Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.5% → 95.9%

Time: 16.7s

Alternatives: 16

Speedup: 3.9×

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: 75.5% 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: 95.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;n \leq 20:\\ \;\;\;\;\cos M \cdot e^{\left(t_0 - \ell\right) + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \left(n \cdot n\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= n 20.0)
     (*
      (cos M)
      (exp
       (+
        (- t_0 l)
        (- (* (+ n (- (* m 0.5) M)) (- M (* m 0.5))) (* (* n n) 0.25)))))
     (exp (- t_0 (+ l (* 0.25 (pow (+ m n) 2.0))))))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if (n <= 20.0) {
		tmp = cos(M) * exp(((t_0 - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	} else {
		tmp = exp((t_0 - (l + (0.25 * pow((m + n), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 = abs((m - n))
    if (n <= 20.0d0) then
        tmp = cos(m_1) * exp(((t_0 - l) + (((n + ((m * 0.5d0) - m_1)) * (m_1 - (m * 0.5d0))) - ((n * n) * 0.25d0))))
    else
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n));
	double tmp;
	if (n <= 20.0) {
		tmp = Math.cos(M) * Math.exp(((t_0 - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	} else {
		tmp = Math.exp((t_0 - (l + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if n <= 20.0:
		tmp = math.cos(M) * math.exp(((t_0 - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))))
	else:
		tmp = math.exp((t_0 - (l + (0.25 * math.pow((m + n), 2.0)))))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (n <= 20.0)
		tmp = Float64(cos(M) * exp(Float64(Float64(t_0 - l) + Float64(Float64(Float64(n + Float64(Float64(m * 0.5) - M)) * Float64(M - Float64(m * 0.5))) - Float64(Float64(n * n) * 0.25)))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if (n <= 20.0)
		tmp = cos(M) * exp(((t_0 - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	else
		tmp = exp((t_0 - (l + (0.25 * ((m + n) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 20.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] + N[(N[(N[(n + N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 20

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r/ 78.3%

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

      3. associate--r- 78.3%

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

      4. +-commutative 78.3%

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

      5. associate-+r- 78.3%

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

      6. unsub-neg 78.3%

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

      7. associate--r+ 78.3%

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

      8. +-commutative 78.3%

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

      9. associate--r+ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in K around 0 96.3%

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

      1. cos-neg 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in n around 0 89.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 89.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 89.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 94.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 94.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 94.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 94.7%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   if 20 < n

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-*r/ 63.4%

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

      3. associate--r- 63.4%

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

      4. +-commutative 63.4%

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

      5. associate-+r- 63.4%

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

      6. unsub-neg 63.4%

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

      7. associate--r+ 63.4%

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

      8. +-commutative 63.4%

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

      9. associate--r+ 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in K around 0 74.6%

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

      1. cos-neg 74.6%

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

      2. associate-*r* 74.6%

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

      3. sin-neg 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 20:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \left(n \cdot n\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]

Alternative 2: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: m and n 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((m - n)) - l) - ((((m + n) / 2.0d0) - m_1) ** 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: m and n 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[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.1%

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

   1. *-commutative 74.1%

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

   2. associate-*r/ 74.1%

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

   3. associate--r- 74.1%

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

   4. +-commutative 74.1%

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

   5. associate-+r- 74.1%

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

   6. unsub-neg 74.1%

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

   7. associate--r+ 74.1%

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

   8. +-commutative 74.1%

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

   9. associate--r+ 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

6. Simplified 97.3%

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

7. Final simplification 97.3%

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

Alternative 3: 95.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} t_0 := \left|m - n\right|\\ \mathbf{if}\;n \leq 10000:\\ \;\;\;\;e^{\left(t_0 - \ell\right) + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \left(n \cdot n\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{t_0 - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (fabs (- m n))))
   (if (<= n 10000.0)
     (exp
      (+
       (- t_0 l)
       (- (* (+ n (- (* m 0.5) M)) (- M (* m 0.5))) (* (* n n) 0.25))))
     (exp (- t_0 (+ l (* 0.25 (pow (+ m n) 2.0))))))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double t_0 = fabs((m - n));
	double tmp;
	if (n <= 10000.0) {
		tmp = exp(((t_0 - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	} else {
		tmp = exp((t_0 - (l + (0.25 * pow((m + n), 2.0)))));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 = abs((m - n))
    if (n <= 10000.0d0) then
        tmp = exp(((t_0 - l) + (((n + ((m * 0.5d0) - m_1)) * (m_1 - (m * 0.5d0))) - ((n * n) * 0.25d0))))
    else
        tmp = exp((t_0 - (l + (0.25d0 * ((m + n) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.abs((m - n));
	double tmp;
	if (n <= 10000.0) {
		tmp = Math.exp(((t_0 - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	} else {
		tmp = Math.exp((t_0 - (l + (0.25 * Math.pow((m + n), 2.0)))));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	t_0 = math.fabs((m - n))
	tmp = 0
	if n <= 10000.0:
		tmp = math.exp(((t_0 - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))))
	else:
		tmp = math.exp((t_0 - (l + (0.25 * math.pow((m + n), 2.0)))))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	t_0 = abs(Float64(m - n))
	tmp = 0.0
	if (n <= 10000.0)
		tmp = exp(Float64(Float64(t_0 - l) + Float64(Float64(Float64(n + Float64(Float64(m * 0.5) - M)) * Float64(M - Float64(m * 0.5))) - Float64(Float64(n * n) * 0.25))));
	else
		tmp = exp(Float64(t_0 - Float64(l + Float64(0.25 * (Float64(m + n) ^ 2.0)))));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = abs((m - n));
	tmp = 0.0;
	if (n <= 10000.0)
		tmp = exp(((t_0 - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	else
		tmp = exp((t_0 - (l + (0.25 * ((m + n) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 10000.0], N[Exp[N[(N[(t$95$0 - l), $MachinePrecision] + N[(N[(N[(n + N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(t$95$0 - N[(l + N[(0.25 * N[Power[N[(m + n), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 1e4

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r/ 78.3%

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

      3. associate--r- 78.3%

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

      4. +-commutative 78.3%

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

      5. associate-+r- 78.3%

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

      6. unsub-neg 78.3%

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

      7. associate--r+ 78.3%

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

      8. +-commutative 78.3%

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

      9. associate--r+ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in K around 0 96.3%

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

      1. cos-neg 96.3%

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

   6. Simplified 96.3%

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

   7. Taylor expanded in n around 0 89.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 89.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 89.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 94.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 94.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 94.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 94.7%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 93.0%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   if 1e4 < n

   1. Initial program 63.4%

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

      1. *-commutative 63.4%

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

      2. associate-*r/ 63.4%

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

      3. associate--r- 63.4%

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

      4. +-commutative 63.4%

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

      5. associate-+r- 63.4%

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

      6. unsub-neg 63.4%

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

      7. associate--r+ 63.4%

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

      8. +-commutative 63.4%

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

      9. associate--r+ 63.4%

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

   3. Simplified 63.4%

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

   4. Taylor expanded in K around 0 74.6%

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

      1. cos-neg 74.6%

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

      2. associate-*r* 74.6%

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

      3. sin-neg 74.6%

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

   6. Simplified 74.6%

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

   7. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(n + m\right)}^{2}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10000:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \left(n \cdot n\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}\\ \end{array} \]

Alternative 4: 96.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \left(n \cdot n\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2e+106)
   (exp
    (+
     (- (fabs (- m n)) l)
     (- (* (+ n (- (* m 0.5) M)) (- M (* m 0.5))) (* (* n n) 0.25))))
   (exp (* (* n n) -0.25))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2e+106) {
		tmp = exp(((fabs((m - n)) - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 (n <= 2d+106) then
        tmp = exp(((abs((m - n)) - l) + (((n + ((m * 0.5d0) - m_1)) * (m_1 - (m * 0.5d0))) - ((n * n) * 0.25d0))))
    else
        tmp = exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2e+106) {
		tmp = Math.exp(((Math.abs((m - n)) - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	tmp = 0
	if n <= 2e+106:
		tmp = math.exp(((math.fabs((m - n)) - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))))
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2e+106)
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) + Float64(Float64(Float64(n + Float64(Float64(m * 0.5) - M)) * Float64(M - Float64(m * 0.5))) - Float64(Float64(n * n) * 0.25))));
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2e+106)
		tmp = exp(((abs((m - n)) - l) + (((n + ((m * 0.5) - M)) * (M - (m * 0.5))) - ((n * n) * 0.25))));
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2e+106], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] + N[(N[(N[(n + N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.00000000000000018e106

   1. Initial program 79.1%

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

      1. *-commutative 79.1%

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

      2. associate-*r/ 79.1%

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

      3. associate--r- 79.1%

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

      4. +-commutative 79.1%

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

      5. associate-+r- 79.1%

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

      6. unsub-neg 79.1%

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

      7. associate--r+ 79.1%

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

      8. +-commutative 79.1%

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

      9. associate--r+ 79.1%

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

   3. Simplified 79.1%

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

   4. Taylor expanded in K around 0 96.6%

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

      1. cos-neg 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in n around 0 89.9%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 89.9%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 89.9%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 95.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 95.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 95.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 95.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 93.8%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   if 2.00000000000000018e106 < n

   1. Initial program 53.1%

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

      1. *-commutative 53.1%

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

      2. associate-*r/ 53.1%

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

      3. associate--r- 53.1%

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

      4. +-commutative 53.1%

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

      5. associate-+r- 53.1%

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

      6. unsub-neg 53.1%

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

      7. associate--r+ 53.1%

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

      8. +-commutative 53.1%

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

      9. associate--r+ 53.1%

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

   3. Simplified 53.1%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 67.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 67.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 67.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 75.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 75.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 75.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 75.5%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 75.5%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in n around inf 100.0%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 100.0%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

   13. Simplified 100.0%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+106}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) + \left(\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) - \left(n \cdot n\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 5: 93.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} t_0 := e^{M \cdot \left(-M\right)}\\ \mathbf{if}\;M \leq -2:\\ \;\;\;\;\cos M \cdot t_0\\ \mathbf{elif}\;M \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(0.5 \cdot \left(m \cdot \left(n + m \cdot 0.5\right)\right) + \left(n \cdot n\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* M (- M)))))
   (if (<= M -2.0)
     (* (cos M) t_0)
     (if (<= M 2.35e+21)
       (exp
        (-
         (- (fabs (- m n)) l)
         (+ (* 0.5 (* m (+ n (* m 0.5)))) (* (* n n) 0.25))))
       t_0))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((M * -M));
	double tmp;
	if (M <= -2.0) {
		tmp = cos(M) * t_0;
	} else if (M <= 2.35e+21) {
		tmp = exp(((fabs((m - n)) - l) - ((0.5 * (m * (n + (m * 0.5)))) + ((n * n) * 0.25))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: m and n 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 = exp((m_1 * -m_1))
    if (m_1 <= (-2.0d0)) then
        tmp = cos(m_1) * t_0
    else if (m_1 <= 2.35d+21) then
        tmp = exp(((abs((m - n)) - l) - ((0.5d0 * (m * (n + (m * 0.5d0)))) + ((n * n) * 0.25d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((M * -M));
	double tmp;
	if (M <= -2.0) {
		tmp = Math.cos(M) * t_0;
	} else if (M <= 2.35e+21) {
		tmp = Math.exp(((Math.abs((m - n)) - l) - ((0.5 * (m * (n + (m * 0.5)))) + ((n * n) * 0.25))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	t_0 = math.exp((M * -M))
	tmp = 0
	if M <= -2.0:
		tmp = math.cos(M) * t_0
	elif M <= 2.35e+21:
		tmp = math.exp(((math.fabs((m - n)) - l) - ((0.5 * (m * (n + (m * 0.5)))) + ((n * n) * 0.25))))
	else:
		tmp = t_0
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	t_0 = exp(Float64(M * Float64(-M)))
	tmp = 0.0
	if (M <= -2.0)
		tmp = Float64(cos(M) * t_0);
	elseif (M <= 2.35e+21)
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(Float64(0.5 * Float64(m * Float64(n + Float64(m * 0.5)))) + Float64(Float64(n * n) * 0.25))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((M * -M));
	tmp = 0.0;
	if (M <= -2.0)
		tmp = cos(M) * t_0;
	elseif (M <= 2.35e+21)
		tmp = exp(((abs((m - n)) - l) - ((0.5 * (m * (n + (m * 0.5)))) + ((n * n) * 0.25))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[M, -2.0], N[(N[Cos[M], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[M, 2.35e+21], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(0.5 * N[(m * N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if M < -2

   1. Initial program 84.5%

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

      1. *-commutative 84.5%

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

      2. associate-*r/ 84.5%

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

      3. associate--r- 84.5%

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

      4. +-commutative 84.5%

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

      5. associate-+r- 84.5%

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

      6. unsub-neg 84.5%

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

      7. associate--r+ 84.5%

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

      8. +-commutative 84.5%

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

      9. associate--r+ 84.5%

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

   3. Simplified 84.5%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in M around inf 96.6%

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

      1. mul-1-neg 96.6%

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

      2. unpow2 96.6%

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

      3. distribute-rgt-neg-in 96.6%

         \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   9. Simplified 96.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if -2 < M < 2.35e21

   1. Initial program 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-*r/ 71.6%

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

      3. associate--r- 71.6%

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

      4. +-commutative 71.6%

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

      5. associate-+r- 71.6%

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

      6. unsub-neg 71.6%

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

      7. associate--r+ 71.6%

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

      8. +-commutative 71.6%

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

      9. associate--r+ 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in K around 0 95.0%

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

      1. cos-neg 95.0%

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

   6. Simplified 95.0%

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

   7. Taylor expanded in n around 0 89.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 89.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 89.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 91.4%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 91.4%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 91.4%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 91.4%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 91.4%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in M around 0 91.4%

       \[\leadsto 1 \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{0.5 \cdot \left(\left(n + 0.5 \cdot m\right) \cdot m\right)} + \left(n \cdot n\right) \cdot 0.25\right)} \]

   if 2.35e21 < M

   1. Initial program 70.0%

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

      1. *-commutative 70.0%

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

      2. associate-*r/ 70.0%

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

      3. associate--r- 70.0%

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

      4. +-commutative 70.0%

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

      5. associate-+r- 70.0%

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

      6. unsub-neg 70.0%

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

      7. associate--r+ 70.0%

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

      8. +-commutative 70.0%

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

      9. associate--r+ 70.0%

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

   3. Simplified 70.0%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 80.0%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 80.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 80.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 86.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 86.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 86.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 86.7%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 85.0%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in M around inf 98.4%

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

       1. neg-mul-1 98.4%

          \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

       2. unpow2 98.4%

          \[\leadsto 1 \cdot e^{-\color{blue}{M \cdot M}} \]

       3. distribute-rgt-neg-in 98.4%

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   13. Simplified 98.4%

       \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(-M\right)}\\ \mathbf{elif}\;M \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(0.5 \cdot \left(m \cdot \left(n + m \cdot 0.5\right)\right) + \left(n \cdot n\right) \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]

Alternative 6: 89.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -80000000000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 3 \cdot 10^{-303}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(m - \left(n + \ell\right)\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= m -80000000000.0)
   (exp (* -0.25 (* m m)))
   (if (<= m 3e-303)
     (* (cos (- (* (+ m n) (/ K 2.0)) M)) (exp (- (- m (+ n l)) (* M M))))
     (exp (- (- (fabs (- m n)) l) (* (* n n) 0.25))))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -80000000000.0) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= 3e-303) {
		tmp = cos((((m + n) * (K / 2.0)) - M)) * exp(((m - (n + l)) - (M * M)));
	} else {
		tmp = exp(((fabs((m - n)) - l) - ((n * n) * 0.25)));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 <= (-80000000000.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= 3d-303) then
        tmp = cos((((m + n) * (k / 2.0d0)) - m_1)) * exp(((m - (n + l)) - (m_1 * m_1)))
    else
        tmp = exp(((abs((m - n)) - l) - ((n * n) * 0.25d0)))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -80000000000.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= 3e-303) {
		tmp = Math.cos((((m + n) * (K / 2.0)) - M)) * Math.exp(((m - (n + l)) - (M * M)));
	} else {
		tmp = Math.exp(((Math.abs((m - n)) - l) - ((n * n) * 0.25)));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -80000000000.0:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= 3e-303:
		tmp = math.cos((((m + n) * (K / 2.0)) - M)) * math.exp(((m - (n + l)) - (M * M)))
	else:
		tmp = math.exp(((math.fabs((m - n)) - l) - ((n * n) * 0.25)))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -80000000000.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= 3e-303)
		tmp = Float64(cos(Float64(Float64(Float64(m + n) * Float64(K / 2.0)) - M)) * exp(Float64(Float64(m - Float64(n + l)) - Float64(M * M))));
	else
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(Float64(n * n) * 0.25)));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -80000000000.0)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= 3e-303)
		tmp = cos((((m + n) * (K / 2.0)) - M)) * exp(((m - (n + l)) - (M * M)));
	else
		tmp = exp(((abs((m - n)) - l) - ((n * n) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[m, -80000000000.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, 3e-303], N[(N[Cos[N[(N[(N[(m + n), $MachinePrecision] * N[(K / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -8e10

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*r/ 62.3%

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

      3. associate--r- 62.3%

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

      4. +-commutative 62.3%

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

      5. associate-+r- 62.3%

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

      6. unsub-neg 62.3%

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

      7. associate--r+ 62.3%

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

      8. +-commutative 62.3%

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

      9. associate--r+ 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in n around 0 73.8%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 73.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 73.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 83.6%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 83.6%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 83.6%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 83.6%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 83.6%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in m around inf 91.9%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 91.9%

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 91.9%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   13. Simplified 91.9%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -8e10 < m < 3.00000000000000028e-303

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

      2. associate-*r/ 81.8%

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

      3. associate--r- 81.8%

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

      4. +-commutative 81.8%

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

      5. associate-+r- 81.8%

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

      6. unsub-neg 81.8%

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

      7. associate--r+ 81.8%

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

      8. +-commutative 81.8%

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

      9. associate--r+ 81.8%

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

   3. Simplified 81.8%

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

      1. exp-diff 37.3%

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

      2. add-sqr-sqrt 7.9%

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

      3. fabs-sqr 7.9%

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

      4. add-sqr-sqrt 55.1%

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

      5. add-sqr-sqrt 55.1%

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

      6. add-sqr-sqrt 55.1%

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

      7. div-inv 55.1%

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

      8. fma-neg 55.1%

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

      9. metadata-eval 55.1%

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

   5. Applied egg-rr 55.1%

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

      1. div-exp 80.5%

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

      2. associate--l- 80.5%

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

      3. fma-neg 80.5%

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

      4. *-commutative 80.5%

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

      5. +-commutative 80.5%

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

   7. Simplified 80.5%

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

   8. Taylor expanded in M around inf 69.6%

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

      1. unpow2 69.6%

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

   10. Simplified 69.6%

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

   if 3.00000000000000028e-303 < m

   1. Initial program 76.0%

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

      1. *-commutative 76.0%

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

      2. associate-*r/ 76.0%

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

      3. associate--r- 76.0%

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

      4. +-commutative 76.0%

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

      5. associate-+r- 76.0%

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

      6. unsub-neg 76.0%

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

      7. associate--r+ 76.0%

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

      8. +-commutative 76.0%

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

      9. associate--r+ 76.0%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in n around inf 50.9%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {n}^{2}}} \]
   5. Step-by-step derivation

      1. *-commutative 50.9%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{n}^{2} \cdot 0.25}} \]

      2. unpow2 50.9%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right)} \cdot 0.25} \]

   6. Simplified 50.9%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right) \cdot 0.25}} \]

   7. Taylor expanded in n around inf 50.2%

      \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(n \cdot K\right)\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25} \]

   8. Taylor expanded in n around 0 69.0%

      \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -80000000000:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq 3 \cdot 10^{-303}:\\ \;\;\;\;\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(m - \left(n + \ell\right)\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

Alternative 7: 84.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} t_0 := e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ t_1 := e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{-236}:\\ \;\;\;\;\cos M \cdot t_1\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-271}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-254}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq 14.2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- (- (fabs (- m n)) l) (* M M))))
        (t_1 (exp (* -0.25 (* m m)))))
   (if (<= n -2.9e-236)
     (* (cos M) t_1)
     (if (<= n 1.75e-271)
       t_0
       (if (<= n 4.8e-254)
         t_1
         (if (<= n 14.2) t_0 (exp (* (* n n) -0.25))))))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((fabs((m - n)) - l) - (M * M)));
	double t_1 = exp((-0.25 * (m * m)));
	double tmp;
	if (n <= -2.9e-236) {
		tmp = cos(M) * t_1;
	} else if (n <= 1.75e-271) {
		tmp = t_0;
	} else if (n <= 4.8e-254) {
		tmp = t_1;
	} else if (n <= 14.2) {
		tmp = t_0;
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	return tmp;
}

Fortran

NOTE: m and n 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) :: t_1
    real(8) :: tmp
    t_0 = exp(((abs((m - n)) - l) - (m_1 * m_1)))
    t_1 = exp(((-0.25d0) * (m * m)))
    if (n <= (-2.9d-236)) then
        tmp = cos(m_1) * t_1
    else if (n <= 1.75d-271) then
        tmp = t_0
    else if (n <= 4.8d-254) then
        tmp = t_1
    else if (n <= 14.2d0) then
        tmp = t_0
    else
        tmp = exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((Math.abs((m - n)) - l) - (M * M)));
	double t_1 = Math.exp((-0.25 * (m * m)));
	double tmp;
	if (n <= -2.9e-236) {
		tmp = Math.cos(M) * t_1;
	} else if (n <= 1.75e-271) {
		tmp = t_0;
	} else if (n <= 4.8e-254) {
		tmp = t_1;
	} else if (n <= 14.2) {
		tmp = t_0;
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	t_0 = math.exp(((math.fabs((m - n)) - l) - (M * M)))
	t_1 = math.exp((-0.25 * (m * m)))
	tmp = 0
	if n <= -2.9e-236:
		tmp = math.cos(M) * t_1
	elif n <= 1.75e-271:
		tmp = t_0
	elif n <= 4.8e-254:
		tmp = t_1
	elif n <= 14.2:
		tmp = t_0
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(M * M)))
	t_1 = exp(Float64(-0.25 * Float64(m * m)))
	tmp = 0.0
	if (n <= -2.9e-236)
		tmp = Float64(cos(M) * t_1);
	elseif (n <= 1.75e-271)
		tmp = t_0;
	elseif (n <= 4.8e-254)
		tmp = t_1;
	elseif (n <= 14.2)
		tmp = t_0;
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((abs((m - n)) - l) - (M * M)));
	t_1 = exp((-0.25 * (m * m)));
	tmp = 0.0;
	if (n <= -2.9e-236)
		tmp = cos(M) * t_1;
	elseif (n <= 1.75e-271)
		tmp = t_0;
	elseif (n <= 4.8e-254)
		tmp = t_1;
	elseif (n <= 14.2)
		tmp = t_0;
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(M * M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -2.9e-236], N[(N[Cos[M], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[n, 1.75e-271], t$95$0, If[LessEqual[n, 4.8e-254], t$95$1, If[LessEqual[n, 14.2], t$95$0, N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -2.9e-236

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. associate-*r/ 72.6%

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

      3. associate--r- 72.6%

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

      4. +-commutative 72.6%

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

      5. associate-+r- 72.6%

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

      6. unsub-neg 72.6%

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

      7. associate--r+ 72.6%

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

      8. +-commutative 72.6%

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

      9. associate--r+ 72.6%

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

   3. Simplified 72.6%

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

   4. Taylor expanded in K around 0 99.1%

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

      1. cos-neg 99.1%

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

   6. Simplified 99.1%

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

   7. Taylor expanded in m around inf 50.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 50.6%

         \[\leadsto \cos M \cdot e^{-0.25 \cdot \color{blue}{\left(m \cdot m\right)}} \]

   9. Simplified 50.6%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -2.9e-236 < n < 1.75e-271 or 4.80000000000000003e-254 < n < 14.199999999999999

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

      2. associate-*r/ 88.3%

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

      3. associate--r- 88.3%

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

      4. +-commutative 88.3%

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

      5. associate-+r- 88.3%

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

      6. unsub-neg 88.3%

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

      7. associate--r+ 88.3%

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

      8. +-commutative 88.3%

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

      9. associate--r+ 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in K around 0 93.2%

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

      1. cos-neg 93.2%

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

   6. Simplified 93.2%

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

   7. Taylor expanded in M around inf 72.9%

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

      1. unpow2 72.9%

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

   9. Simplified 72.9%

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

   10. Taylor expanded in M around 0 71.5%

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

   if 1.75e-271 < n < 4.80000000000000003e-254

   1. Initial program 66.7%

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

      1. *-commutative 66.7%

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

      2. associate-*r/ 66.7%

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

      3. associate--r- 66.7%

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

      4. +-commutative 66.7%

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

      5. associate-+r- 66.7%

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

      6. unsub-neg 66.7%

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

      7. associate--r+ 66.7%

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

      8. +-commutative 66.7%

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

      9. associate--r+ 66.7%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 100.0%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 100.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 100.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 100.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 100.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 100.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 100.0%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 100.0%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in m around inf 100.0%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 100.0%

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 100.0%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   13. Simplified 100.0%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if 14.199999999999999 < n

   1. Initial program 63.2%

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

      1. *-commutative 63.2%

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

      2. associate-*r/ 63.2%

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

      3. associate--r- 63.2%

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

      4. +-commutative 63.2%

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

      5. associate-+r- 63.2%

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

      6. unsub-neg 63.2%

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

      7. associate--r+ 63.2%

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

      8. +-commutative 63.2%

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

      9. associate--r+ 63.2%

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

   3. Simplified 63.2%

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

   4. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in n around 0 75.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 75.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 75.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 82.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 82.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 82.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 82.1%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 82.1%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in n around inf 94.7%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 94.7%

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 94.7%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

   13. Simplified 94.7%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{-236}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 1.75 \cdot 10^{-271}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-254}:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;n \leq 14.2:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - M \cdot M}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 8: 86.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -56:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= m -56.0)
   (exp (* -0.25 (* m m)))
   (exp (- (- (fabs (- m n)) l) (* (* n n) 0.25)))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -56.0) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp(((fabs((m - n)) - l) - ((n * n) * 0.25)));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 <= (-56.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp(((abs((m - n)) - l) - ((n * n) * 0.25d0)))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -56.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp(((Math.abs((m - n)) - l) - ((n * n) * 0.25)));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -56.0:
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp(((math.fabs((m - n)) - l) - ((n * n) * 0.25)))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -56.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(Float64(abs(Float64(m - n)) - l) - Float64(Float64(n * n) * 0.25)));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -56.0)
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp(((abs((m - n)) - l) - ((n * n) * 0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[m, -56.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(n * n), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -56

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*r/ 62.3%

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

      3. associate--r- 62.3%

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

      4. +-commutative 62.3%

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

      5. associate-+r- 62.3%

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

      6. unsub-neg 62.3%

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

      7. associate--r+ 62.3%

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

      8. +-commutative 62.3%

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

      9. associate--r+ 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in n around 0 74.8%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 74.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 74.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 84.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 84.2%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in m around inf 90.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 90.8%

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 90.8%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   13. Simplified 90.8%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -56 < m

   1. Initial program 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*r/ 78.1%

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

      3. associate--r- 78.1%

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

      4. +-commutative 78.1%

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

      5. associate-+r- 78.1%

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

      6. unsub-neg 78.1%

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

      7. associate--r+ 78.1%

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

      8. +-commutative 78.1%

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

      9. associate--r+ 78.1%

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

   3. Simplified 78.1%

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

   4. Taylor expanded in n around inf 55.3%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{0.25 \cdot {n}^{2}}} \]
   5. Step-by-step derivation

      1. *-commutative 55.3%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{{n}^{2} \cdot 0.25}} \]

      2. unpow2 55.3%

         \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right)} \cdot 0.25} \]

   6. Simplified 55.3%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(n \cdot n\right) \cdot 0.25}} \]

   7. Taylor expanded in n around inf 54.1%

      \[\leadsto \cos \color{blue}{\left(0.5 \cdot \left(n \cdot K\right)\right)} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25} \]

   8. Taylor expanded in n around 0 71.1%

      \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -56:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\left|m - n\right| - \ell\right) - \left(n \cdot n\right) \cdot 0.25}\\ \end{array} \]

Alternative 9: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -2.85 \cdot 10^{-130}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= m -53.0)
   (exp (* -0.25 (* m m)))
   (if (<= m -2.85e-130)
     (* (cos M) (exp (- l)))
     (* (cos M) (exp (* n (* n -0.25)))))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= -2.85e-130) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = cos(M) * exp((n * (n * -0.25)));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 <= (-53.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= (-2.85d-130)) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = cos(m_1) * exp((n * (n * (-0.25d0))))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= -2.85e-130) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.cos(M) * Math.exp((n * (n * -0.25)));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= -2.85e-130:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.cos(M) * math.exp((n * (n * -0.25)))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -53.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= -2.85e-130)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = Float64(cos(M) * exp(Float64(n * Float64(n * -0.25))));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -53.0)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= -2.85e-130)
		tmp = cos(M) * exp(-l);
	else
		tmp = cos(M) * exp((n * (n * -0.25)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[m, -53.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -2.85e-130], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(n * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*r/ 62.3%

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

      3. associate--r- 62.3%

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

      4. +-commutative 62.3%

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

      5. associate-+r- 62.3%

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

      6. unsub-neg 62.3%

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

      7. associate--r+ 62.3%

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

      8. +-commutative 62.3%

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

      9. associate--r+ 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in n around 0 74.8%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 74.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 74.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 84.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 84.2%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in m around inf 90.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 90.8%

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 90.8%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   13. Simplified 90.8%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -53 < m < -2.8499999999999999e-130

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-*r/ 76.8%

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

      3. associate--r- 76.8%

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

      4. +-commutative 76.8%

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

      5. associate-+r- 76.8%

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

      6. unsub-neg 76.8%

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

      7. associate--r+ 76.8%

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

      8. +-commutative 76.8%

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

      9. associate--r+ 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in K around 0 92.2%

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

      1. cos-neg 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in l around inf 54.1%

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

      1. mul-1-neg 54.1%

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

   9. Simplified 54.1%

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

   if -2.8499999999999999e-130 < m

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r/ 78.3%

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

      3. associate--r- 78.3%

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

      4. +-commutative 78.3%

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

      5. associate-+r- 78.3%

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

      6. unsub-neg 78.3%

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

      7. associate--r+ 78.3%

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

      8. +-commutative 78.3%

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

      9. associate--r+ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in n around inf 58.8%

      \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   8. Step-by-step derivation

      1. metadata-eval 58.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{\left(-0.25\right)} \cdot {n}^{2}} \]

      2. distribute-lft-neg-in 58.8%

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

      3. *-commutative 58.8%

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

      4. unpow2 58.8%

         \[\leadsto \cos M \cdot e^{-\color{blue}{\left(n \cdot n\right)} \cdot 0.25} \]

      5. associate-*r* 58.8%

         \[\leadsto \cos M \cdot e^{-\color{blue}{n \cdot \left(n \cdot 0.25\right)}} \]

      6. distribute-rgt-neg-in 58.8%

         \[\leadsto \cos M \cdot e^{\color{blue}{n \cdot \left(-n \cdot 0.25\right)}} \]

      7. distribute-rgt-neg-in 58.8%

         \[\leadsto \cos M \cdot e^{n \cdot \color{blue}{\left(n \cdot \left(-0.25\right)\right)}} \]

      8. metadata-eval 58.8%

         \[\leadsto \cos M \cdot e^{n \cdot \left(n \cdot \color{blue}{-0.25}\right)} \]

   9. Simplified 58.8%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -2.85 \cdot 10^{-130}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(n \cdot -0.25\right)}\\ \end{array} \]

Alternative 10: 77.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.82 \cdot 10^{-127}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= m -53.0)
   (exp (* -0.25 (* m m)))
   (if (<= m -1.82e-127) (* (cos M) (exp (- l))) (exp (* (* n n) -0.25)))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= -1.82e-127) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 <= (-53.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= (-1.82d-127)) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= -1.82e-127) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= -1.82e-127:
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -53.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= -1.82e-127)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -53.0)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= -1.82e-127)
		tmp = cos(M) * exp(-l);
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[m, -53.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -1.82e-127], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*r/ 62.3%

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

      3. associate--r- 62.3%

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

      4. +-commutative 62.3%

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

      5. associate-+r- 62.3%

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

      6. unsub-neg 62.3%

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

      7. associate--r+ 62.3%

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

      8. +-commutative 62.3%

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

      9. associate--r+ 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in n around 0 74.8%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 74.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 74.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 84.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 84.2%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in m around inf 90.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 90.8%

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 90.8%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   13. Simplified 90.8%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -53 < m < -1.82000000000000002e-127

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-*r/ 76.8%

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

      3. associate--r- 76.8%

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

      4. +-commutative 76.8%

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

      5. associate-+r- 76.8%

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

      6. unsub-neg 76.8%

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

      7. associate--r+ 76.8%

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

      8. +-commutative 76.8%

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

      9. associate--r+ 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in K around 0 92.2%

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

      1. cos-neg 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in l around inf 54.1%

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

      1. mul-1-neg 54.1%

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

   9. Simplified 54.1%

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

   if -1.82000000000000002e-127 < m

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r/ 78.3%

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

      3. associate--r- 78.3%

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

      4. +-commutative 78.3%

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

      5. associate-+r- 78.3%

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

      6. unsub-neg 78.3%

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

      7. associate--r+ 78.3%

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

      8. +-commutative 78.3%

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

      9. associate--r+ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in n around 0 90.5%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 90.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 90.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 95.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 95.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 95.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 95.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 94.1%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in n around inf 58.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 58.8%

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 58.8%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

   13. Simplified 58.8%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.82 \cdot 10^{-127}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 11: 76.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -53 \lor \neg \left(m \leq 4.3 \cdot 10^{-39}\right):\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -53.0) (not (<= m 4.3e-39)))
   (exp (* -0.25 (* m m)))
   (exp (* M (- M)))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -53.0) || !(m <= 4.3e-39)) {
		tmp = exp((-0.25 * (m * m)));
	} else {
		tmp = exp((M * -M));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 <= (-53.0d0)) .or. (.not. (m <= 4.3d-39))) then
        tmp = exp(((-0.25d0) * (m * m)))
    else
        tmp = exp((m_1 * -m_1))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -53.0) || !(m <= 4.3e-39)) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else {
		tmp = Math.exp((M * -M));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	tmp = 0
	if (m <= -53.0) or not (m <= 4.3e-39):
		tmp = math.exp((-0.25 * (m * m)))
	else:
		tmp = math.exp((M * -M))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -53.0) || !(m <= 4.3e-39))
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	else
		tmp = exp(Float64(M * Float64(-M)));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -53.0) || ~((m <= 4.3e-39)))
		tmp = exp((-0.25 * (m * m)));
	else
		tmp = exp((M * -M));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -53.0], N[Not[LessEqual[m, 4.3e-39]], $MachinePrecision]], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -53 or 4.2999999999999999e-39 < m

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-*r/ 69.3%

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

      3. associate--r- 69.3%

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

      4. +-commutative 69.3%

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

      5. associate-+r- 69.3%

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

      6. unsub-neg 69.3%

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

      7. associate--r+ 69.3%

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

      8. +-commutative 69.3%

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

      9. associate--r+ 69.3%

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

   3. Simplified 69.3%

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

   4. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in n around 0 82.7%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 82.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 82.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 90.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 90.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 90.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 90.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 89.5%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in m around inf 89.3%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 89.3%

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 89.3%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   13. Simplified 89.3%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -53 < m < 4.2999999999999999e-39

   1. Initial program 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*r/ 79.4%

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

      3. associate--r- 79.4%

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

      4. +-commutative 79.4%

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

      5. associate-+r- 79.4%

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

      6. unsub-neg 79.4%

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

      7. associate--r+ 79.4%

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

      8. +-commutative 79.4%

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

      9. associate--r+ 79.4%

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

   3. Simplified 79.4%

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

   4. Taylor expanded in K around 0 96.9%

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

      1. cos-neg 96.9%

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

   6. Simplified 96.9%

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

   7. Taylor expanded in n around 0 88.7%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 88.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 88.7%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 92.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 92.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 92.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 92.8%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 91.2%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in M around inf 55.6%

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

       1. neg-mul-1 55.6%

          \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

       2. unpow2 55.6%

          \[\leadsto 1 \cdot e^{-\color{blue}{M \cdot M}} \]

       3. distribute-rgt-neg-in 55.6%

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   13. Simplified 55.6%

       \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53 \lor \neg \left(m \leq 4.3 \cdot 10^{-39}\right):\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \end{array} \]

Alternative 12: 76.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.72 \cdot 10^{-128}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= m -53.0)
   (exp (* -0.25 (* m m)))
   (if (<= m -1.72e-128) (exp (- l)) (exp (* (* n n) -0.25)))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = exp((-0.25 * (m * m)));
	} else if (m <= -1.72e-128) {
		tmp = exp(-l);
	} else {
		tmp = exp(((n * n) * -0.25));
	}
	return tmp;
}

Fortran

NOTE: m and n 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 <= (-53.0d0)) then
        tmp = exp(((-0.25d0) * (m * m)))
    else if (m <= (-1.72d-128)) then
        tmp = exp(-l)
    else
        tmp = exp(((n * n) * (-0.25d0)))
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -53.0) {
		tmp = Math.exp((-0.25 * (m * m)));
	} else if (m <= -1.72e-128) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.exp(((n * n) * -0.25));
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	tmp = 0
	if m <= -53.0:
		tmp = math.exp((-0.25 * (m * m)))
	elif m <= -1.72e-128:
		tmp = math.exp(-l)
	else:
		tmp = math.exp(((n * n) * -0.25))
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -53.0)
		tmp = exp(Float64(-0.25 * Float64(m * m)));
	elseif (m <= -1.72e-128)
		tmp = exp(Float64(-l));
	else
		tmp = exp(Float64(Float64(n * n) * -0.25));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -53.0)
		tmp = exp((-0.25 * (m * m)));
	elseif (m <= -1.72e-128)
		tmp = exp(-l);
	else
		tmp = exp(((n * n) * -0.25));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[m, -53.0], N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -1.72e-128], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(N[(n * n), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -53

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*r/ 62.3%

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

      3. associate--r- 62.3%

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

      4. +-commutative 62.3%

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

      5. associate-+r- 62.3%

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

      6. unsub-neg 62.3%

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

      7. associate--r+ 62.3%

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

      8. +-commutative 62.3%

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

      9. associate--r+ 62.3%

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

   3. Simplified 62.3%

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

   4. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in n around 0 74.8%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 74.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 74.8%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 84.2%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 84.2%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 84.2%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in m around inf 90.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 90.8%

          \[\leadsto 1 \cdot e^{\color{blue}{{m}^{2} \cdot -0.25}} \]

       2. unpow2 90.8%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right)} \cdot -0.25} \]

   13. Simplified 90.8%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(m \cdot m\right) \cdot -0.25}} \]

   if -53 < m < -1.71999999999999992e-128

   1. Initial program 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-*r/ 76.8%

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

      3. associate--r- 76.8%

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

      4. +-commutative 76.8%

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

      5. associate-+r- 76.8%

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

      6. unsub-neg 76.8%

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

      7. associate--r+ 76.8%

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

      8. +-commutative 76.8%

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

      9. associate--r+ 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in K around 0 92.2%

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

      1. cos-neg 92.2%

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

   6. Simplified 92.2%

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

   7. Taylor expanded in l around inf 54.1%

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

      1. mul-1-neg 54.1%

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

   9. Simplified 54.1%

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

   10. Taylor expanded in M around 0 49.8%

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

   if -1.71999999999999992e-128 < m

   1. Initial program 78.3%

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

      1. *-commutative 78.3%

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

      2. associate-*r/ 78.3%

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

      3. associate--r- 78.3%

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

      4. +-commutative 78.3%

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

      5. associate-+r- 78.3%

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

      6. unsub-neg 78.3%

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

      7. associate--r+ 78.3%

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

      8. +-commutative 78.3%

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

      9. associate--r+ 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in n around 0 90.5%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 90.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 90.5%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 95.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 95.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 95.3%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 95.3%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 94.1%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in n around inf 58.8%

       \[\leadsto 1 \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   12. Step-by-step derivation

       1. *-commutative 58.8%

          \[\leadsto 1 \cdot e^{\color{blue}{{n}^{2} \cdot -0.25}} \]

       2. unpow2 58.8%

          \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right)} \cdot -0.25} \]

   13. Simplified 58.8%

       \[\leadsto 1 \cdot e^{\color{blue}{\left(n \cdot n\right) \cdot -0.25}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -53:\\ \;\;\;\;e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;m \leq -1.72 \cdot 10^{-128}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n \cdot n\right) \cdot -0.25}\\ \end{array} \]

Alternative 13: 70.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -2 \lor \neg \left(M \leq 26\right):\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -2.0) (not (<= M 26.0))) (exp (* M (- M))) (exp (- l))))

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.0) || !(M <= 26.0)) {
		tmp = exp((M * -M));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

NOTE: m and n 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_1 <= (-2.0d0)) .or. (.not. (m_1 <= 26.0d0))) then
        tmp = exp((m_1 * -m_1))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.0) || !(M <= 26.0)) {
		tmp = Math.exp((M * -M));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	tmp = 0
	if (M <= -2.0) or not (M <= 26.0):
		tmp = math.exp((M * -M))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -2.0) || !(M <= 26.0))
		tmp = exp(Float64(M * Float64(-M)));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -2.0) || ~((M <= 26.0)))
		tmp = exp((M * -M));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -2.0], N[Not[LessEqual[M, 26.0]], $MachinePrecision]], N[Exp[N[(M * (-M)), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2 or 26 < M

   1. Initial program 77.9%

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

      1. *-commutative 77.9%

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

      2. associate-*r/ 77.9%

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

      3. associate--r- 77.9%

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

      4. +-commutative 77.9%

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

      5. associate-+r- 77.9%

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

      6. unsub-neg 77.9%

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

      7. associate--r+ 77.9%

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

      8. +-commutative 77.9%

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

      9. associate--r+ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 81.1%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + \left(n \cdot \left(0.5 \cdot m - M\right) + 0.25 \cdot {n}^{2}\right)\right)}} \]
   8. Step-by-step derivation

      1. associate-+r+ 81.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)}} \]

      2. unpow2 81.1%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right) + 0.25 \cdot {n}^{2}\right)} \]

      3. distribute-rgt-out 91.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)} + 0.25 \cdot {n}^{2}\right)} \]

      4. *-commutative 91.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{{n}^{2} \cdot 0.25}\right)} \]

      5. unpow2 91.0%

         \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \color{blue}{\left(n \cdot n\right)} \cdot 0.25\right)} \]

   9. Simplified 91.0%

      \[\leadsto \cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \color{blue}{\left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)}} \]

   10. Taylor expanded in M around 0 88.5%

       \[\leadsto \color{blue}{1} \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right) + \left(n \cdot n\right) \cdot 0.25\right)} \]

   11. Taylor expanded in M around inf 96.8%

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

       1. neg-mul-1 96.8%

          \[\leadsto 1 \cdot e^{\color{blue}{-{M}^{2}}} \]

       2. unpow2 96.8%

          \[\leadsto 1 \cdot e^{-\color{blue}{M \cdot M}} \]

       3. distribute-rgt-neg-in 96.8%

          \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   13. Simplified 96.8%

       \[\leadsto 1 \cdot e^{\color{blue}{M \cdot \left(-M\right)}} \]

   if -2 < M < 26

   1. Initial program 70.7%

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

      1. *-commutative 70.7%

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

      2. associate-*r/ 70.7%

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

      3. associate--r- 70.7%

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

      4. +-commutative 70.7%

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

      5. associate-+r- 70.7%

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

      6. unsub-neg 70.7%

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

      7. associate--r+ 70.7%

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

      8. +-commutative 70.7%

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

      9. associate--r+ 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in K around 0 94.8%

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

      1. cos-neg 94.8%

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

   6. Simplified 94.8%

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

   7. Taylor expanded in l around inf 44.3%

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

      1. mul-1-neg 44.3%

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

   9. Simplified 44.3%

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

   10. Taylor expanded in M around 0 44.3%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2 \lor \neg \left(M \leq 26\right):\\ \;\;\;\;e^{M \cdot \left(-M\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 14: 35.5% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. *-commutative 74.1%

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

   2. associate-*r/ 74.1%

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

   3. associate--r- 74.1%

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

   4. +-commutative 74.1%

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

   5. associate-+r- 74.1%

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

   6. unsub-neg 74.1%

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

   7. associate--r+ 74.1%

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

   8. +-commutative 74.1%

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

   9. associate--r+ 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

6. Simplified 97.3%

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

7. Taylor expanded in l around inf 40.1%

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

   1. mul-1-neg 40.1%

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

9. Simplified 40.1%

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

10. Taylor expanded in M around 0 38.9%

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

11. Final simplification 38.9%

    \[\leadsto e^{-\ell} \]

Alternative 15: 6.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.1%

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

   1. *-commutative 74.1%

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

   2. associate-*r/ 74.1%

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

   3. associate--r- 74.1%

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

   4. +-commutative 74.1%

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

   5. associate-+r- 74.1%

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

   6. unsub-neg 74.1%

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

   7. associate--r+ 74.1%

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

   8. +-commutative 74.1%

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

   9. associate--r+ 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

6. Simplified 97.3%

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

7. Taylor expanded in l around inf 40.1%

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

   1. mul-1-neg 40.1%

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

9. Simplified 40.1%

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

10. Taylor expanded in l around 0 6.6%

    \[\leadsto \color{blue}{\cos M} \]

11. Final simplification 6.6%

    \[\leadsto \cos M \]

Alternative 16: 6.9% accurate, 425.0× speedup

Math

\[\begin{array}{l} [m, n] = \mathsf{sort}([m, n])\\ \\ 1 \end{array} \]

FPCore

NOTE: m and n should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 1.0)

C

assert(m < n);
double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Fortran

NOTE: m and n 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 = 1.0d0
end function

Java

assert m < n;
public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

[m, n] = sort([m, n])
def code(K, m, n, M, l):
	return 1.0

Julia

m, n = sort([m, n])
function code(K, m, n, M, l)
	return 1.0
end

MATLAB

m, n = num2cell(sort([m, n])){:}
function tmp = code(K, m, n, M, l)
	tmp = 1.0;
end

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 74.1%

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

   1. *-commutative 74.1%

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

   2. associate-*r/ 74.1%

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

   3. associate--r- 74.1%

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

   4. +-commutative 74.1%

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

   5. associate-+r- 74.1%

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

   6. unsub-neg 74.1%

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

   7. associate--r+ 74.1%

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

   8. +-commutative 74.1%

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

   9. associate--r+ 74.1%

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

3. Simplified 74.1%

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

4. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

6. Simplified 97.3%

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

7. Taylor expanded in l around inf 40.1%

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

   1. mul-1-neg 40.1%

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

9. Simplified 40.1%

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

10. Taylor expanded in l around 0 6.6%

    \[\leadsto \color{blue}{\cos M} \]

11. Taylor expanded in M around 0 6.6%

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

12. Final simplification 6.6%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023229 
(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)))))))