Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.7%

Time: 10.8s

Alternatives: 16

Speedup: 1.4×

• 49.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -0.14 \lor \neg \left(\ell \leq 880\right):\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (or (<= l -0.14) (not (<= l 880.0)))
     (+ U (* t_0 (* J (- (exp l) (exp (- l))))))
     (+ U (* t_0 (* J (* l (+ 2.0 (* (pow l 2.0) 0.3333333333333333)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((l <= -0.14) || !(l <= 880.0)) {
		tmp = U + (t_0 * (J * (exp(l) - exp(-l))));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (pow(l, 2.0) * 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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if ((l <= (-0.14d0)) .or. (.not. (l <= 880.0d0))) then
        tmp = u + (t_0 * (j * (exp(l) - exp(-l))))
    else
        tmp = u + (t_0 * (j * (l * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0)))))
    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 ((l <= -0.14) || !(l <= 880.0)) {
		tmp = U + (t_0 * (J * (Math.exp(l) - Math.exp(-l))));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (l <= -0.14) or not (l <= 880.0):
		tmp = U + (t_0 * (J * (math.exp(l) - math.exp(-l))))
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if ((l <= -0.14) || !(l <= 880.0))
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(exp(l) - exp(Float64(-l))))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((l <= -0.14) || ~((l <= 880.0)))
		tmp = U + (t_0 * (J * (exp(l) - exp(-l))));
	else
		tmp = U + (t_0 * (J * (l * (2.0 + ((l ^ 2.0) * 0.3333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[l, -0.14], N[Not[LessEqual[l, 880.0]], $MachinePrecision]], N[(U + N[(t$95$0 * N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -0.14000000000000001 or 880 < 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 -0.14000000000000001 < l < 880

   1. Initial program 75.7%

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

   3. Taylor expanded in l around 0 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.14 \lor \neg \left(\ell \leq 880\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -\infty \lor \neg \left(t\_1 \leq 0.1\right):\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.1)))
     (+ U (* t_0 (* J (* 0.0003968253968253968 (pow l 7.0)))))
     (+ U (* t_0 (* J (* l 2.0)))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.1)) {
		tmp = U + (t_0 * (J * (0.0003968253968253968 * pow(l, 7.0))));
	} else {
		tmp = U + (t_0 * (J * (l * 2.0)));
	}
	return tmp;
}

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.1)) {
		tmp = U + (t_0 * (J * (0.0003968253968253968 * Math.pow(l, 7.0))));
	} else {
		tmp = U + (t_0 * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.1):
		tmp = U + (t_0 * (J * (0.0003968253968253968 * math.pow(l, 7.0))))
	else:
		tmp = U + (t_0 * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 0.1))
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(0.0003968253968253968 * (l ^ 7.0)))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 0.1)))
		tmp = U + (t_0 * (J * (0.0003968253968253968 * (l ^ 7.0))));
	else
		tmp = U + (t_0 * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.1]], $MachinePrecision]], N[(U + N[(t$95$0 * N[(J * N[(0.0003968253968253968 * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * 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 0.10000000000000001 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.3%

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

   3. Taylor expanded in l around 0 93.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 93.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 93.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 93.8%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. associate-*r* 93.8%

         \[\leadsto \color{blue}{\left(\left(0.0003968253968253968 \cdot J\right) \cdot {\ell}^{7}\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 93.8%

         \[\leadsto \left(\color{blue}{\left(J \cdot 0.0003968253968253968\right)} \cdot {\ell}^{7}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*l* 93.8%

         \[\leadsto \color{blue}{\left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 93.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -inf.0 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 0.10000000000000001

   1. Initial program 76.1%

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

   3. Taylor expanded in l around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r* 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -\infty \lor \neg \left(e^{\ell} - e^{-\ell} \leq 0.1\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq 880:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= l 880.0)
     (+
      (*
       (*
        J
        (*
         l
         (+
          2.0
          (*
           (pow l 2.0)
           (+
            0.3333333333333333
            (*
             (pow l 2.0)
             (+
              0.016666666666666666
              (* (pow l 2.0) 0.0003968253968253968))))))))
       t_0)
      U)
     (+ U (* t_0 (* J (- (exp l) (exp (- l)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (l <= 880.0) {
		tmp = ((J * (l * (2.0 + (pow(l, 2.0) * (0.3333333333333333 + (pow(l, 2.0) * (0.016666666666666666 + (pow(l, 2.0) * 0.0003968253968253968)))))))) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (exp(l) - exp(-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 (l <= 880.0d0) then
        tmp = ((j * (l * (2.0d0 + ((l ** 2.0d0) * (0.3333333333333333d0 + ((l ** 2.0d0) * (0.016666666666666666d0 + ((l ** 2.0d0) * 0.0003968253968253968d0)))))))) * t_0) + u
    else
        tmp = u + (t_0 * (j * (exp(l) - exp(-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 (l <= 880.0) {
		tmp = ((J * (l * (2.0 + (Math.pow(l, 2.0) * (0.3333333333333333 + (Math.pow(l, 2.0) * (0.016666666666666666 + (Math.pow(l, 2.0) * 0.0003968253968253968)))))))) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (Math.exp(l) - Math.exp(-l))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if l <= 880.0:
		tmp = ((J * (l * (2.0 + (math.pow(l, 2.0) * (0.3333333333333333 + (math.pow(l, 2.0) * (0.016666666666666666 + (math.pow(l, 2.0) * 0.0003968253968253968)))))))) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (math.exp(l) - math.exp(-l))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (l <= 880.0)
		tmp = Float64(Float64(Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * Float64(0.3333333333333333 + Float64((l ^ 2.0) * Float64(0.016666666666666666 + Float64((l ^ 2.0) * 0.0003968253968253968)))))))) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(exp(l) - exp(Float64(-l))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if (l <= 880.0)
		tmp = ((J * (l * (2.0 + ((l ^ 2.0) * (0.3333333333333333 + ((l ^ 2.0) * (0.016666666666666666 + ((l ^ 2.0) * 0.0003968253968253968)))))))) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (exp(l) - exp(-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[l, 880.0], N[(N[(N[(J * N[(l * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[Power[l, 2.0], $MachinePrecision] * N[(0.016666666666666666 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 880

   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 97.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 97.9%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 97.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   if 880 < 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. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 880:\\ \;\;\;\;\left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;\ell \leq -5.6 \lor \neg \left(\ell \leq 2.2\right):\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (or (<= l -5.6) (not (<= l 2.2)))
     (+ U (* t_0 (* J (* 0.0003968253968253968 (pow l 7.0)))))
     (+ U (* t_0 (* J (* l (+ 2.0 (* (pow l 2.0) 0.3333333333333333)))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if ((l <= -5.6) || !(l <= 2.2)) {
		tmp = U + (t_0 * (J * (0.0003968253968253968 * pow(l, 7.0))));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (pow(l, 2.0) * 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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if ((l <= (-5.6d0)) .or. (.not. (l <= 2.2d0))) then
        tmp = u + (t_0 * (j * (0.0003968253968253968d0 * (l ** 7.0d0))))
    else
        tmp = u + (t_0 * (j * (l * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0)))))
    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 ((l <= -5.6) || !(l <= 2.2)) {
		tmp = U + (t_0 * (J * (0.0003968253968253968 * Math.pow(l, 7.0))));
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if (l <= -5.6) or not (l <= 2.2):
		tmp = U + (t_0 * (J * (0.0003968253968253968 * math.pow(l, 7.0))))
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333)))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if ((l <= -5.6) || !(l <= 2.2))
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(0.0003968253968253968 * (l ^ 7.0)))));
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	tmp = 0.0;
	if ((l <= -5.6) || ~((l <= 2.2)))
		tmp = U + (t_0 * (J * (0.0003968253968253968 * (l ^ 7.0))));
	else
		tmp = U + (t_0 * (J * (l * (2.0 + ((l ^ 2.0) * 0.3333333333333333)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[l, -5.6], N[Not[LessEqual[l, 2.2]], $MachinePrecision]], N[(U + N[(t$95$0 * N[(J * N[(0.0003968253968253968 * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < -5.5999999999999996 or 2.2000000000000002 < l

   1. Initial program 99.3%

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

   3. Taylor expanded in l around 0 93.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 93.8%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 93.8%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 93.8%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   7. Step-by-step derivation

      1. associate-*r* 93.8%

         \[\leadsto \color{blue}{\left(\left(0.0003968253968253968 \cdot J\right) \cdot {\ell}^{7}\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. *-commutative 93.8%

         \[\leadsto \left(\color{blue}{\left(J \cdot 0.0003968253968253968\right)} \cdot {\ell}^{7}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      3. associate-*l* 93.8%

         \[\leadsto \color{blue}{\left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Simplified 93.8%

      \[\leadsto \color{blue}{\left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -5.5999999999999996 < l < 2.2000000000000002

   1. Initial program 76.1%

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

   3. Taylor expanded in l around 0 99.6%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5.6 \lor \neg \left(\ell \leq 2.2\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(0.0003968253968253968 \cdot {\ell}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.345:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left({K}^{2} \cdot -8 + 5\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.345)
   (+ U (* (* J (* l 2.0)) (+ (* (pow K 2.0) -8.0) 5.0)))
   (+ U (* J (* l (+ 2.0 (* (pow l 2.0) 0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.345) {
		tmp = U + ((J * (l * 2.0)) * ((pow(K, 2.0) * -8.0) + 5.0));
	} else {
		tmp = U + (J * (l * (2.0 + (pow(l, 2.0) * 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 (cos((k / 2.0d0)) <= (-0.345d0)) then
        tmp = u + ((j * (l * 2.0d0)) * (((k ** 2.0d0) * (-8.0d0)) + 5.0d0))
    else
        tmp = u + (j * (l * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0))))
    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)) <= -0.345) {
		tmp = U + ((J * (l * 2.0)) * ((Math.pow(K, 2.0) * -8.0) + 5.0));
	} else {
		tmp = U + (J * (l * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.345:
		tmp = U + ((J * (l * 2.0)) * ((math.pow(K, 2.0) * -8.0) + 5.0))
	else:
		tmp = U + (J * (l * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.345)
		tmp = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(Float64((K ^ 2.0) * -8.0) + 5.0)));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.345)
		tmp = U + ((J * (l * 2.0)) * (((K ^ 2.0) * -8.0) + 5.0));
	else
		tmp = U + (J * (l * (2.0 + ((l ^ 2.0) * 0.3333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.345], N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[K, 2.0], $MachinePrecision] * -8.0), $MachinePrecision] + 5.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.34499999999999997

   1. Initial program 93.7%

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

   3. Taylor expanded in l around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*r* 63.3%

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

   5. Simplified 63.3%

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

   7. Applied egg-rr 33.5%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 33.5%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 33.5%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 33.5%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 33.5%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 33.5%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 33.5%

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

   9. Simplified 33.5%

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

   10. Taylor expanded in K around 0 73.5%

       \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(5 + -8 \cdot {K}^{2}\right)} + U \]
   11. Step-by-step derivation

       1. +-commutative 73.5%

          \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(-8 \cdot {K}^{2} + 5\right)} + U \]

       2. *-commutative 73.5%

          \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{{K}^{2} \cdot -8} + 5\right) + U \]

   12. Simplified 73.5%

       \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left({K}^{2} \cdot -8 + 5\right)} + U \]

   if -0.34499999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.2%

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

   3. Taylor expanded in l around 0 96.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 96.6%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 96.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 90.5%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \color{blue}{1} + U \]

   7. Taylor expanded in l around 0 85.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.3333333333333333 \cdot {\ell}^{2}}\right)\right)\right) \cdot 1 + U \]
   8. Step-by-step derivation

      1. *-commutative 85.0%

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

   9. Simplified 85.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.345:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left({K}^{2} \cdot -8 + 5\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.345:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + {K}^{2} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.345)
   (+ U (* (* J (* l 2.0)) (+ 1.0 (* (pow K 2.0) -0.125))))
   (+ U (* J (* l (+ 2.0 (* (pow l 2.0) 0.3333333333333333)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.345) {
		tmp = U + ((J * (l * 2.0)) * (1.0 + (pow(K, 2.0) * -0.125)));
	} else {
		tmp = U + (J * (l * (2.0 + (pow(l, 2.0) * 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 (cos((k / 2.0d0)) <= (-0.345d0)) then
        tmp = u + ((j * (l * 2.0d0)) * (1.0d0 + ((k ** 2.0d0) * (-0.125d0))))
    else
        tmp = u + (j * (l * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0))))
    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)) <= -0.345) {
		tmp = U + ((J * (l * 2.0)) * (1.0 + (Math.pow(K, 2.0) * -0.125)));
	} else {
		tmp = U + (J * (l * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.345:
		tmp = U + ((J * (l * 2.0)) * (1.0 + (math.pow(K, 2.0) * -0.125)))
	else:
		tmp = U + (J * (l * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.345)
		tmp = Float64(U + Float64(Float64(J * Float64(l * 2.0)) * Float64(1.0 + Float64((K ^ 2.0) * -0.125))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.345)
		tmp = U + ((J * (l * 2.0)) * (1.0 + ((K ^ 2.0) * -0.125)));
	else
		tmp = U + (J * (l * (2.0 + ((l ^ 2.0) * 0.3333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.345], N[(U + N[(N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[K, 2.0], $MachinePrecision] * -0.125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.34499999999999997

   1. Initial program 93.7%

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

   3. Taylor expanded in l around 0 63.3%

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

      1. *-commutative 63.3%

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

      2. associate-*r* 63.3%

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

   5. Simplified 63.3%

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

   6. Taylor expanded in K around 0 73.5%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(1 + -0.125 \cdot {K}^{2}\right)} + U \]

   if -0.34499999999999997 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.2%

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

   3. Taylor expanded in l around 0 96.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 96.6%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 96.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 90.5%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \color{blue}{1} + U \]

   7. Taylor expanded in l around 0 85.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.3333333333333333 \cdot {\ell}^{2}}\right)\right)\right) \cdot 1 + U \]
   8. Step-by-step derivation

      1. *-commutative 85.0%

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

   9. Simplified 85.0%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.345:\\ \;\;\;\;U + \left(J \cdot \left(\ell \cdot 2\right)\right) \cdot \left(1 + {K}^{2} \cdot -0.125\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \mathbf{if}\;t\_0 \leq 0.22:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (if (<= t_0 0.22)
     (+ U (* t_0 (* J (* l 2.0))))
     (+ U (* J (* l (+ 2.0 (* (pow l 2.0) 0.3333333333333333))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double tmp;
	if (t_0 <= 0.22) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (pow(l, 2.0) * 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) :: t_0
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    if (t_0 <= 0.22d0) then
        tmp = u + (t_0 * (j * (l * 2.0d0)))
    else
        tmp = u + (j * (l * (2.0d0 + ((l ** 2.0d0) * 0.3333333333333333d0))))
    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 <= 0.22) {
		tmp = U + (t_0 * (J * (l * 2.0)));
	} else {
		tmp = U + (J * (l * (2.0 + (Math.pow(l, 2.0) * 0.3333333333333333))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	tmp = 0
	if t_0 <= 0.22:
		tmp = U + (t_0 * (J * (l * 2.0)))
	else:
		tmp = U + (J * (l * (2.0 + (math.pow(l, 2.0) * 0.3333333333333333))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	tmp = 0.0
	if (t_0 <= 0.22)
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * 2.0))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64((l ^ 2.0) * 0.3333333333333333)))));
	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 <= 0.22)
		tmp = U + (t_0 * (J * (l * 2.0)));
	else
		tmp = U + (J * (l * (2.0 + ((l ^ 2.0) * 0.3333333333333333))));
	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, 0.22], 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[(N[Power[l, 2.0], $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < 0.220000000000000001

   1. Initial program 92.1%

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

   3. Taylor expanded in l around 0 69.0%

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

      1. *-commutative 69.0%

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

      2. associate-*r* 69.0%

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

   5. Simplified 69.0%

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

   if 0.220000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 87.0%

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

   3. Taylor expanded in l around 0 96.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 96.2%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 96.2%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 94.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \color{blue}{1} + U \]

   7. Taylor expanded in l around 0 87.7%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.3333333333333333 \cdot {\ell}^{2}}\right)\right)\right) \cdot 1 + U \]
   8. Step-by-step derivation

      1. *-commutative 87.7%

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

   9. Simplified 87.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq 0.22:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+20} \lor \neg \left(\ell \leq 880\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4.9e+20) (not (<= l 880.0)))
   (+ U (* J (- (exp l) (exp (- l)))))
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.9e+20) || !(l <= 880.0)) {
		tmp = U + (J * (exp(l) - exp(-l)));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4.9d+20)) .or. (.not. (l <= 880.0d0))) then
        tmp = u + (j * (exp(l) - exp(-l)))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.9e+20) || !(l <= 880.0)) {
		tmp = U + (J * (Math.exp(l) - Math.exp(-l)));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4.9e+20) or not (l <= 880.0):
		tmp = U + (J * (math.exp(l) - math.exp(-l)))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4.9e+20) || !(l <= 880.0))
		tmp = Float64(U + Float64(J * Float64(exp(l) - exp(Float64(-l)))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4.9e+20) || ~((l <= 880.0)))
		tmp = U + (J * (exp(l) - exp(-l)));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4.9e+20], N[Not[LessEqual[l, 880.0]], $MachinePrecision]], N[(U + N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.9e20 or 880 < 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 72.4%

      \[\leadsto \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \color{blue}{1} + U \]

   if -4.9e20 < l < 880

   1. Initial program 76.1%

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

   3. Taylor expanded in l around 0 97.9%

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

      1. *-commutative 97.9%

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

      2. associate-*r* 97.9%

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

   5. Simplified 97.9%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+20} \lor \neg \left(\ell \leq 880\right):\\ \;\;\;\;U + J \cdot \left(e^{\ell} - e^{-\ell}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ U (* J (* l -6.0)))
   (+ U (* 0.0003968253968253968 (* J (pow l 7.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = U + (J * (l * -6.0));
	} else {
		tmp = U + (0.0003968253968253968 * (J * pow(l, 7.0)));
	}
	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)) <= (-0.005d0)) then
        tmp = u + (j * (l * (-6.0d0)))
    else
        tmp = u + (0.0003968253968253968d0 * (j * (l ** 7.0d0)))
    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)) <= -0.005) {
		tmp = U + (J * (l * -6.0));
	} else {
		tmp = U + (0.0003968253968253968 * (J * Math.pow(l, 7.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = U + (J * (l * -6.0))
	else:
		tmp = U + (0.0003968253968253968 * (J * math.pow(l, 7.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(U + Float64(J * Float64(l * -6.0)));
	else
		tmp = Float64(U + Float64(0.0003968253968253968 * Float64(J * (l ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = U + (J * (l * -6.0));
	else
		tmp = U + (0.0003968253968253968 * (J * (l ^ 7.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(U + N[(J * N[(l * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(0.0003968253968253968 * N[(J * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 94.9%

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

   3. Taylor expanded in l around 0 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*r* 66.7%

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

   5. Simplified 66.7%

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

   7. Applied egg-rr 49.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) \cdot -3\right)} + U \]
   8. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} \cdot -3\right) + U \]

   9. Simplified 49.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) \cdot -3\right)} + U \]

   10. Taylor expanded in K around 0 61.9%

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

       1. *-commutative 61.9%

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

       2. associate-*l* 61.9%

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   12. Simplified 61.9%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   3. Taylor expanded in l around 0 96.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 96.4%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 96.4%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 93.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \color{blue}{1} + U \]

   7. Taylor expanded in l around inf 82.7%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot 1 + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;U + 0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+20} \lor \neg \left(\ell \leq 1.12 \cdot 10^{+31}\right):\\ \;\;\;\;U + 0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4.9e+20) (not (<= l 1.12e+31)))
   (+ U (* 0.0003968253968253968 (* J (pow l 7.0))))
   (+ U (* (cos (/ K 2.0)) (* J (* l 2.0))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.9e+20) || !(l <= 1.12e+31)) {
		tmp = U + (0.0003968253968253968 * (J * pow(l, 7.0)));
	} else {
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4.9d+20)) .or. (.not. (l <= 1.12d+31))) then
        tmp = u + (0.0003968253968253968d0 * (j * (l ** 7.0d0)))
    else
        tmp = u + (cos((k / 2.0d0)) * (j * (l * 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.9e+20) || !(l <= 1.12e+31)) {
		tmp = U + (0.0003968253968253968 * (J * Math.pow(l, 7.0)));
	} else {
		tmp = U + (Math.cos((K / 2.0)) * (J * (l * 2.0)));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4.9e+20) or not (l <= 1.12e+31):
		tmp = U + (0.0003968253968253968 * (J * math.pow(l, 7.0)))
	else:
		tmp = U + (math.cos((K / 2.0)) * (J * (l * 2.0)))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4.9e+20) || !(l <= 1.12e+31))
		tmp = Float64(U + Float64(0.0003968253968253968 * Float64(J * (l ^ 7.0))));
	else
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4.9e+20) || ~((l <= 1.12e+31)))
		tmp = U + (0.0003968253968253968 * (J * (l ^ 7.0)));
	else
		tmp = U + (cos((K / 2.0)) * (J * (l * 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4.9e+20], N[Not[LessEqual[l, 1.12e+31]], $MachinePrecision]], N[(U + N[(0.0003968253968253968 * N[(J * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.9e20 or 1.11999999999999993e31 < 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 97.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 97.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 69.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \color{blue}{1} + U \]

   7. Taylor expanded in l around inf 69.9%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot 1 + U \]

   if -4.9e20 < l < 1.11999999999999993e31

   1. Initial program 77.0%

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

   3. Taylor expanded in l around 0 94.9%

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

      1. *-commutative 94.9%

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

      2. associate-*r* 94.9%

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

   5. Simplified 94.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+20} \lor \neg \left(\ell \leq 1.12 \cdot 10^{+31}\right):\\ \;\;\;\;U + 0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+20} \lor \neg \left(\ell \leq 1.6 \cdot 10^{+30}\right):\\ \;\;\;\;U + 0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4.9e+20) (not (<= l 1.6e+30)))
   (+ U (* 0.0003968253968253968 (* J (pow l 7.0))))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.9e+20) || !(l <= 1.6e+30)) {
		tmp = U + (0.0003968253968253968 * (J * pow(l, 7.0)));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4.9d+20)) .or. (.not. (l <= 1.6d+30))) then
        tmp = u + (0.0003968253968253968d0 * (j * (l ** 7.0d0)))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.9e+20) || !(l <= 1.6e+30)) {
		tmp = U + (0.0003968253968253968 * (J * Math.pow(l, 7.0)));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4.9e+20) or not (l <= 1.6e+30):
		tmp = U + (0.0003968253968253968 * (J * math.pow(l, 7.0)))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4.9e+20) || !(l <= 1.6e+30))
		tmp = Float64(U + Float64(0.0003968253968253968 * Float64(J * (l ^ 7.0))));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4.9e+20) || ~((l <= 1.6e+30)))
		tmp = U + (0.0003968253968253968 * (J * (l ^ 7.0)));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4.9e+20], N[Not[LessEqual[l, 1.6e+30]], $MachinePrecision]], N[(U + N[(0.0003968253968253968 * N[(J * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.9e20 or 1.59999999999999986e30 < 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 97.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 97.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 69.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)\right) \cdot \color{blue}{1} + U \]

   7. Taylor expanded in l around inf 69.9%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot 1 + U \]

   if -4.9e20 < l < 1.59999999999999986e30

   1. Initial program 77.0%

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

   3. Taylor expanded in l around 0 94.9%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.9 \cdot 10^{+20} \lor \neg \left(\ell \leq 1.6 \cdot 10^{+30}\right):\\ \;\;\;\;U + 0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.005)
   (+ U (* J (* l -6.0)))
   (* U (+ 1.0 (* 10.0 (* J (/ l U)))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.005) {
		tmp = U + (J * (l * -6.0));
	} else {
		tmp = U * (1.0 + (10.0 * (J * (l / U))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.005d0)) then
        tmp = u + (j * (l * (-6.0d0)))
    else
        tmp = u * (1.0d0 + (10.0d0 * (j * (l / u))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.005) {
		tmp = U + (J * (l * -6.0));
	} else {
		tmp = U * (1.0 + (10.0 * (J * (l / U))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.005:
		tmp = U + (J * (l * -6.0))
	else:
		tmp = U * (1.0 + (10.0 * (J * (l / U))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.005)
		tmp = Float64(U + Float64(J * Float64(l * -6.0)));
	else
		tmp = Float64(U * Float64(1.0 + Float64(10.0 * Float64(J * Float64(l / U)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.005)
		tmp = U + (J * (l * -6.0));
	else
		tmp = U * (1.0 + (10.0 * (J * (l / U))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.005], N[(U + N[(J * N[(l * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U * N[(1.0 + N[(10.0 * N[(J * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.0050000000000000001

   1. Initial program 94.9%

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

   3. Taylor expanded in l around 0 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*r* 66.7%

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

   5. Simplified 66.7%

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

   7. Applied egg-rr 49.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) \cdot -3\right)} + U \]
   8. Step-by-step derivation

      1. *-commutative 49.4%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} \cdot -3\right) + U \]

   9. Simplified 49.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) \cdot -3\right)} + U \]

   10. Taylor expanded in K around 0 61.9%

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

       1. *-commutative 61.9%

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

       2. associate-*l* 61.9%

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   12. Simplified 61.9%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   if -0.0050000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.3%

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

   3. Taylor expanded in l around 0 62.4%

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

      1. *-commutative 62.4%

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

      2. associate-*r* 62.4%

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

   5. Simplified 62.4%

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

   7. Applied egg-rr 50.7%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 50.7%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 50.7%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 50.7%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 50.7%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 50.7%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 50.7%

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

   9. Simplified 50.7%

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

   10. Taylor expanded in K around 0 50.7%

       \[\leadsto \color{blue}{10 \cdot \left(J \cdot \ell\right)} + U \]
   11. Step-by-step derivation

       1. associate-*r* 50.7%

          \[\leadsto \color{blue}{\left(10 \cdot J\right) \cdot \ell} + U \]

   12. Simplified 50.7%

       \[\leadsto \color{blue}{\left(10 \cdot J\right) \cdot \ell} + U \]

   13. Taylor expanded in U around inf 58.6%

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

       1. associate-/l* 64.8%

          \[\leadsto U \cdot \left(1 + 10 \cdot \color{blue}{\left(J \cdot \frac{\ell}{U}\right)}\right) \]

   15. Simplified 64.8%

       \[\leadsto \color{blue}{U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.005:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \mathbf{else}:\\ \;\;\;\;U \cdot \left(1 + 10 \cdot \left(J \cdot \frac{\ell}{U}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.7% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2.05 \cdot 10^{+34}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2.05e+34) (+ U (* J (* l 2.0))) (+ U (* J (* l -6.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2.05e+34) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U + (J * (l * -6.0));
	}
	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 (k <= 2.05d+34) then
        tmp = u + (j * (l * 2.0d0))
    else
        tmp = u + (j * (l * (-6.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2.05e+34) {
		tmp = U + (J * (l * 2.0));
	} else {
		tmp = U + (J * (l * -6.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 2.05e+34:
		tmp = U + (J * (l * 2.0))
	else:
		tmp = U + (J * (l * -6.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 2.05e+34)
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	else
		tmp = Float64(U + Float64(J * Float64(l * -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 2.05e+34)
		tmp = U + (J * (l * 2.0));
	else
		tmp = U + (J * (l * -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 2.05e+34], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 2.0499999999999999e34

   1. Initial program 87.7%

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

   3. Taylor expanded in l around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*r* 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in K around 0 54.6%

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

   if 2.0499999999999999e34 < K

   1. Initial program 91.5%

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

   3. Taylor expanded in l around 0 71.2%

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

      1. *-commutative 71.2%

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

      2. associate-*r* 71.2%

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

   5. Simplified 71.2%

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

   7. Applied egg-rr 54.8%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) \cdot -3\right)} + U \]
   8. Step-by-step derivation

      1. *-commutative 54.8%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} \cdot -3\right) + U \]

   9. Simplified 54.8%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) \cdot -3\right)} + U \]

   10. Taylor expanded in K around 0 55.8%

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

       1. *-commutative 55.8%

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

       2. associate-*l* 55.8%

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   12. Simplified 55.8%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 2.05 \cdot 10^{+34}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.5% accurate, 26.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;K \leq 2.05 \cdot 10^{+34}:\\ \;\;\;\;U + 10 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= K 2.05e+34) (+ U (* 10.0 (* l J))) (+ U (* J (* l -6.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2.05e+34) {
		tmp = U + (10.0 * (l * J));
	} else {
		tmp = U + (J * (l * -6.0));
	}
	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 (k <= 2.05d+34) then
        tmp = u + (10.0d0 * (l * j))
    else
        tmp = u + (j * (l * (-6.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (K <= 2.05e+34) {
		tmp = U + (10.0 * (l * J));
	} else {
		tmp = U + (J * (l * -6.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if K <= 2.05e+34:
		tmp = U + (10.0 * (l * J))
	else:
		tmp = U + (J * (l * -6.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (K <= 2.05e+34)
		tmp = Float64(U + Float64(10.0 * Float64(l * J)));
	else
		tmp = Float64(U + Float64(J * Float64(l * -6.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (K <= 2.05e+34)
		tmp = U + (10.0 * (l * J));
	else
		tmp = U + (J * (l * -6.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[K, 2.05e+34], N[(U + N[(10.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * -6.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if K < 2.0499999999999999e34

   1. Initial program 87.7%

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

   3. Taylor expanded in l around 0 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*r* 61.1%

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

   5. Simplified 61.1%

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

   7. Applied egg-rr 46.6%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
   8. Step-by-step derivation

      1. log1p-undefine 46.6%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

      2. rem-exp-log 46.6%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

      3. +-commutative 46.6%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

      4. associate--l+ 46.6%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

      5. *-commutative 46.6%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

      6. metadata-eval 46.6%

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

   9. Simplified 46.6%

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

   10. Taylor expanded in K around 0 46.6%

       \[\leadsto \color{blue}{10 \cdot \left(J \cdot \ell\right)} + U \]

   if 2.0499999999999999e34 < K

   1. Initial program 91.5%

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

   3. Taylor expanded in l around 0 71.2%

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

      1. *-commutative 71.2%

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

      2. associate-*r* 71.2%

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

   5. Simplified 71.2%

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

   7. Applied egg-rr 54.8%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) \cdot -3\right)} + U \]
   8. Step-by-step derivation

      1. *-commutative 54.8%

         \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} \cdot -3\right) + U \]

   9. Simplified 54.8%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(K \cdot -4\right) \cdot -3\right)} + U \]

   10. Taylor expanded in K around 0 55.8%

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

       1. *-commutative 55.8%

          \[\leadsto \color{blue}{\left(J \cdot \ell\right) \cdot -6} + U \]

       2. associate-*l* 55.8%

          \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]

   12. Simplified 55.8%

       \[\leadsto \color{blue}{J \cdot \left(\ell \cdot -6\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;K \leq 2.05 \cdot 10^{+34}:\\ \;\;\;\;U + 10 \cdot \left(\ell \cdot J\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot -6\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 47.6% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + 10 \cdot \left(\ell \cdot J\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* 10.0 (* l J))))

C

double code(double J, double l, double K, double U) {
	return U + (10.0 * (l * J));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (10.0d0 * (l * j))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (10.0 * (l * J));
}

Python

def code(J, l, K, U):
	return U + (10.0 * (l * J))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(10.0 * Float64(l * J)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (10.0 * (l * J));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(10.0 * N[(l * J), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 88.6%

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

3. Taylor expanded in l around 0 63.5%

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

   1. *-commutative 63.5%

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

   2. associate-*r* 63.5%

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

5. Simplified 63.5%

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

7. Applied egg-rr 47.4%

   \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(-4 \cdot K\right)\right)} - -3\right)} + U \]
8. Step-by-step derivation

   1. log1p-undefine 47.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(e^{\color{blue}{\log \left(1 + \cos \left(-4 \cdot K\right)\right)}} - -3\right) + U \]

   2. rem-exp-log 47.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{\left(1 + \cos \left(-4 \cdot K\right)\right)} - -3\right) + U \]

   3. +-commutative 47.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\color{blue}{\left(\cos \left(-4 \cdot K\right) + 1\right)} - -3\right) + U \]

   4. associate--l+ 47.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \color{blue}{\left(\cos \left(-4 \cdot K\right) + \left(1 - -3\right)\right)} + U \]

   5. *-commutative 47.4%

      \[\leadsto \left(\left(2 \cdot \ell\right) \cdot J\right) \cdot \left(\cos \color{blue}{\left(K \cdot -4\right)} + \left(1 - -3\right)\right) + U \]

   6. metadata-eval 47.4%

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

9. Simplified 47.4%

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

10. Taylor expanded in K around 0 47.4%

    \[\leadsto \color{blue}{10 \cdot \left(J \cdot \ell\right)} + U \]

11. Final simplification 47.4%

    \[\leadsto U + 10 \cdot \left(\ell \cdot J\right) \]
12. Add Preprocessing

Alternative 16: 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 88.6%

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

4. Applied egg-rr 27.2%

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

5. Taylor expanded in J around 0 36.4%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))