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

      1. associate-+r+ 1.8%

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

      2. distribute-rgt1-in 1.8%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.8%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.8%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.8%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.7%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.7%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.7%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.7%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.7%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around 0 36.9%

      \[\leadsto \color{blue}{U \cdot \left(0.4444444444444444 + U \cdot \left(0.7901234567901234 \cdot U - 0.5925925925925926\right)\right) - 0.3333333333333333} \]
3. Recombined 6 regimes into one program.

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{+197}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq -1.05 \cdot 10^{+62}:\\ \;\;\;\;U + J \cdot \left(8 - {K}^{2}\right)\\ \mathbf{elif}\;\ell \leq -72000000000000:\\ \;\;\;\;{U}^{-4}\\ \mathbf{elif}\;\ell \leq 12200:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 6.7 \cdot 10^{+140} \lor \neg \left(\ell \leq 4.1 \cdot 10^{+273}\right):\\ \;\;\;\;\frac{{U}^{3}}{U + -4}\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(0.4444444444444444 + U \cdot \left(U \cdot 0.7901234567901234 - 0.5925925925925926\right)\right) - 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+84} \lor \neg \left(\ell \leq 0.47\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.8e+84) (not (<= l 0.47)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.8e+84) || !(l <= 0.47)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.8d+84)) .or. (.not. (l <= 0.47d0))) then
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.8e+84) || !(l <= 0.47)) {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.8e+84) or not (l <= 0.47):
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.8e+84) || !(l <= 0.47))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.8e+84) || ~((l <= 0.47)))
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.8e+84], N[Not[LessEqual[l, 0.47]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $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 < -2.79999999999999982e84 or 0.46999999999999997 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 88.7%

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

   4. Taylor expanded in l around inf 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-*r* 88.7%

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

      3. associate-*l* 88.7%

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

      4. *-commutative 88.7%

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

      5. associate-*r* 88.7%

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

      6. *-commutative 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in K around 0 70.7%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

      3. *-commutative 70.7%

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

      4. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   if -2.79999999999999982e84 < l < 0.46999999999999997

   1. Initial program 74.3%

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

   3. Taylor expanded in l around 0 91.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+84} \lor \neg \left(\ell \leq 0.47\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+84} \lor \neg \left(\ell \leq 0.47\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.8e+84) (not (<= l 0.47)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (+ U (* l (* 2.0 (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.8e+84) || !(l <= 0.47)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} 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 <= (-2.8d+84)) .or. (.not. (l <= 0.47d0))) then
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    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 <= -2.8e+84) || !(l <= 0.47)) {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	} 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 <= -2.8e+84) or not (l <= 0.47):
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	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 <= -2.8e+84) || !(l <= 0.47))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = Float64(U + Float64(l * Float64(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 <= -2.8e+84) || ~((l <= 0.47)))
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	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, -2.8e+84], N[Not[LessEqual[l, 0.47]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.79999999999999982e84 or 0.46999999999999997 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 88.7%

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

   4. Taylor expanded in l around inf 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-*r* 88.7%

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

      3. associate-*l* 88.7%

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

      4. *-commutative 88.7%

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

      5. associate-*r* 88.7%

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

      6. *-commutative 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in K around 0 70.7%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

      3. *-commutative 70.7%

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

      4. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   if -2.79999999999999982e84 < l < 0.46999999999999997

   1. Initial program 74.3%

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

   3. Taylor expanded in l around 0 92.3%

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

      1. associate-*r* 92.3%

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

      2. *-commutative 92.3%

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

      3. associate-*r* 92.3%

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

      4. associate-*r* 92.3%

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

      5. *-commutative 92.3%

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

      6. associate-*l* 92.3%

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

      7. *-commutative 92.3%

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

      8. distribute-lft-out 92.3%

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

      9. fma-define 92.3%

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

   5. Simplified 92.3%

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

   6. Taylor expanded in l around 0 91.4%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.8 \cdot 10^{+84} \lor \neg \left(\ell \leq 0.47\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+84} \lor \neg \left(\ell \leq 0.47\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.9e+84) (not (<= l 0.47)))
   (+ U (* (pow l 3.0) (* J 0.3333333333333333)))
   (+ U (* (cos (/ K 2.0)) (* l (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.9e+84) || !(l <= 0.47)) {
		tmp = U + (pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.9d+84)) .or. (.not. (l <= 0.47d0))) then
        tmp = u + ((l ** 3.0d0) * (j * 0.3333333333333333d0))
    else
        tmp = u + (cos((k / 2.0d0)) * (l * (j * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.9e+84) || !(l <= 0.47)) {
		tmp = U + (Math.pow(l, 3.0) * (J * 0.3333333333333333));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (l * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.9e+84) or not (l <= 0.47):
		tmp = U + (math.pow(l, 3.0) * (J * 0.3333333333333333))
	else:
		tmp = U + (math.cos((K / 2.0)) * (l * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.9e+84) || !(l <= 0.47))
		tmp = Float64(U + Float64((l ^ 3.0) * Float64(J * 0.3333333333333333)));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(l * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.9e+84) || ~((l <= 0.47)))
		tmp = U + ((l ^ 3.0) * (J * 0.3333333333333333));
	else
		tmp = U + (cos((K / 2.0)) * (l * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.9e+84], N[Not[LessEqual[l, 0.47]], $MachinePrecision]], N[(U + N[(N[Power[l, 3.0], $MachinePrecision] * N[(J * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(l * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.89999999999999989e84 or 0.46999999999999997 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 88.7%

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

   4. Taylor expanded in l around inf 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-*r* 88.7%

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

      3. associate-*l* 88.7%

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

      4. *-commutative 88.7%

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

      5. associate-*r* 88.7%

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

      6. *-commutative 88.7%

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

   6. Simplified 88.7%

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

   7. Taylor expanded in K around 0 70.7%

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

      1. associate-*r* 70.7%

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

      2. *-commutative 70.7%

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

      3. *-commutative 70.7%

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

      4. *-commutative 70.7%

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

   9. Simplified 70.7%

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

   if -2.89999999999999989e84 < l < 0.46999999999999997

   1. Initial program 74.3%

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

   3. Taylor expanded in l around 0 91.5%

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

      1. associate-*r* 91.5%

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

   5. Simplified 91.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.9 \cdot 10^{+84} \lor \neg \left(\ell \leq 0.47\right):\\ \;\;\;\;U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(\ell \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -940:\\ \;\;\;\;\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}\\ \mathbf{elif}\;\ell \leq 12200:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\frac{{U}^{3}}{U + -4}\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -940.0)
   (/ (- (/ (+ 0.1875 (/ -0.140625 U)) U) 0.25) U)
   (if (<= l 12200.0) U (/ (pow U 3.0) (+ U -4.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -940.0) {
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	} else if (l <= 12200.0) {
		tmp = U;
	} else {
		tmp = pow(U, 3.0) / (U + -4.0);
	}
	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 <= (-940.0d0)) then
        tmp = (((0.1875d0 + ((-0.140625d0) / u)) / u) - 0.25d0) / u
    else if (l <= 12200.0d0) then
        tmp = u
    else
        tmp = (u ** 3.0d0) / (u + (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -940.0) {
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	} else if (l <= 12200.0) {
		tmp = U;
	} else {
		tmp = Math.pow(U, 3.0) / (U + -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -940.0:
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U
	elif l <= 12200.0:
		tmp = U
	else:
		tmp = math.pow(U, 3.0) / (U + -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -940.0)
		tmp = Float64(Float64(Float64(Float64(0.1875 + Float64(-0.140625 / U)) / U) - 0.25) / U);
	elseif (l <= 12200.0)
		tmp = U;
	else
		tmp = Float64((U ^ 3.0) / Float64(U + -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -940.0)
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	elseif (l <= 12200.0)
		tmp = U;
	else
		tmp = (U ^ 3.0) / (U + -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -940.0], N[(N[(N[(N[(0.1875 + N[(-0.140625 / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision] - 0.25), $MachinePrecision] / U), $MachinePrecision], If[LessEqual[l, 12200.0], U, N[(N[Power[U, 3.0], $MachinePrecision] / N[(U + -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -940

   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 1.6%

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

      1. associate-+r+ 1.6%

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

      2. distribute-rgt1-in 1.6%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.6%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.6%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.6%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.5%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.5%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.5%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.5%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.5%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around inf 18.8%

      \[\leadsto \color{blue}{\frac{0.1875 \cdot \frac{1}{U} - \left(0.25 + \frac{0.140625}{{U}^{2}}\right)}{U}} \]
   8. Step-by-step derivation

      1. +-commutative 18.8%

         \[\leadsto \frac{0.1875 \cdot \frac{1}{U} - \color{blue}{\left(\frac{0.140625}{{U}^{2}} + 0.25\right)}}{U} \]

      2. associate--r+ 18.8%

         \[\leadsto \frac{\color{blue}{\left(0.1875 \cdot \frac{1}{U} - \frac{0.140625}{{U}^{2}}\right) - 0.25}}{U} \]

      3. associate-*r/ 18.8%

         \[\leadsto \frac{\left(\color{blue}{\frac{0.1875 \cdot 1}{U}} - \frac{0.140625}{{U}^{2}}\right) - 0.25}{U} \]

      4. metadata-eval 18.8%

         \[\leadsto \frac{\left(\frac{\color{blue}{0.1875}}{U} - \frac{0.140625}{{U}^{2}}\right) - 0.25}{U} \]

      5. unpow2 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{0.140625}{\color{blue}{U \cdot U}}\right) - 0.25}{U} \]

      6. associate-/r* 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \color{blue}{\frac{\frac{0.140625}{U}}{U}}\right) - 0.25}{U} \]

      7. metadata-eval 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{\frac{\color{blue}{0.140625 \cdot 1}}{U}}{U}\right) - 0.25}{U} \]

      8. associate-*r/ 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{\color{blue}{0.140625 \cdot \frac{1}{U}}}{U}\right) - 0.25}{U} \]

      9. div-sub 18.8%

         \[\leadsto \frac{\color{blue}{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}} - 0.25}{U} \]

      10. sub-neg 18.8%

          \[\leadsto \frac{\frac{\color{blue}{0.1875 + \left(-0.140625 \cdot \frac{1}{U}\right)}}{U} - 0.25}{U} \]

      11. associate-*r/ 18.8%

          \[\leadsto \frac{\frac{0.1875 + \left(-\color{blue}{\frac{0.140625 \cdot 1}{U}}\right)}{U} - 0.25}{U} \]

      12. metadata-eval 18.8%

          \[\leadsto \frac{\frac{0.1875 + \left(-\frac{\color{blue}{0.140625}}{U}\right)}{U} - 0.25}{U} \]

      13. distribute-neg-frac 18.8%

          \[\leadsto \frac{\frac{0.1875 + \color{blue}{\frac{-0.140625}{U}}}{U} - 0.25}{U} \]

      14. metadata-eval 18.8%

          \[\leadsto \frac{\frac{0.1875 + \frac{\color{blue}{-0.140625}}{U}}{U} - 0.25}{U} \]

   9. Simplified 18.8%

      \[\leadsto \color{blue}{\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}} \]

   if -940 < l < 12200

   1. Initial program 72.2%

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

   3. Taylor expanded in J around 0 70.3%

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

   if 12200 < 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 21.9%

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

      1. *-commutative 21.9%

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

      2. flip-- 21.9%

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

      3. associate-*l/ 27.7%

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

      4. fmm-def 27.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(U, U, --4 \cdot -4\right)} \cdot U}{U + -4} \]

      5. metadata-eval 27.7%

         \[\leadsto \frac{\mathsf{fma}\left(U, U, -\color{blue}{16}\right) \cdot U}{U + -4} \]

      6. metadata-eval 27.7%

         \[\leadsto \frac{\mathsf{fma}\left(U, U, \color{blue}{-16}\right) \cdot U}{U + -4} \]

   6. Applied egg-rr 27.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(U, U, -16\right) \cdot U}{U + -4}} \]

   7. Taylor expanded in U around inf 27.6%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -940:\\ \;\;\;\;\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}\\ \mathbf{elif}\;\ell \leq 12200:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;\frac{{U}^{3}}{U + -4}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in l around 0 90.8%

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

4. Taylor expanded in l around inf 76.0%

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

   1. *-commutative 76.0%

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

   2. associate-*r* 76.0%

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

   3. associate-*l* 76.0%

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

   4. *-commutative 76.0%

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

   5. associate-*r* 76.0%

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

   6. *-commutative 76.0%

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

6. Simplified 76.0%

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

7. Taylor expanded in K around 0 67.3%

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

   1. associate-*r* 67.3%

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

   2. *-commutative 67.3%

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

   3. *-commutative 67.3%

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

   4. *-commutative 67.3%

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

9. Simplified 67.3%

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

10. Final simplification 67.3%

    \[\leadsto U + {\ell}^{3} \cdot \left(J \cdot 0.3333333333333333\right) \]
11. Add Preprocessing

Alternative 12: 44.5% accurate, 11.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -600:\\ \;\;\;\;\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}\\ \mathbf{elif}\;\ell \leq 28500:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+141}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(0.4444444444444444 + U \cdot \left(U \cdot 0.7901234567901234 - 0.5925925925925926\right)\right) - 0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -600.0)
   (/ (- (/ (+ 0.1875 (/ -0.140625 U)) U) 0.25) U)
   (if (<= l 28500.0)
     U
     (if (<= l 4e+141)
       (* U U)
       (-
        (*
         U
         (+
          0.4444444444444444
          (* U (- (* U 0.7901234567901234) 0.5925925925925926))))
        0.3333333333333333)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -600.0) {
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	} else if (l <= 28500.0) {
		tmp = U;
	} else if (l <= 4e+141) {
		tmp = U * U;
	} else {
		tmp = (U * (0.4444444444444444 + (U * ((U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333;
	}
	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 <= (-600.0d0)) then
        tmp = (((0.1875d0 + ((-0.140625d0) / u)) / u) - 0.25d0) / u
    else if (l <= 28500.0d0) then
        tmp = u
    else if (l <= 4d+141) then
        tmp = u * u
    else
        tmp = (u * (0.4444444444444444d0 + (u * ((u * 0.7901234567901234d0) - 0.5925925925925926d0)))) - 0.3333333333333333d0
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -600.0) {
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	} else if (l <= 28500.0) {
		tmp = U;
	} else if (l <= 4e+141) {
		tmp = U * U;
	} else {
		tmp = (U * (0.4444444444444444 + (U * ((U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -600.0:
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U
	elif l <= 28500.0:
		tmp = U
	elif l <= 4e+141:
		tmp = U * U
	else:
		tmp = (U * (0.4444444444444444 + (U * ((U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -600.0)
		tmp = Float64(Float64(Float64(Float64(0.1875 + Float64(-0.140625 / U)) / U) - 0.25) / U);
	elseif (l <= 28500.0)
		tmp = U;
	elseif (l <= 4e+141)
		tmp = Float64(U * U);
	else
		tmp = Float64(Float64(U * Float64(0.4444444444444444 + Float64(U * Float64(Float64(U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -600.0)
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	elseif (l <= 28500.0)
		tmp = U;
	elseif (l <= 4e+141)
		tmp = U * U;
	else
		tmp = (U * (0.4444444444444444 + (U * ((U * 0.7901234567901234) - 0.5925925925925926)))) - 0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -600.0], N[(N[(N[(N[(0.1875 + N[(-0.140625 / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision] - 0.25), $MachinePrecision] / U), $MachinePrecision], If[LessEqual[l, 28500.0], U, If[LessEqual[l, 4e+141], N[(U * U), $MachinePrecision], N[(N[(U * N[(0.4444444444444444 + N[(U * N[(N[(U * 0.7901234567901234), $MachinePrecision] - 0.5925925925925926), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -600

   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 1.6%

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

      1. associate-+r+ 1.6%

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

      2. distribute-rgt1-in 1.6%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.6%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.6%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.6%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.5%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.5%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.5%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.5%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.5%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around inf 18.8%

      \[\leadsto \color{blue}{\frac{0.1875 \cdot \frac{1}{U} - \left(0.25 + \frac{0.140625}{{U}^{2}}\right)}{U}} \]
   8. Step-by-step derivation

      1. +-commutative 18.8%

         \[\leadsto \frac{0.1875 \cdot \frac{1}{U} - \color{blue}{\left(\frac{0.140625}{{U}^{2}} + 0.25\right)}}{U} \]

      2. associate--r+ 18.8%

         \[\leadsto \frac{\color{blue}{\left(0.1875 \cdot \frac{1}{U} - \frac{0.140625}{{U}^{2}}\right) - 0.25}}{U} \]

      3. associate-*r/ 18.8%

         \[\leadsto \frac{\left(\color{blue}{\frac{0.1875 \cdot 1}{U}} - \frac{0.140625}{{U}^{2}}\right) - 0.25}{U} \]

      4. metadata-eval 18.8%

         \[\leadsto \frac{\left(\frac{\color{blue}{0.1875}}{U} - \frac{0.140625}{{U}^{2}}\right) - 0.25}{U} \]

      5. unpow2 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{0.140625}{\color{blue}{U \cdot U}}\right) - 0.25}{U} \]

      6. associate-/r* 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \color{blue}{\frac{\frac{0.140625}{U}}{U}}\right) - 0.25}{U} \]

      7. metadata-eval 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{\frac{\color{blue}{0.140625 \cdot 1}}{U}}{U}\right) - 0.25}{U} \]

      8. associate-*r/ 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{\color{blue}{0.140625 \cdot \frac{1}{U}}}{U}\right) - 0.25}{U} \]

      9. div-sub 18.8%

         \[\leadsto \frac{\color{blue}{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}} - 0.25}{U} \]

      10. sub-neg 18.8%

          \[\leadsto \frac{\frac{\color{blue}{0.1875 + \left(-0.140625 \cdot \frac{1}{U}\right)}}{U} - 0.25}{U} \]

      11. associate-*r/ 18.8%

          \[\leadsto \frac{\frac{0.1875 + \left(-\color{blue}{\frac{0.140625 \cdot 1}{U}}\right)}{U} - 0.25}{U} \]

      12. metadata-eval 18.8%

          \[\leadsto \frac{\frac{0.1875 + \left(-\frac{\color{blue}{0.140625}}{U}\right)}{U} - 0.25}{U} \]

      13. distribute-neg-frac 18.8%

          \[\leadsto \frac{\frac{0.1875 + \color{blue}{\frac{-0.140625}{U}}}{U} - 0.25}{U} \]

      14. metadata-eval 18.8%

          \[\leadsto \frac{\frac{0.1875 + \frac{\color{blue}{-0.140625}}{U}}{U} - 0.25}{U} \]

   9. Simplified 18.8%

      \[\leadsto \color{blue}{\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}} \]

   if -600 < l < 28500

   1. Initial program 72.2%

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

   3. Taylor expanded in J around 0 70.3%

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

   if 28500 < l < 4.00000000000000007e141

   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 31.3%

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

   if 4.00000000000000007e141 < 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 1.7%

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

      1. associate-+r+ 1.7%

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

      2. distribute-rgt1-in 1.7%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.7%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.7%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.7%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.6%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.6%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.6%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.6%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.6%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around 0 33.1%

      \[\leadsto \color{blue}{U \cdot \left(0.4444444444444444 + U \cdot \left(0.7901234567901234 \cdot U - 0.5925925925925926\right)\right) - 0.3333333333333333} \]
3. Recombined 4 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -600:\\ \;\;\;\;\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}\\ \mathbf{elif}\;\ell \leq 28500:\\ \;\;\;\;U\\ \mathbf{elif}\;\ell \leq 4 \cdot 10^{+141}:\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(0.4444444444444444 + U \cdot \left(U \cdot 0.7901234567901234 - 0.5925925925925926\right)\right) - 0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.0% accurate, 19.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -740:\\ \;\;\;\;\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}\\ \mathbf{elif}\;\ell \leq 485:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -740.0)
   (/ (- (/ (+ 0.1875 (/ -0.140625 U)) U) 0.25) U)
   (if (<= l 485.0) U (* U (- U -4.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -740.0) {
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	} else if (l <= 485.0) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	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 <= (-740.0d0)) then
        tmp = (((0.1875d0 + ((-0.140625d0) / u)) / u) - 0.25d0) / u
    else if (l <= 485.0d0) then
        tmp = u
    else
        tmp = u * (u - (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -740.0) {
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	} else if (l <= 485.0) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -740.0:
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U
	elif l <= 485.0:
		tmp = U
	else:
		tmp = U * (U - -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -740.0)
		tmp = Float64(Float64(Float64(Float64(0.1875 + Float64(-0.140625 / U)) / U) - 0.25) / U);
	elseif (l <= 485.0)
		tmp = U;
	else
		tmp = Float64(U * Float64(U - -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -740.0)
		tmp = (((0.1875 + (-0.140625 / U)) / U) - 0.25) / U;
	elseif (l <= 485.0)
		tmp = U;
	else
		tmp = U * (U - -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -740.0], N[(N[(N[(N[(0.1875 + N[(-0.140625 / U), $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision] - 0.25), $MachinePrecision] / U), $MachinePrecision], If[LessEqual[l, 485.0], U, N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -740

   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 1.6%

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

      1. associate-+r+ 1.6%

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

      2. distribute-rgt1-in 1.6%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.6%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.6%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.6%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.5%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.5%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.5%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.5%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.5%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around inf 18.8%

      \[\leadsto \color{blue}{\frac{0.1875 \cdot \frac{1}{U} - \left(0.25 + \frac{0.140625}{{U}^{2}}\right)}{U}} \]
   8. Step-by-step derivation

      1. +-commutative 18.8%

         \[\leadsto \frac{0.1875 \cdot \frac{1}{U} - \color{blue}{\left(\frac{0.140625}{{U}^{2}} + 0.25\right)}}{U} \]

      2. associate--r+ 18.8%

         \[\leadsto \frac{\color{blue}{\left(0.1875 \cdot \frac{1}{U} - \frac{0.140625}{{U}^{2}}\right) - 0.25}}{U} \]

      3. associate-*r/ 18.8%

         \[\leadsto \frac{\left(\color{blue}{\frac{0.1875 \cdot 1}{U}} - \frac{0.140625}{{U}^{2}}\right) - 0.25}{U} \]

      4. metadata-eval 18.8%

         \[\leadsto \frac{\left(\frac{\color{blue}{0.1875}}{U} - \frac{0.140625}{{U}^{2}}\right) - 0.25}{U} \]

      5. unpow2 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{0.140625}{\color{blue}{U \cdot U}}\right) - 0.25}{U} \]

      6. associate-/r* 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \color{blue}{\frac{\frac{0.140625}{U}}{U}}\right) - 0.25}{U} \]

      7. metadata-eval 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{\frac{\color{blue}{0.140625 \cdot 1}}{U}}{U}\right) - 0.25}{U} \]

      8. associate-*r/ 18.8%

         \[\leadsto \frac{\left(\frac{0.1875}{U} - \frac{\color{blue}{0.140625 \cdot \frac{1}{U}}}{U}\right) - 0.25}{U} \]

      9. div-sub 18.8%

         \[\leadsto \frac{\color{blue}{\frac{0.1875 - 0.140625 \cdot \frac{1}{U}}{U}} - 0.25}{U} \]

      10. sub-neg 18.8%

          \[\leadsto \frac{\frac{\color{blue}{0.1875 + \left(-0.140625 \cdot \frac{1}{U}\right)}}{U} - 0.25}{U} \]

      11. associate-*r/ 18.8%

          \[\leadsto \frac{\frac{0.1875 + \left(-\color{blue}{\frac{0.140625 \cdot 1}{U}}\right)}{U} - 0.25}{U} \]

      12. metadata-eval 18.8%

          \[\leadsto \frac{\frac{0.1875 + \left(-\frac{\color{blue}{0.140625}}{U}\right)}{U} - 0.25}{U} \]

      13. distribute-neg-frac 18.8%

          \[\leadsto \frac{\frac{0.1875 + \color{blue}{\frac{-0.140625}{U}}}{U} - 0.25}{U} \]

      14. metadata-eval 18.8%

          \[\leadsto \frac{\frac{0.1875 + \frac{\color{blue}{-0.140625}}{U}}{U} - 0.25}{U} \]

   9. Simplified 18.8%

      \[\leadsto \color{blue}{\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}} \]

   if -740 < l < 485

   1. Initial program 72.0%

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

   3. Taylor expanded in J around 0 70.9%

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

   if 485 < 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 21.6%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -740:\\ \;\;\;\;\frac{\frac{0.1875 + \frac{-0.140625}{U}}{U} - 0.25}{U}\\ \mathbf{elif}\;\ell \leq 485:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.1% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+65}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq 950:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -2.3e+65) (- -4.0 (* U U)) (if (<= l 950.0) U (* U (- U -4.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.3e+65) {
		tmp = -4.0 - (U * U);
	} else if (l <= 950.0) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= (-2.3d+65)) then
        tmp = (-4.0d0) - (u * u)
    else if (l <= 950.0d0) then
        tmp = u
    else
        tmp = u * (u - (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -2.3e+65) {
		tmp = -4.0 - (U * U);
	} else if (l <= 950.0) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -2.3e+65:
		tmp = -4.0 - (U * U)
	elif l <= 950.0:
		tmp = U
	else:
		tmp = U * (U - -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -2.3e+65)
		tmp = Float64(-4.0 - Float64(U * U));
	elseif (l <= 950.0)
		tmp = U;
	else
		tmp = Float64(U * Float64(U - -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -2.3e+65)
		tmp = -4.0 - (U * U);
	elseif (l <= 950.0)
		tmp = U;
	else
		tmp = U * (U - -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -2.3e+65], N[(-4.0 - N[(U * U), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 950.0], U, N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.3e65

   1. Initial program 100.0%

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

      1. associate-*l* 100.0%

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

      2. fma-define 100.0%

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

   3. Simplified 100.0%

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

   6. Applied egg-rr 14.2%

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

      1. cancel-sign-sub-inv 14.2%

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

   8. Simplified 14.2%

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

   if -2.3e65 < l < 950

   1. Initial program 73.8%

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

   3. Taylor expanded in J around 0 66.5%

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

   if 950 < 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 21.6%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.3 \cdot 10^{+65}:\\ \;\;\;\;-4 - U \cdot U\\ \mathbf{elif}\;\ell \leq 950:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.7% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -580000000000:\\ \;\;\;\;\frac{\frac{0.1875}{U} + -0.25}{U}\\ \mathbf{elif}\;\ell \leq 620:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= l -580000000000.0)
   (/ (+ (/ 0.1875 U) -0.25) U)
   (if (<= l 620.0) U (* U (- U -4.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -580000000000.0) {
		tmp = ((0.1875 / U) + -0.25) / U;
	} else if (l <= 620.0) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	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 <= (-580000000000.0d0)) then
        tmp = ((0.1875d0 / u) + (-0.25d0)) / u
    else if (l <= 620.0d0) then
        tmp = u
    else
        tmp = u * (u - (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= -580000000000.0) {
		tmp = ((0.1875 / U) + -0.25) / U;
	} else if (l <= 620.0) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= -580000000000.0:
		tmp = ((0.1875 / U) + -0.25) / U
	elif l <= 620.0:
		tmp = U
	else:
		tmp = U * (U - -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= -580000000000.0)
		tmp = Float64(Float64(Float64(0.1875 / U) + -0.25) / U);
	elseif (l <= 620.0)
		tmp = U;
	else
		tmp = Float64(U * Float64(U - -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= -580000000000.0)
		tmp = ((0.1875 / U) + -0.25) / U;
	elseif (l <= 620.0)
		tmp = U;
	else
		tmp = U * (U - -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, -580000000000.0], N[(N[(N[(0.1875 / U), $MachinePrecision] + -0.25), $MachinePrecision] / U), $MachinePrecision], If[LessEqual[l, 620.0], U, N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -5.8e11

   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 1.5%

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

      1. associate-+r+ 1.5%

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

      2. distribute-rgt1-in 1.5%

         \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

      3. metadata-eval 1.5%

         \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

      4. *-commutative 1.5%

         \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

      5. distribute-lft-out 1.5%

         \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

      6. associate-/r* 1.5%

         \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

      7. *-inverses 1.5%

         \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

      8. +-commutative 1.5%

         \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

      9. *-commutative 1.5%

         \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

   6. Simplified 1.5%

      \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

   7. Taylor expanded in U around inf 15.4%

      \[\leadsto \color{blue}{\frac{0.1875 \cdot \frac{1}{U} - 0.25}{U}} \]
   8. Step-by-step derivation

      1. sub-neg 15.4%

         \[\leadsto \frac{\color{blue}{0.1875 \cdot \frac{1}{U} + \left(-0.25\right)}}{U} \]

      2. associate-*r/ 15.4%

         \[\leadsto \frac{\color{blue}{\frac{0.1875 \cdot 1}{U}} + \left(-0.25\right)}{U} \]

      3. metadata-eval 15.4%

         \[\leadsto \frac{\frac{\color{blue}{0.1875}}{U} + \left(-0.25\right)}{U} \]

      4. metadata-eval 15.4%

         \[\leadsto \frac{\frac{0.1875}{U} + \color{blue}{-0.25}}{U} \]

   9. Simplified 15.4%

      \[\leadsto \color{blue}{\frac{\frac{0.1875}{U} + -0.25}{U}} \]

   if -5.8e11 < l < 620

   1. Initial program 72.2%

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

   3. Taylor expanded in J around 0 70.3%

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

   if 620 < 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 21.6%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -580000000000:\\ \;\;\;\;\frac{\frac{0.1875}{U} + -0.25}{U}\\ \mathbf{elif}\;\ell \leq 620:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 39.7% accurate, 31.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 950:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l 950.0) U (* U (- U -4.0))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 950.0) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	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 <= 950.0d0) then
        tmp = u
    else
        tmp = u * (u - (-4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 950.0) {
		tmp = U;
	} else {
		tmp = U * (U - -4.0);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 950.0:
		tmp = U
	else:
		tmp = U * (U - -4.0)
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 950.0)
		tmp = U;
	else
		tmp = Float64(U * Float64(U - -4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 950.0)
		tmp = U;
	else
		tmp = U * (U - -4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 950.0], U, N[(U * N[(U - -4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 950

   1. Initial program 80.5%

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

   3. Taylor expanded in J around 0 50.0%

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

   if 950 < 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 21.6%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 950:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(U - -4\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 39.7% accurate, 38.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 19000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (if (<= l 19000.0) U (* U U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 19000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (l <= 19000.0d0) then
        tmp = u
    else
        tmp = u * u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (l <= 19000.0) {
		tmp = U;
	} else {
		tmp = U * U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if l <= 19000.0:
		tmp = U
	else:
		tmp = U * U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (l <= 19000.0)
		tmp = U;
	else
		tmp = Float64(U * U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (l <= 19000.0)
		tmp = U;
	else
		tmp = U * U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[l, 19000.0], U, N[(U * U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 19000

   1. Initial program 80.6%

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

   3. Taylor expanded in J around 0 49.7%

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

   if 19000 < 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 21.8%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 19000:\\ \;\;\;\;U\\ \mathbf{else}:\\ \;\;\;\;U \cdot U\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 2.7% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -0.3333333333333333)

C

double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

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 = -0.3333333333333333d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -0.3333333333333333;
}

Python

def code(J, l, K, U):
	return -0.3333333333333333

Julia

function code(J, l, K, U)
	return -0.3333333333333333
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -0.3333333333333333;
end

Wolfram

code[J_, l_, K_, U_] := -0.3333333333333333

Derivation

1. Initial program 85.5%

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

4. Applied egg-rr 2.1%

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

   1. associate-+r+ 2.1%

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

   2. distribute-rgt1-in 2.1%

      \[\leadsto \frac{U}{\color{blue}{\left(-4 + 1\right) \cdot U} + U \cdot \left(-4 \cdot U\right)} \]

   3. metadata-eval 2.1%

      \[\leadsto \frac{U}{\color{blue}{-3} \cdot U + U \cdot \left(-4 \cdot U\right)} \]

   4. *-commutative 2.1%

      \[\leadsto \frac{U}{\color{blue}{U \cdot -3} + U \cdot \left(-4 \cdot U\right)} \]

   5. distribute-lft-out 2.1%

      \[\leadsto \frac{U}{\color{blue}{U \cdot \left(-3 + -4 \cdot U\right)}} \]

   6. associate-/r* 2.0%

      \[\leadsto \color{blue}{\frac{\frac{U}{U}}{-3 + -4 \cdot U}} \]

   7. *-inverses 2.0%

      \[\leadsto \frac{\color{blue}{1}}{-3 + -4 \cdot U} \]

   8. +-commutative 2.0%

      \[\leadsto \frac{1}{\color{blue}{-4 \cdot U + -3}} \]

   9. *-commutative 2.0%

      \[\leadsto \frac{1}{\color{blue}{U \cdot -4} + -3} \]

6. Simplified 2.0%

   \[\leadsto \color{blue}{\frac{1}{U \cdot -4 + -3}} \]

7. Taylor expanded in U around 0 2.6%

   \[\leadsto \color{blue}{-0.3333333333333333} \]

8. Final simplification 2.6%

   \[\leadsto -0.3333333333333333 \]
9. Add Preprocessing

Alternative 19: 2.7% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

4. Applied egg-rr 2.8%

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

   1. *-inverses 2.8%

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

6. Simplified 2.8%

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

7. Final simplification 2.8%

   \[\leadsto 1 \]
8. Add Preprocessing

Alternative 20: 36.9% accurate, 312.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.5%

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

3. Taylor expanded in J around 0 37.9%

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

4. Final simplification 37.9%

   \[\leadsto U \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024131 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))