Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.3%

Time: 31.9s

Alternatives: 10

Speedup: 1.8×

Specification

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (/ (* K (+ m n)) 2.0) M))
  (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	return cos((((K * (m + n)) / 2.0) - M)) * exp((-pow((((m + n) / 2.0) - M), 2.0) - (l - fabs((m - n)))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos((((k * (m + n)) / 2.0d0) - m_1)) * exp((-((((m + n) / 2.0d0) - m_1) ** 2.0d0) - (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos((((K * (m + n)) / 2.0) - M)) * Math.exp((-Math.pow((((m + n) / 2.0) - M), 2.0) - (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	return math.cos((((K * (m + n)) / 2.0) - M)) * math.exp((-math.pow((((m + n) / 2.0) - M), 2.0) - (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(Float64(K * Float64(m + n)) / 2.0) - M)) * exp(Float64(Float64(-(Float64(Float64(Float64(m + n) / 2.0) - M) ^ 2.0)) - Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos((((K * (m + n)) / 2.0) - M)) * exp((-((((m + n) / 2.0) - M) ^ 2.0) - (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(N[(K * N[(m + n), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]) - N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right)\\ \log \left({\left(e^{e^{\left(n - m\right) + \left(\left(t\_0 - M\right) \cdot \left(M - t\_0\right) - \ell\right)}}\right)}^{\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right)}\right) \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ n m))))
   (log
    (pow
     (exp (exp (+ (- n m) (- (* (- t_0 M) (- M t_0)) l))))
     (cos (- (* 0.5 (* m K)) M))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	return log(pow(exp(exp(((n - m) + (((t_0 - M) * (M - t_0)) - l)))), cos(((0.5 * (m * K)) - M))));
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = 0.5d0 * (n + m)
    code = log((exp(exp(((n - m) + (((t_0 - m_1) * (m_1 - t_0)) - l)))) ** cos(((0.5d0 * (m * k)) - m_1))))
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	return Math.log(Math.pow(Math.exp(Math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l)))), Math.cos(((0.5 * (m * K)) - M))));
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = 0.5 * (n + m)
	return math.log(math.pow(math.exp(math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l)))), math.cos(((0.5 * (m * K)) - M))))

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(0.5 * Float64(n + m))
	return log((exp(exp(Float64(Float64(n - m) + Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) - l)))) ^ cos(Float64(Float64(0.5 * Float64(m * K)) - M))))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	t_0 = 0.5 * (n + m);
	tmp = log((exp(exp(((n - m) + (((t_0 - M) * (M - t_0)) - l)))) ^ cos(((0.5 * (m * K)) - M))));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision]}, N[Log[N[Power[N[Exp[N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], N[Cos[N[(N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 76.3%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-log-exp 76.3%

      \[\leadsto \color{blue}{\log \left(e^{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}}\right)} \]

   2. *-commutative 76.3%

      \[\leadsto \log \left(e^{\color{blue}{e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)}}\right) \]

   3. exp-prod 96.2%

      \[\leadsto \log \color{blue}{\left({\left(e^{e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}}\right)}^{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)}\right)} \]

6. Applied egg-rr 96.2%

   \[\leadsto \color{blue}{\log \left({\left(e^{e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right)} \]
7. Step-by-step derivation

   1. unpow2 96.2%

      \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)} + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

   2. +-commutative 96.2%

      \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

   3. +-commutative 96.2%

      \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

8. Applied egg-rr 96.2%

   \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\color{blue}{\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\left(n + m\right) \cdot 0.5 - M\right)} + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

9. Taylor expanded in n around 0 96.2%

   \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\left(n + m\right) \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\color{blue}{\cos \left(0.5 \cdot \left(K \cdot m\right) - M\right)}}\right) \]
10. Step-by-step derivation

    1. *-commutative 96.2%

       \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\left(n + m\right) \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(0.5 \cdot \color{blue}{\left(m \cdot K\right)} - M\right)}\right) \]

11. Simplified 96.2%

    \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\left(n + m\right) \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\color{blue}{\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right)}}\right) \]

12. Final simplification 96.2%

    \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) + \left(\left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(M - 0.5 \cdot \left(n + m\right)\right) - \ell\right)}}\right)}^{\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right)}\right) \]
13. Add Preprocessing

Alternative 2: 96.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- (fabs (- n m)) l) (pow (- (/ (+ n m) 2.0) M) 2.0)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp(((fabs((n - m)) - l) - pow((((n + m) / 2.0) - M), 2.0)));
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp(((abs((n - m)) - l) - ((((n + m) / 2.0d0) - m_1) ** 2.0d0)))
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp(((Math.abs((n - m)) - l) - Math.pow((((n + m) / 2.0) - M), 2.0)));
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.cos(M) * math.exp(((math.fabs((n - m)) - l) - math.pow((((n + m) / 2.0) - M), 2.0)))

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(Float64(abs(Float64(n - m)) - l) - (Float64(Float64(Float64(n + m) / 2.0) - M) ^ 2.0))))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(((abs((n - m)) - l) - ((((n + m) / 2.0) - M) ^ 2.0)));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0 96.7%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
6. Step-by-step derivation

   1. cos-neg 96.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

7. Simplified 96.7%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

8. Final simplification 96.7%

   \[\leadsto \cos M \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{n + m}{2} - M\right)}^{2}} \]
9. Add Preprocessing

Alternative 3: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-274}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right)}{e^{\left(m + \ell\right) + {\left(m \cdot 0.5 - M\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(m \cdot \left(0.5 \cdot K\right) - M\right) \cdot e^{n - \left(m + {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= m -5e+42)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m 2.8e-274)
     (/
      (cos (- (* 0.5 (* m K)) M))
      (exp (+ (+ m l) (pow (- (* m 0.5) M) 2.0))))
     (*
      (cos (- (* m (* 0.5 K)) M))
      (exp (- n (+ m (pow (- (* 0.5 (+ n m)) M) 2.0))))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -5e+42) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= 2.8e-274) {
		tmp = cos(((0.5 * (m * K)) - M)) / exp(((m + l) + pow(((m * 0.5) - M), 2.0)));
	} else {
		tmp = cos(((m * (0.5 * K)) - M)) * exp((n - (m + pow(((0.5 * (n + m)) - M), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-5d+42)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= 2.8d-274) then
        tmp = cos(((0.5d0 * (m * k)) - m_1)) / exp(((m + l) + (((m * 0.5d0) - m_1) ** 2.0d0)))
    else
        tmp = cos(((m * (0.5d0 * k)) - m_1)) * exp((n - (m + (((0.5d0 * (n + m)) - m_1) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -5e+42) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= 2.8e-274) {
		tmp = Math.cos(((0.5 * (m * K)) - M)) / Math.exp(((m + l) + Math.pow(((m * 0.5) - M), 2.0)));
	} else {
		tmp = Math.cos(((m * (0.5 * K)) - M)) * Math.exp((n - (m + Math.pow(((0.5 * (n + m)) - M), 2.0))));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -5e+42:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= 2.8e-274:
		tmp = math.cos(((0.5 * (m * K)) - M)) / math.exp(((m + l) + math.pow(((m * 0.5) - M), 2.0)))
	else:
		tmp = math.cos(((m * (0.5 * K)) - M)) * math.exp((n - (m + math.pow(((0.5 * (n + m)) - M), 2.0))))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -5e+42)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= 2.8e-274)
		tmp = Float64(cos(Float64(Float64(0.5 * Float64(m * K)) - M)) / exp(Float64(Float64(m + l) + (Float64(Float64(m * 0.5) - M) ^ 2.0))));
	else
		tmp = Float64(cos(Float64(Float64(m * Float64(0.5 * K)) - M)) * exp(Float64(n - Float64(m + (Float64(Float64(0.5 * Float64(n + m)) - M) ^ 2.0)))));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -5e+42)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= 2.8e-274)
		tmp = cos(((0.5 * (m * K)) - M)) / exp(((m + l) + (((m * 0.5) - M) ^ 2.0)));
	else
		tmp = cos(((m * (0.5 * K)) - M)) * exp((n - (m + (((0.5 * (n + m)) - M) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[m, -5e+42], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 2.8e-274], N[(N[Cos[N[(N[(0.5 * N[(m * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[(m + l), $MachinePrecision] + N[Power[N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[(m * N[(0.5 * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(n - N[(m + N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -5.00000000000000007e42

   1. Initial program 71.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 71.2%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 98.1%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   6. Step-by-step derivation

      1. cos-neg 98.1%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   7. Simplified 98.1%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   8. Taylor expanded in m around inf 98.1%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]

   if -5.00000000000000007e42 < m < 2.79999999999999975e-274

   1. Initial program 83.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 83.6%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. expm1-log1p-u 51.1%

         \[\leadsto \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(m + n\right) \cdot \frac{K}{2} - M\right)\right)\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      2. expm1-undefine 51.0%

         \[\leadsto \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\left(m + n\right) \cdot \frac{K}{2} - M\right)} - 1\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      3. div-inv 51.0%

         \[\leadsto \cos \left(e^{\mathsf{log1p}\left(\left(m + n\right) \cdot \color{blue}{\left(K \cdot \frac{1}{2}\right)} - M\right)} - 1\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      4. metadata-eval 51.0%

         \[\leadsto \cos \left(e^{\mathsf{log1p}\left(\left(m + n\right) \cdot \left(K \cdot \color{blue}{0.5}\right) - M\right)} - 1\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   6. Applied egg-rr 51.0%

      \[\leadsto \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)} - 1\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   7. Step-by-step derivation

      1. expm1-define 51.1%

         \[\leadsto \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)\right)\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      2. fma-neg 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(m + n, K \cdot 0.5, -M\right)}\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      3. *-commutative 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(m + n, \color{blue}{0.5 \cdot K}, -M\right)\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      4. fma-define 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(m + n\right) \cdot \left(0.5 \cdot K\right) + \left(-M\right)}\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      5. *-commutative 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot K\right) \cdot \left(m + n\right)} + \left(-M\right)\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      6. fma-define 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(0.5 \cdot K, m + n, -M\right)}\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      7. +-commutative 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5 \cdot K, \color{blue}{n + m}, -M\right)\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      8. remove-double-neg 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5 \cdot K, n + \color{blue}{\left(-\left(-m\right)\right)}, -M\right)\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      9. mul-1-neg 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5 \cdot K, n + \left(-\color{blue}{-1 \cdot m}\right), -M\right)\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      10. sub-neg 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(0.5 \cdot K, \color{blue}{n - -1 \cdot m}, -M\right)\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      11. fma-neg 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(0.5 \cdot K\right) \cdot \left(n - -1 \cdot m\right) - M}\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      12. cancel-sign-sub-inv 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot K\right) \cdot \color{blue}{\left(n + \left(--1\right) \cdot m\right)} - M\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      13. metadata-eval 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot K\right) \cdot \left(n + \color{blue}{1} \cdot m\right) - M\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      14. *-lft-identity 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot K\right) \cdot \left(n + \color{blue}{m}\right) - M\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      15. +-commutative 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot K\right) \cdot \color{blue}{\left(m + n\right)} - M\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      16. *-commutative 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(m + n\right) \cdot \left(0.5 \cdot K\right)} - M\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      17. +-commutative 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(n + m\right)} \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   8. Simplified 51.1%

      \[\leadsto \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   9. Step-by-step derivation

      1. associate--l- 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot e^{\color{blue}{\left|n - m\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \]

      2. div-inv 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot e^{\left|n - m\right| - \left(\ell + {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}\right)} \]

      3. metadata-eval 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot e^{\left|n - m\right| - \left(\ell + {\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2}\right)} \]

      4. +-commutative 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot e^{\left|n - m\right| - \color{blue}{\left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)}} \]

      5. exp-diff 30.8%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot \color{blue}{\frac{e^{\left|n - m\right|}}{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell}}} \]

      6. add-sqr-sqrt 20.4%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot \frac{e^{\left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right|}}{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell}} \]

      7. fabs-sqr 20.4%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot \frac{e^{\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}}}{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell}} \]

      8. add-sqr-sqrt 41.0%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot \frac{e^{\color{blue}{n - m}}}{e^{{\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell}} \]

      9. exp-diff 51.1%

         \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot \color{blue}{e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)}} \]

      10. associate--l- 51.1%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot e^{\color{blue}{n - \left(m + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)\right)}} \]

      11. exp-diff 44.3%

          \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot \color{blue}{\frac{e^{n}}{e^{m + \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)}}} \]

   10. Applied egg-rr 44.3%

       \[\leadsto \cos \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\right)\right) \cdot \color{blue}{\frac{e^{n}}{e^{m + \left(\ell + {\left(\left(n + m\right) \cdot 0.5 - M\right)}^{2}\right)}}} \]

   11. Taylor expanded in n around 0 81.8%

       \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot \left(K \cdot m\right) - M\right)}{e^{\ell + \left(m + {\left(0.5 \cdot m - M\right)}^{2}\right)}}} \]
   12. Step-by-step derivation

       1. associate-+r+ 81.8%

          \[\leadsto \frac{\cos \left(0.5 \cdot \left(K \cdot m\right) - M\right)}{e^{\color{blue}{\left(\ell + m\right) + {\left(0.5 \cdot m - M\right)}^{2}}}} \]

       2. +-commutative 81.8%

          \[\leadsto \frac{\cos \left(0.5 \cdot \left(K \cdot m\right) - M\right)}{e^{\color{blue}{\left(m + \ell\right)} + {\left(0.5 \cdot m - M\right)}^{2}}} \]

   13. Simplified 81.8%

       \[\leadsto \color{blue}{\frac{\cos \left(0.5 \cdot \left(K \cdot m\right) - M\right)}{e^{\left(m + \ell\right) + {\left(0.5 \cdot m - M\right)}^{2}}}} \]

   if 2.79999999999999975e-274 < m

   1. Initial program 73.1%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 73.1%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 73.1%

         \[\leadsto \color{blue}{e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)} \]

      2. associate--l- 73.1%

         \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      3. add-sqr-sqrt 23.5%

         \[\leadsto e^{\left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      4. fabs-sqr 23.5%

         \[\leadsto e^{\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}} - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      5. add-sqr-sqrt 73.1%

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

      6. +-commutative 73.1%

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

      7. div-inv 73.1%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      8. metadata-eval 73.1%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

   6. Applied egg-rr 73.1%

      \[\leadsto \color{blue}{e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)} \]

   7. Taylor expanded in l around 0 65.5%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot \left(m + n\right)\right) - M\right) \cdot e^{n - \left(m + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)}} \]

   8. Taylor expanded in m around inf 74.2%

      \[\leadsto \cos \left(\color{blue}{0.5 \cdot \left(K \cdot m\right)} - M\right) \cdot e^{n - \left(m + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 74.2%

         \[\leadsto \cos \left(\color{blue}{\left(0.5 \cdot K\right) \cdot m} - M\right) \cdot e^{n - \left(m + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \]

   10. Simplified 74.2%

       \[\leadsto \cos \left(\color{blue}{\left(0.5 \cdot K\right) \cdot m} - M\right) \cdot e^{n - \left(m + {\left(0.5 \cdot \left(m + n\right) - M\right)}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+42}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 2.8 \cdot 10^{-274}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(m \cdot K\right) - M\right)}{e^{\left(m + \ell\right) + {\left(m \cdot 0.5 - M\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(m \cdot \left(0.5 \cdot K\right) - M\right) \cdot e^{n - \left(m + {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \cos M \cdot {e}^{\left(\left(\left(n - m\right) - \ell\right) - {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}\right)} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (pow E (- (- (- n m) l) (pow (- (* 0.5 (+ n m)) M) 2.0)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return cos(M) * pow(((double) M_E), (((n - m) - l) - pow(((0.5 * (n + m)) - M), 2.0)));
}

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.pow(Math.E, (((n - m) - l) - Math.pow(((0.5 * (n + m)) - M), 2.0)));
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.cos(M) * math.pow(math.e, (((n - m) - l) - math.pow(((0.5 * (n + m)) - M), 2.0)))

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(cos(M) * (exp(1) ^ Float64(Float64(Float64(n - m) - l) - (Float64(Float64(0.5 * Float64(n + m)) - M) ^ 2.0))))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = cos(M) * (2.71828182845904523536 ^ (((n - m) - l) - (((0.5 * (n + m)) - M) ^ 2.0)));
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Power[E, N[(N[(N[(n - m), $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0 96.7%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
6. Step-by-step derivation

   1. cos-neg 96.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

7. Simplified 96.7%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
8. Step-by-step derivation

   1. *-un-lft-identity 96.7%

      \[\leadsto \cos M \cdot e^{\color{blue}{1 \cdot \left(\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \]

   2. exp-prod 96.7%

      \[\leadsto \cos M \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \]

   3. add-sqr-sqrt 52.3%

      \[\leadsto \cos M \cdot {\left(e^{1}\right)}^{\left(\left(\left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \]

   4. fabs-sqr 52.3%

      \[\leadsto \cos M \cdot {\left(e^{1}\right)}^{\left(\left(\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \]

   5. add-sqr-sqrt 96.7%

      \[\leadsto \cos M \cdot {\left(e^{1}\right)}^{\left(\left(\color{blue}{\left(n - m\right)} - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \]

   6. div-inv 96.7%

      \[\leadsto \cos M \cdot {\left(e^{1}\right)}^{\left(\left(\left(n - m\right) - \ell\right) - {\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2}\right)} \]

   7. metadata-eval 96.7%

      \[\leadsto \cos M \cdot {\left(e^{1}\right)}^{\left(\left(\left(n - m\right) - \ell\right) - {\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2}\right)} \]

   8. +-commutative 96.7%

      \[\leadsto \cos M \cdot {\left(e^{1}\right)}^{\left(\left(\left(n - m\right) - \ell\right) - {\left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right)}^{2}\right)} \]

9. Applied egg-rr 96.7%

   \[\leadsto \cos M \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\left(\left(n - m\right) - \ell\right) - {\left(\left(n + m\right) \cdot 0.5 - M\right)}^{2}\right)}} \]

10. Final simplification 96.7%

    \[\leadsto \cos M \cdot {e}^{\left(\left(\left(n - m\right) - \ell\right) - {\left(0.5 \cdot \left(n + m\right) - M\right)}^{2}\right)} \]
11. Add Preprocessing

Alternative 5: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right)\\ \mathbf{if}\;m \leq -5 \cdot 10^{+41} \lor \neg \left(m \leq 54\right):\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(t\_0 - M\right) \cdot \left(M - t\_0\right) - \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ n m))))
   (if (or (<= m -5e+41) (not (<= m 54.0)))
     (* (cos M) (exp (* -0.25 (pow m 2.0))))
     (*
      (exp (+ (- n m) (- (* (- t_0 M) (- M t_0)) l)))
      (cos (- (* (+ n m) (* 0.5 K)) M))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	double tmp;
	if ((m <= -5e+41) || !(m <= 54.0)) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos((((n + m) * (0.5 * K)) - M));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (n + m)
    if ((m <= (-5d+41)) .or. (.not. (m <= 54.0d0))) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = exp(((n - m) + (((t_0 - m_1) * (m_1 - t_0)) - l))) * cos((((n + m) * (0.5d0 * k)) - m_1))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	double tmp;
	if ((m <= -5e+41) || !(m <= 54.0)) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * Math.cos((((n + m) * (0.5 * K)) - M));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = 0.5 * (n + m)
	tmp = 0
	if (m <= -5e+41) or not (m <= 54.0):
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * math.cos((((n + m) * (0.5 * K)) - M))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(0.5 * Float64(n + m))
	tmp = 0.0
	if ((m <= -5e+41) || !(m <= 54.0))
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(exp(Float64(Float64(n - m) + Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) - l))) * cos(Float64(Float64(Float64(n + m) * Float64(0.5 * K)) - M)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = 0.5 * (n + m);
	tmp = 0.0;
	if ((m <= -5e+41) || ~((m <= 54.0)))
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos((((n + m) * (0.5 * K)) - M));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[m, -5e+41], N[Not[LessEqual[m, 54.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(n + m), $MachinePrecision] * N[(0.5 * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -5.00000000000000022e41 or 54 < m

   1. Initial program 69.2%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 98.3%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   6. Step-by-step derivation

      1. cos-neg 98.3%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   7. Simplified 98.3%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   8. Taylor expanded in m around inf 95.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]

   if -5.00000000000000022e41 < m < 54

   1. Initial program 82.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 82.3%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 82.3%

         \[\leadsto \color{blue}{e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)} \]

      2. associate--l- 82.3%

         \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      3. add-sqr-sqrt 46.3%

         \[\leadsto e^{\left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      4. fabs-sqr 46.3%

         \[\leadsto e^{\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}} - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      5. add-sqr-sqrt 82.3%

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

      6. +-commutative 82.3%

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

      7. div-inv 82.3%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      8. metadata-eval 82.3%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

   6. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)} \]
   7. Step-by-step derivation

      1. unpow2 96.0%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)} + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

      2. +-commutative 96.0%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

      3. +-commutative 96.0%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

   8. Applied egg-rr 82.3%

      \[\leadsto e^{\left(n - m\right) - \left(\color{blue}{\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\left(n + m\right) \cdot 0.5 - M\right)} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5 \cdot 10^{+41} \lor \neg \left(m \leq 54\right):\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(M - 0.5 \cdot \left(n + m\right)\right) - \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right)\\ \mathbf{if}\;n \leq -8 \cdot 10^{-94}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 1200:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(t\_0 - M\right) \cdot \left(M - t\_0\right) - \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ n m))))
   (if (<= n -8e-94)
     (* (cos M) (exp (* -0.25 (pow m 2.0))))
     (if (<= n 1200.0)
       (*
        (exp (+ (- n m) (- (* (- t_0 M) (- M t_0)) l)))
        (cos (- (* (+ n m) (* 0.5 K)) M)))
       (* (cos M) (exp (* -0.25 (pow n 2.0))))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	double tmp;
	if (n <= -8e-94) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (n <= 1200.0) {
		tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos((((n + m) * (0.5 * K)) - M));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (n + m)
    if (n <= (-8d-94)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (n <= 1200.0d0) then
        tmp = exp(((n - m) + (((t_0 - m_1) * (m_1 - t_0)) - l))) * cos((((n + m) * (0.5d0 * k)) - m_1))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	double tmp;
	if (n <= -8e-94) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (n <= 1200.0) {
		tmp = Math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * Math.cos((((n + m) * (0.5 * K)) - M));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = 0.5 * (n + m)
	tmp = 0
	if n <= -8e-94:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif n <= 1200.0:
		tmp = math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * math.cos((((n + m) * (0.5 * K)) - M))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(0.5 * Float64(n + m))
	tmp = 0.0
	if (n <= -8e-94)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (n <= 1200.0)
		tmp = Float64(exp(Float64(Float64(n - m) + Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) - l))) * cos(Float64(Float64(Float64(n + m) * Float64(0.5 * K)) - M)));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = 0.5 * (n + m);
	tmp = 0.0;
	if (n <= -8e-94)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (n <= 1200.0)
		tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos((((n + m) * (0.5 * K)) - M));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -8e-94], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1200.0], N[(N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(n + m), $MachinePrecision] * N[(0.5 * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -7.9999999999999996e-94

   1. Initial program 76.6%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 97.8%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   6. Step-by-step derivation

      1. cos-neg 97.8%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   7. Simplified 97.8%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   8. Taylor expanded in m around inf 52.4%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]

   if -7.9999999999999996e-94 < n < 1200

   1. Initial program 83.3%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 83.3%

         \[\leadsto \color{blue}{e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)} \]

      2. associate--l- 83.3%

         \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      3. add-sqr-sqrt 51.8%

         \[\leadsto e^{\left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      4. fabs-sqr 51.8%

         \[\leadsto e^{\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}} - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      5. add-sqr-sqrt 83.3%

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

      6. +-commutative 83.3%

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

      7. div-inv 83.3%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      8. metadata-eval 83.3%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

   6. Applied egg-rr 83.3%

      \[\leadsto \color{blue}{e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)} \]
   7. Step-by-step derivation

      1. unpow2 92.7%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)} + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

      2. +-commutative 92.7%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

      3. +-commutative 92.7%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

   8. Applied egg-rr 83.3%

      \[\leadsto e^{\left(n - m\right) - \left(\color{blue}{\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\left(n + m\right) \cdot 0.5 - M\right)} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \]

   if 1200 < n

   1. Initial program 65.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   6. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   8. Taylor expanded in n around inf 98.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8 \cdot 10^{-94}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 1200:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(M - 0.5 \cdot \left(n + m\right)\right) - \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right)\\ \mathbf{if}\;M \leq -8.4 \cdot 10^{+18} \lor \neg \left(M \leq 1.4 \cdot 10^{+41}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(t\_0 - M\right) \cdot \left(M - t\_0\right) - \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ n m))))
   (if (or (<= M -8.4e+18) (not (<= M 1.4e+41)))
     (* (cos M) (exp (- (pow M 2.0))))
     (*
      (exp (+ (- n m) (- (* (- t_0 M) (- M t_0)) l)))
      (cos (- (* (+ n m) (* 0.5 K)) M))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	double tmp;
	if ((M <= -8.4e+18) || !(M <= 1.4e+41)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos((((n + m) * (0.5 * K)) - M));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (n + m)
    if ((m_1 <= (-8.4d+18)) .or. (.not. (m_1 <= 1.4d+41))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((n - m) + (((t_0 - m_1) * (m_1 - t_0)) - l))) * cos((((n + m) * (0.5d0 * k)) - m_1))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	double tmp;
	if ((M <= -8.4e+18) || !(M <= 1.4e+41)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * Math.cos((((n + m) * (0.5 * K)) - M));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = 0.5 * (n + m)
	tmp = 0
	if (M <= -8.4e+18) or not (M <= 1.4e+41):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * math.cos((((n + m) * (0.5 * K)) - M))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(0.5 * Float64(n + m))
	tmp = 0.0
	if ((M <= -8.4e+18) || !(M <= 1.4e+41))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(exp(Float64(Float64(n - m) + Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) - l))) * cos(Float64(Float64(Float64(n + m) * Float64(0.5 * K)) - M)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = 0.5 * (n + m);
	tmp = 0.0;
	if ((M <= -8.4e+18) || ~((M <= 1.4e+41)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos((((n + m) * (0.5 * K)) - M));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[M, -8.4e+18], N[Not[LessEqual[M, 1.4e+41]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(n + m), $MachinePrecision] * N[(0.5 * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -8.4e18 or 1.4e41 < M

   1. Initial program 78.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   6. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   8. Taylor expanded in M around inf 99.3%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 99.3%

         \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   10. Simplified 99.3%

       \[\leadsto \cos M \cdot e^{\color{blue}{-{M}^{2}}} \]

   if -8.4e18 < M < 1.4e41

   1. Initial program 73.7%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 73.7%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 73.7%

         \[\leadsto \color{blue}{e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)} \]

      2. associate--l- 73.7%

         \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      3. add-sqr-sqrt 35.8%

         \[\leadsto e^{\left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      4. fabs-sqr 35.8%

         \[\leadsto e^{\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}} - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      5. add-sqr-sqrt 73.7%

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

      6. +-commutative 73.7%

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

      7. div-inv 73.7%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      8. metadata-eval 73.7%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

   6. Applied egg-rr 73.7%

      \[\leadsto \color{blue}{e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)} \]
   7. Step-by-step derivation

      1. unpow2 92.2%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)} + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

      2. +-commutative 92.2%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

      3. +-commutative 92.2%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

   8. Applied egg-rr 73.7%

      \[\leadsto e^{\left(n - m\right) - \left(\color{blue}{\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\left(n + m\right) \cdot 0.5 - M\right)} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -8.4 \cdot 10^{+18} \lor \neg \left(M \leq 1.4 \cdot 10^{+41}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(M - 0.5 \cdot \left(n + m\right)\right) - \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := 0.5 \cdot \left(n + m\right)\\ \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(t\_0 - M\right) \cdot \left(M - t\_0\right) - \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* 0.5 (+ n m))))
   (if (<= l 740.0)
     (*
      (exp (+ (- n m) (- (* (- t_0 M) (- M t_0)) l)))
      (cos (- (* (+ n m) (* 0.5 K)) M)))
     (exp (- l)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	double tmp;
	if (l <= 740.0) {
		tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos((((n + m) * (0.5 * K)) - M));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (n + m)
    if (l <= 740.0d0) then
        tmp = exp(((n - m) + (((t_0 - m_1) * (m_1 - t_0)) - l))) * cos((((n + m) * (0.5d0 * k)) - m_1))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = 0.5 * (n + m);
	double tmp;
	if (l <= 740.0) {
		tmp = Math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * Math.cos((((n + m) * (0.5 * K)) - M));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = 0.5 * (n + m)
	tmp = 0
	if l <= 740.0:
		tmp = math.exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * math.cos((((n + m) * (0.5 * K)) - M))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(0.5 * Float64(n + m))
	tmp = 0.0
	if (l <= 740.0)
		tmp = Float64(exp(Float64(Float64(n - m) + Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) - l))) * cos(Float64(Float64(Float64(n + m) * Float64(0.5 * K)) - M)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = 0.5 * (n + m);
	tmp = 0.0;
	if (l <= 740.0)
		tmp = exp(((n - m) + (((t_0 - M) * (M - t_0)) - l))) * cos((((n + m) * (0.5 * K)) - M));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, 740.0], N[(N[Exp[N[(N[(n - m), $MachinePrecision] + N[(N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(N[(n + m), $MachinePrecision] * N[(0.5 * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 740

   1. Initial program 76.5%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 76.5%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 76.5%

         \[\leadsto \color{blue}{e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)} \]

      2. associate--l- 76.5%

         \[\leadsto e^{\color{blue}{\left|n - m\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)}} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      3. add-sqr-sqrt 44.5%

         \[\leadsto e^{\left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right| - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      4. fabs-sqr 44.5%

         \[\leadsto e^{\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}} - \left(\ell + {\left(\frac{m + n}{2} - M\right)}^{2}\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      5. add-sqr-sqrt 76.5%

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

      6. +-commutative 76.5%

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

      7. div-inv 76.5%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\color{blue}{\left(m + n\right) \cdot \frac{1}{2}} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

      8. metadata-eval 76.5%

         \[\leadsto e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot \color{blue}{0.5} - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \]

   6. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{e^{\left(n - m\right) - \left({\left(\left(m + n\right) \cdot 0.5 - M\right)}^{2} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)} \]
   7. Step-by-step derivation

      1. unpow2 95.0%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\color{blue}{\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right)} + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

      2. +-commutative 95.0%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) \cdot \left(\left(m + n\right) \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

      3. +-commutative 95.0%

         \[\leadsto \log \left({\left(e^{e^{\left(n - m\right) - \left(\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\color{blue}{\left(n + m\right)} \cdot 0.5 - M\right) + \ell\right)}}\right)}^{\cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right)}\right) \]

   8. Applied egg-rr 76.5%

      \[\leadsto e^{\left(n - m\right) - \left(\color{blue}{\left(\left(n + m\right) \cdot 0.5 - M\right) \cdot \left(\left(n + m\right) \cdot 0.5 - M\right)} + \ell\right)} \cdot \cos \left(\left(m + n\right) \cdot \left(K \cdot 0.5\right) - M\right) \]

   if 740 < l

   1. Initial program 75.8%

      \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
   4. Add Preprocessing

   5. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
   6. Step-by-step derivation

      1. cos-neg 100.0%

         \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   8. Taylor expanded in l around inf 100.0%

      \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   10. Simplified 100.0%

       \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

   11. Taylor expanded in M around 0 100.0%

       \[\leadsto \color{blue}{e^{-\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 740:\\ \;\;\;\;e^{\left(n - m\right) + \left(\left(0.5 \cdot \left(n + m\right) - M\right) \cdot \left(M - 0.5 \cdot \left(n + m\right)\right) - \ell\right)} \cdot \cos \left(\left(n + m\right) \cdot \left(0.5 \cdot K\right) - M\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ e^{-\ell} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 (exp (- l)))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return exp(-l);
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp(-l)
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(-l);
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.exp(-l)

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return exp(Float64(-l))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp(-l);
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[Exp[(-l)], $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0 96.7%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
6. Step-by-step derivation

   1. cos-neg 96.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

7. Simplified 96.7%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

8. Taylor expanded in l around inf 34.9%

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
9. Step-by-step derivation

   1. neg-mul-1 34.9%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

10. Simplified 34.9%

    \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

11. Taylor expanded in M around 0 35.3%

    \[\leadsto \color{blue}{e^{-\ell}} \]
12. Add Preprocessing

Alternative 10: 7.1% accurate, 4.2× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \cos M \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 (cos M))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return cos(M);
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1)
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M);
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.cos(M)

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return cos(M)
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = cos(M);
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}}} \]
4. Add Preprocessing

5. Taylor expanded in K around 0 96.7%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]
6. Step-by-step derivation

   1. cos-neg 96.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

7. Simplified 96.7%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

8. Taylor expanded in l around inf 34.9%

   \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
9. Step-by-step derivation

   1. neg-mul-1 34.9%

      \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

10. Simplified 34.9%

    \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

11. Taylor expanded in l around 0 6.4%

    \[\leadsto \color{blue}{\cos M} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024095 
(FPCore (K m n M l)
  :name "Maksimov and Kolovsky, Equation (32)"
  :precision binary64
  (* (cos (- (/ (* K (+ m n)) 2.0) M)) (exp (- (- (pow (- (/ (+ m n) 2.0) M) 2.0)) (- l (fabs (- m n)))))))