Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 85.8% → 99.6%

Time: 11.2s

Alternatives: 16

Speedup: 2.7×

• 49.4% 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: 85.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -1 \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \end{array} \]

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 -1.0) (not (<= t_1 0.0)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0)))))))))

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 <= -1.0) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-1.0d0)) .or. (.not. (t_1 <= 0.0d0))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -1.0) || !(t_1 <= 0.0)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0)))));
	}
	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 <= -1.0) or not (t_1 <= 0.0):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0)))))
	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 <= -1.0) || !(t_1 <= 0.0))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -1.0) || ~((t_1 <= 0.0)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -1.0], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -1 or 0.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 100.0%

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

   if -1 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.0

   1. Initial program 68.3%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. distribute-rgt-in 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. unpow2 99.9%

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

      5. pow3 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -1 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+101} \lor \neg \left(\ell \leq -0.006 \lor \neg \left(\ell \leq 9 \cdot 10^{-19}\right) \land \ell \leq 1.05 \cdot 10^{+100}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.1e+101)
         (not (or (<= l -0.006) (and (not (<= l 9e-19)) (<= l 1.05e+100)))))
   (+ U (* (cos (/ K 2.0)) (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))
   (+ (* (- (exp l) (exp (- l))) J) U)))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.1e+101) || !((l <= -0.006) || (!(l <= 9e-19) && (l <= 1.05e+100)))) {
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = ((exp(l) - exp(-l)) * J) + U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-1.1d+101)) .or. (.not. (l <= (-0.006d0)) .or. (.not. (l <= 9d-19)) .and. (l <= 1.05d+100))) then
        tmp = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    else
        tmp = ((exp(l) - exp(-l)) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.1e+101) || !((l <= -0.006) || (!(l <= 9e-19) && (l <= 1.05e+100)))) {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = ((Math.exp(l) - Math.exp(-l)) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.1e+101) or not ((l <= -0.006) or (not (l <= 9e-19) and (l <= 1.05e+100))):
		tmp = U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	else:
		tmp = ((math.exp(l) - math.exp(-l)) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.1e+101) || !((l <= -0.006) || (!(l <= 9e-19) && (l <= 1.05e+100))))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	else
		tmp = Float64(Float64(Float64(exp(l) - exp(Float64(-l))) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.1e+101) || ~(((l <= -0.006) || (~((l <= 9e-19)) && (l <= 1.05e+100)))))
		tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	else
		tmp = ((exp(l) - exp(-l)) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.1e+101], N[Not[Or[LessEqual[l, -0.006], And[N[Not[LessEqual[l, 9e-19]], $MachinePrecision], LessEqual[l, 1.05e+100]]]], $MachinePrecision]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.1e101 or -0.0060000000000000001 < l < 9.00000000000000026e-19 or 1.0499999999999999e100 < l

   1. Initial program 81.5%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. unpow2 99.9%

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

   5. Applied egg-rr 99.9%

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

   if -1.1e101 < l < -0.0060000000000000001 or 9.00000000000000026e-19 < l < 1.0499999999999999e100

   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 82.2%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.1 \cdot 10^{+101} \lor \neg \left(\ell \leq -0.006 \lor \neg \left(\ell \leq 9 \cdot 10^{-19}\right) \land \ell \leq 1.05 \cdot 10^{+100}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -85000:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-19}:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - t\_0\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -85000.0)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (<= l 9e-19)
       (+ U (* t_1 (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0))))))
       (if (<= l 1.05e+100)
         (+ (* (- (exp l) t_0) J) U)
         (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -85000.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 9e-19) {
		tmp = U + (t_1 * (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0)))));
	} else if (l <= 1.05e+100) {
		tmp = ((exp(l) - t_0) * J) + U;
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-85000.0d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if (l <= 9d-19) then
        tmp = u + (t_1 * (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0)))))
    else if (l <= 1.05d+100) then
        tmp = ((exp(l) - t_0) * j) + u
    else
        tmp = u + (t_1 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -85000.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if (l <= 9e-19) {
		tmp = U + (t_1 * (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0)))));
	} else if (l <= 1.05e+100) {
		tmp = ((Math.exp(l) - t_0) * J) + U;
	} else {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -85000.0:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif l <= 9e-19:
		tmp = U + (t_1 * (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0)))))
	elif l <= 1.05e+100:
		tmp = ((math.exp(l) - t_0) * J) + U
	else:
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -85000.0)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif (l <= 9e-19)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0))))));
	elseif (l <= 1.05e+100)
		tmp = Float64(Float64(Float64(exp(l) - t_0) * J) + U);
	else
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -85000.0)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif (l <= 9e-19)
		tmp = U + (t_1 * (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0)))));
	elseif (l <= 1.05e+100)
		tmp = ((exp(l) - t_0) * J) + U;
	else
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -85000.0], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 9e-19], N[(U + N[(t$95$1 * N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.05e+100], N[(N[(N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -85000

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -85000 < l < 9.00000000000000026e-19

   1. Initial program 68.5%

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

   3. Taylor expanded in l around 0 99.4%

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

      1. distribute-rgt-in 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*l* 99.4%

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

      4. unpow2 99.4%

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

      5. pow3 99.4%

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

   5. Applied egg-rr 99.4%

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

   if 9.00000000000000026e-19 < l < 1.0499999999999999e100

   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 78.6%

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

   if 1.0499999999999999e100 < l

   1. Initial program 100.0%

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

   3. Taylor expanded in l around 0 100.0%

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

      1. unpow2 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -85000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-19}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\right)\\ \mathbf{elif}\;\ell \leq 1.05 \cdot 10^{+100}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -85000:\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(27 - t\_0\right)\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-19} \lor \neg \left(\ell \leq 2.1 \cdot 10^{+99}\right):\\ \;\;\;\;U + t\_1 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - t\_0\right) \cdot J + U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (cos (/ K 2.0))))
   (if (<= l -85000.0)
     (+ U (* t_1 (* J (- 27.0 t_0))))
     (if (or (<= l 9e-19) (not (<= l 2.1e+99)))
       (+ U (* t_1 (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))
       (+ (* (- (exp l) t_0) J) U)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(-l);
	double t_1 = cos((K / 2.0));
	double tmp;
	if (l <= -85000.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if ((l <= 9e-19) || !(l <= 2.1e+99)) {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = ((exp(l) - t_0) * J) + U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos((k / 2.0d0))
    if (l <= (-85000.0d0)) then
        tmp = u + (t_1 * (j * (27.0d0 - t_0)))
    else if ((l <= 9d-19) .or. (.not. (l <= 2.1d+99))) then
        tmp = u + (t_1 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l))))))
    else
        tmp = ((exp(l) - t_0) * j) + u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.exp(-l);
	double t_1 = Math.cos((K / 2.0));
	double tmp;
	if (l <= -85000.0) {
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	} else if ((l <= 9e-19) || !(l <= 2.1e+99)) {
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	} else {
		tmp = ((Math.exp(l) - t_0) * J) + U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.exp(-l)
	t_1 = math.cos((K / 2.0))
	tmp = 0
	if l <= -85000.0:
		tmp = U + (t_1 * (J * (27.0 - t_0)))
	elif (l <= 9e-19) or not (l <= 2.1e+99):
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))))
	else:
		tmp = ((math.exp(l) - t_0) * J) + U
	return tmp

Julia

function code(J, l, K, U)
	t_0 = exp(Float64(-l))
	t_1 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= -85000.0)
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(27.0 - t_0))));
	elseif ((l <= 9e-19) || !(l <= 2.1e+99))
		tmp = Float64(U + Float64(t_1 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l)))))));
	else
		tmp = Float64(Float64(Float64(exp(l) - t_0) * J) + U);
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = exp(-l);
	t_1 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= -85000.0)
		tmp = U + (t_1 * (J * (27.0 - t_0)));
	elseif ((l <= 9e-19) || ~((l <= 2.1e+99)))
		tmp = U + (t_1 * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
	else
		tmp = ((exp(l) - t_0) * J) + U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -85000.0], N[(U + N[(t$95$1 * N[(J * N[(27.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[l, 9e-19], N[Not[LessEqual[l, 2.1e+99]], $MachinePrecision]], N[(U + N[(t$95$1 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Exp[l], $MachinePrecision] - t$95$0), $MachinePrecision] * J), $MachinePrecision] + U), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -85000

   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 100.0%

      \[\leadsto \left(J \cdot \left(\color{blue}{27} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -85000 < l < 9.00000000000000026e-19 or 2.1000000000000001e99 < l

   1. Initial program 76.6%

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

   3. Taylor expanded in l around 0 99.5%

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

      1. unpow2 99.5%

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

   5. Applied egg-rr 99.5%

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

   if 9.00000000000000026e-19 < l < 2.1000000000000001e99

   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 78.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -85000:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(27 - e^{-\ell}\right)\right)\\ \mathbf{elif}\;\ell \leq 9 \cdot 10^{-19} \lor \neg \left(\ell \leq 2.1 \cdot 10^{+99}\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(e^{\ell} - e^{-\ell}\right) \cdot J + U\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.65:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{\left(\ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right) + J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) 0.65)
   (* U (+ 1.0 (* J (/ (* (* l 2.0) (cos (* K 0.5))) U))))
   (+ U (+ (* J (* 0.3333333333333333 (pow l 3.0))) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= 0.65) {
		tmp = U * (1.0 + (J * (((l * 2.0) * cos((K * 0.5))) / U)));
	} else {
		tmp = U + ((J * (0.3333333333333333 * pow(l, 3.0))) + (J * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= 0.65d0) then
        tmp = u * (1.0d0 + (j * (((l * 2.0d0) * cos((k * 0.5d0))) / u)))
    else
        tmp = u + ((j * (0.3333333333333333d0 * (l ** 3.0d0))) + (j * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= 0.65) {
		tmp = U * (1.0 + (J * (((l * 2.0) * Math.cos((K * 0.5))) / U)));
	} else {
		tmp = U + ((J * (0.3333333333333333 * Math.pow(l, 3.0))) + (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= 0.65:
		tmp = U * (1.0 + (J * (((l * 2.0) * math.cos((K * 0.5))) / U)))
	else:
		tmp = U + ((J * (0.3333333333333333 * math.pow(l, 3.0))) + (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= 0.65)
		tmp = Float64(U * Float64(1.0 + Float64(J * Float64(Float64(Float64(l * 2.0) * cos(Float64(K * 0.5))) / U))));
	else
		tmp = Float64(U + Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) + Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= 0.65)
		tmp = U * (1.0 + (J * (((l * 2.0) * cos((K * 0.5))) / U)));
	else
		tmp = U + ((J * (0.3333333333333333 * (l ^ 3.0))) + (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], 0.65], N[(U * N[(1.0 + N[(J * N[(N[(N[(l * 2.0), $MachinePrecision] * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.650000000000000022

   1. Initial program 83.6%

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

   3. Taylor expanded in l around 0 87.1%

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

      1. distribute-rgt-in 87.1%

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

      2. *-commutative 87.1%

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

      3. associate-*l* 87.1%

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

      4. unpow2 87.1%

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

      5. pow3 87.1%

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

   5. Applied egg-rr 87.1%

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

   6. Taylor expanded in U around inf 89.9%

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

      1. associate-/l* 89.9%

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

      2. *-commutative 89.9%

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

      3. fma-define 89.9%

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

   8. Simplified 89.9%

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

   9. Taylor expanded in l around 0 69.7%

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

       1. associate-*r/ 69.7%

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

       2. associate-*r* 69.7%

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

       3. *-commutative 69.7%

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

   11. Simplified 69.7%

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

   if 0.650000000000000022 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   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 90.3%

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

   4. Taylor expanded in K around 0 89.4%

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

      1. distribute-lft-in 89.4%

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

      2. distribute-rgt-in 89.4%

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

      3. *-commutative 89.4%

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

      4. associate-*l* 89.4%

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

      5. unpow2 89.4%

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

      6. pow3 89.4%

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

   6. Applied egg-rr 89.4%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.65:\\ \;\;\;\;U \cdot \left(1 + J \cdot \frac{\left(\ell \cdot 2\right) \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right) + J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := J \cdot \left(\ell \cdot 2\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;U + t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right) + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (* J (* l 2.0))))
   (if (<= t_0 -0.02)
     (+ U (* t_0 t_1))
     (+ U (+ (* J (* 0.3333333333333333 (pow l 3.0))) t_1)))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = J * (l * 2.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = U + (t_0 * t_1);
	} else {
		tmp = U + ((J * (0.3333333333333333 * pow(l, 3.0))) + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = j * (l * 2.0d0)
    if (t_0 <= (-0.02d0)) then
        tmp = u + (t_0 * t_1)
    else
        tmp = u + ((j * (0.3333333333333333d0 * (l ** 3.0d0))) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = J * (l * 2.0);
	double tmp;
	if (t_0 <= -0.02) {
		tmp = U + (t_0 * t_1);
	} else {
		tmp = U + ((J * (0.3333333333333333 * Math.pow(l, 3.0))) + t_1);
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = J * (l * 2.0)
	tmp = 0
	if t_0 <= -0.02:
		tmp = U + (t_0 * t_1)
	else:
		tmp = U + ((J * (0.3333333333333333 * math.pow(l, 3.0))) + t_1)
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(J * Float64(l * 2.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(U + Float64(t_0 * t_1));
	else
		tmp = Float64(U + Float64(Float64(J * Float64(0.3333333333333333 * (l ^ 3.0))) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = J * (l * 2.0);
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = U + (t_0 * t_1);
	else
		tmp = U + ((J * (0.3333333333333333 * (l ^ 3.0))) + t_1);
	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[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(U + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[(J * N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 55.1%

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

   5. Simplified 55.1%

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

   if -0.0200000000000000004 < (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 89.3%

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

   4. Taylor expanded in K around 0 86.6%

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

      1. distribute-lft-in 86.6%

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

      2. distribute-rgt-in 86.6%

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

      3. *-commutative 86.6%

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

      4. associate-*l* 86.6%

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

      5. unpow2 86.6%

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

      6. pow3 86.6%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{3}\right) + J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.02)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (+ (* l 2.0) (* 0.3333333333333333 (pow l 3.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * pow(l, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.02d0)) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else
        tmp = u + (j * ((l * 2.0d0) + (0.3333333333333333d0 * (l ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * Math.pow(l, 3.0))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.02:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * math.pow(l, 3.0))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(Float64(l * 2.0) + Float64(0.3333333333333333 * (l ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * ((l * 2.0) + (0.3333333333333333 * (l ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(N[(l * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[Power[l, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 55.1%

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

   5. Simplified 55.1%

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

   if -0.0200000000000000004 < (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 89.3%

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

   4. Taylor expanded in K around 0 86.6%

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

      1. distribute-rgt-in 89.4%

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

      2. *-commutative 89.4%

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

      3. associate-*l* 89.4%

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

      4. unpow2 89.4%

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

      5. pow3 89.4%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2 + 0.3333333333333333 \cdot {\ell}^{3}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.02:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 -0.02)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= (-0.02d0)) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= -0.02:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -0.02], N[(U + N[(t$95$0 * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0200000000000000004

   1. Initial program 79.9%

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

   3. Taylor expanded in l around 0 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 55.1%

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

   5. Simplified 55.1%

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

   if -0.0200000000000000004 < (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 89.3%

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

   4. Taylor expanded in K around 0 86.6%

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

      1. unpow2 89.3%

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

   6. Applied egg-rr 86.6%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.02:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-7} \lor \neg \left(\ell \leq 1.66 \cdot 10^{-31}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -1.85e-7) (not (<= l 1.66e-31)))
   (+ U (* J (* l (+ 2.0 (* 0.3333333333333333 (* l l))))))
   (+ U (* l (* (cos (* K 0.5)) (* J 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -1.85e-7) || !(l <= 1.66e-31)) {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (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 <= (-1.85d-7)) .or. (.not. (l <= 1.66d-31))) then
        tmp = u + (j * (l * (2.0d0 + (0.3333333333333333d0 * (l * l)))))
    else
        tmp = u + (l * (cos((k * 0.5d0)) * (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 <= -1.85e-7) || !(l <= 1.66e-31)) {
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	} else {
		tmp = U + (l * (Math.cos((K * 0.5)) * (J * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -1.85e-7) or not (l <= 1.66e-31):
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))
	else:
		tmp = U + (l * (math.cos((K * 0.5)) * (J * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -1.85e-7) || !(l <= 1.66e-31))
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * Float64(l * l))))));
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -1.85e-7) || ~((l <= 1.66e-31)))
		tmp = U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
	else
		tmp = U + (l * (cos((K * 0.5)) * (J * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -1.85e-7], N[Not[LessEqual[l, 1.66e-31]], $MachinePrecision]], N[(U + N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.85000000000000002e-7 or 1.66000000000000002e-31 < l

   1. Initial program 98.4%

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

   3. Taylor expanded in l around 0 80.1%

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

   4. Taylor expanded in K around 0 61.4%

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

      1. unpow2 80.1%

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

   6. Applied egg-rr 61.4%

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

   if -1.85000000000000002e-7 < l < 1.66000000000000002e-31

   1. Initial program 68.3%

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

   3. Taylor expanded in l around 0 99.9%

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

      1. *-commutative 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*l* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

      7. associate-*l* 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.85 \cdot 10^{-7} \lor \neg \left(\ell \leq 1.66 \cdot 10^{-31}\right):\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 87.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

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 * l))))))
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 * (l * l))))));
}

Python

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

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 * Float64(l * l)))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.3333333333333333 * (l * l))))));
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[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in l around 0 89.1%

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

   1. unpow2 89.1%

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

5. Applied egg-rr 89.1%

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

6. Final simplification 89.1%

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

Alternative 11: 42.6% accurate, 23.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.66 \lor \neg \left(\ell \leq 3.5 \cdot 10^{+39}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -0.66) (not (<= l 3.5e+39))) (* U U) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.66) || !(l <= 3.5e+39)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-0.66d0)) .or. (.not. (l <= 3.5d+39))) then
        tmp = u * u
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -0.66) || !(l <= 3.5e+39)) {
		tmp = U * U;
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -0.66) or not (l <= 3.5e+39):
		tmp = U * U
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -0.66) || !(l <= 3.5e+39))
		tmp = Float64(U * U);
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -0.66) || ~((l <= 3.5e+39)))
		tmp = U * U;
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -0.66], N[Not[LessEqual[l, 3.5e+39]], $MachinePrecision]], N[(U * U), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -0.660000000000000031 or 3.5000000000000002e39 < 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 18.5%

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

   if -0.660000000000000031 < l < 3.5000000000000002e39

   1. Initial program 70.7%

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

   3. Taylor expanded in J around 0 63.0%

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

4. Final simplification 41.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.66 \lor \neg \left(\ell \leq 3.5 \cdot 10^{+39}\right):\\ \;\;\;\;U \cdot U\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.2% accurate, 24.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))));
}

Python

def code(J, l, K, U):
	return U + (J * (l * (2.0 + (0.3333333333333333 * (l * l)))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in l around 0 89.1%

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

4. Taylor expanded in K around 0 72.1%

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

   1. unpow2 89.1%

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

6. Applied egg-rr 72.1%

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

7. Final simplification 72.1%

   \[\leadsto U + J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot \left(\ell \cdot \ell\right)\right)\right) \]
8. Add Preprocessing

Alternative 13: 53.5% accurate, 44.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.8%

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

3. Taylor expanded in l around 0 89.1%

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

4. Taylor expanded in K around 0 72.1%

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

5. Taylor expanded in l around 0 52.9%

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

   1. associate-*r* 52.9%

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

   2. *-commutative 52.9%

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

7. Simplified 52.9%

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

8. Final simplification 52.9%

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

Alternative 14: 36.5% 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 84.8%

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

3. Taylor expanded in J around 0 33.8%

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

Alternative 15: 2.8% 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 84.8%

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

4. Applied egg-rr 2.5%

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

   1. associate-+r+ 2.5%

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

   2. +-commutative 2.5%

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

   3. distribute-rgt1-in 2.5%

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

   4. metadata-eval 2.5%

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

   5. *-commutative 2.5%

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

   6. distribute-lft-out 2.5%

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

   7. associate-/r* 2.4%

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

   8. *-inverses 2.4%

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

   9. *-commutative 2.4%

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

6. Simplified 2.4%

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

7. Taylor expanded in U around 0 2.7%

   \[\leadsto \color{blue}{-0.3333333333333333} \]
8. Add Preprocessing

Alternative 16: 2.8% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ -4 \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 -4.0)

C

double code(double J, double l, double K, double U) {
	return -4.0;
}

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 = -4.0d0
end function

Java

public static double code(double J, double l, double K, double U) {
	return -4.0;
}

Python

def code(J, l, K, U):
	return -4.0

Julia

function code(J, l, K, U)
	return -4.0
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = -4.0;
end

Wolfram

code[J_, l_, K_, U_] := -4.0

Derivation

1. Initial program 84.8%

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

4. Applied egg-rr 2.7%

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

5. Taylor expanded in U around 0 2.7%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))