Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 95.4%

Time: 17.5s

Alternatives: 8

Speedup: 3.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \cos \left(-M\right)\\ \mathbf{if}\;M \leq -65000 \lor \neg \left(M \leq 22000000\right):\\ \;\;\;\;t\_0 \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot {\left(n + m\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- M))))
   (if (or (<= M -65000.0) (not (<= M 22000000.0)))
     (* t_0 (exp (- (pow M 2.0))))
     (* t_0 (exp (+ (- n (+ m l)) (* -0.25 (pow (+ n m) 2.0))))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(-M);
	double tmp;
	if ((M <= -65000.0) || !(M <= 22000000.0)) {
		tmp = t_0 * exp(-pow(M, 2.0));
	} else {
		tmp = t_0 * exp(((n - (m + l)) + (-0.25 * pow((n + m), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(-m_1)
    if ((m_1 <= (-65000.0d0)) .or. (.not. (m_1 <= 22000000.0d0))) then
        tmp = t_0 * exp(-(m_1 ** 2.0d0))
    else
        tmp = t_0 * exp(((n - (m + l)) + ((-0.25d0) * ((n + m) ** 2.0d0))))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(-M);
	double tmp;
	if ((M <= -65000.0) || !(M <= 22000000.0)) {
		tmp = t_0 * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = t_0 * Math.exp(((n - (m + l)) + (-0.25 * Math.pow((n + m), 2.0))));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = math.cos(-M)
	tmp = 0
	if (M <= -65000.0) or not (M <= 22000000.0):
		tmp = t_0 * math.exp(-math.pow(M, 2.0))
	else:
		tmp = t_0 * math.exp(((n - (m + l)) + (-0.25 * math.pow((n + m), 2.0))))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = cos(Float64(-M))
	tmp = 0.0
	if ((M <= -65000.0) || !(M <= 22000000.0))
		tmp = Float64(t_0 * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(t_0 * exp(Float64(Float64(n - Float64(m + l)) + Float64(-0.25 * (Float64(n + m) ^ 2.0)))));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(-M);
	tmp = 0.0;
	if ((M <= -65000.0) || ~((M <= 22000000.0)))
		tmp = t_0 * exp(-(M ^ 2.0));
	else
		tmp = t_0 * exp(((n - (m + l)) + (-0.25 * ((n + m) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[(-M)], $MachinePrecision]}, If[Or[LessEqual[M, -65000.0], N[Not[LessEqual[M, 22000000.0]], $MachinePrecision]], N[(t$95$0 * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Exp[N[(N[(n - N[(m + l), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[Power[N[(n + m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if M < -65000 or 2.2e7 < M

   1. Initial program 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in K around 0 99.3%

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

   6. Taylor expanded in M around inf 99.3%

      \[\leadsto \cos \left(-M\right) \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.3%

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

   8. Simplified 99.3%

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

   if -65000 < M < 2.2e7

   1. Initial program 72.8%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in K around 0 93.5%

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

   6. Taylor expanded in M around 0 93.5%

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

      1. fabs-sub 93.5%

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

      2. associate--r+ 93.5%

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

      3. cancel-sign-sub-inv 93.5%

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

      4. fabs-sub 93.5%

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

      5. sub-neg 93.5%

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

      6. neg-mul-1 93.5%

         \[\leadsto \cos \left(-M\right) \cdot e^{\left(\left|n + \color{blue}{-1 \cdot m}\right| - \ell\right) + \left(-0.25\right) \cdot {\left(m + n\right)}^{2}} \]

      7. neg-mul-1 93.5%

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

      8. sub-neg 93.5%

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

      9. rem-square-sqrt 47.6%

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

      10. fabs-sqr 47.6%

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

      11. rem-square-sqrt 93.0%

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

      12. associate--r+ 93.0%

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

      13. metadata-eval 93.0%

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

      14. +-commutative 93.0%

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

   8. Simplified 93.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -65000 \lor \neg \left(M \leq 22000000\right):\\ \;\;\;\;\cos \left(-M\right) \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-M\right) \cdot e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot {\left(n + m\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(-m_1) * exp((abs((n - m)) - (l + (((0.5d0 * (n + m)) - m_1) ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Simplified 79.1%

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

5. Taylor expanded in K around 0 96.7%

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

6. Final simplification 96.7%

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

Alternative 3: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;M \leq -2050 \lor \neg \left(M \leq 38\right):\\ \;\;\;\;\cos \left(-M\right) \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -2050.0) (not (<= M 38.0)))
   (* (cos (- M)) (exp (- (pow M 2.0))))
   (exp (+ (- n (+ m l)) (* -0.25 (* (+ n m) (+ n m)))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2050.0) || !(M <= 38.0)) {
		tmp = cos(-M) * exp(-pow(M, 2.0));
	} else {
		tmp = exp(((n - (m + l)) + (-0.25 * ((n + m) * (n + m)))));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-2050.0d0)) .or. (.not. (m_1 <= 38.0d0))) then
        tmp = cos(-m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = exp(((n - (m + l)) + ((-0.25d0) * ((n + m) * (n + m)))))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2050.0) || !(M <= 38.0)) {
		tmp = Math.cos(-M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.exp(((n - (m + l)) + (-0.25 * ((n + m) * (n + m)))));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if (M <= -2050.0) or not (M <= 38.0):
		tmp = math.cos(-M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.exp(((n - (m + l)) + (-0.25 * ((n + m) * (n + m)))))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -2050.0) || !(M <= 38.0))
		tmp = Float64(cos(Float64(-M)) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = exp(Float64(Float64(n - Float64(m + l)) + Float64(-0.25 * Float64(Float64(n + m) * Float64(n + m)))));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -2050.0) || ~((M <= 38.0)))
		tmp = cos(-M) * exp(-(M ^ 2.0));
	else
		tmp = exp(((n - (m + l)) + (-0.25 * ((n + m) * (n + m)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -2050.0], N[Not[LessEqual[M, 38.0]], $MachinePrecision]], N[(N[Cos[(-M)], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(n - N[(m + l), $MachinePrecision]), $MachinePrecision] + N[(-0.25 * N[(N[(n + m), $MachinePrecision] * N[(n + m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2050 or 38 < M

   1. Initial program 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in K around 0 99.3%

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

   6. Taylor expanded in M around inf 99.3%

      \[\leadsto \cos \left(-M\right) \cdot e^{\color{blue}{-1 \cdot {M}^{2}}} \]
   7. Step-by-step derivation

      1. mul-1-neg 99.3%

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

   8. Simplified 99.3%

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

   if -2050 < M < 38

   1. Initial program 72.8%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in K around 0 93.5%

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

   6. Taylor expanded in M around 0 93.5%

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

      1. fabs-sub 93.5%

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

      2. associate--r+ 93.5%

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

      3. cancel-sign-sub-inv 93.5%

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

      4. fabs-sub 93.5%

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

      5. sub-neg 93.5%

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

      6. neg-mul-1 93.5%

         \[\leadsto \cos \left(-M\right) \cdot e^{\left(\left|n + \color{blue}{-1 \cdot m}\right| - \ell\right) + \left(-0.25\right) \cdot {\left(m + n\right)}^{2}} \]

      7. neg-mul-1 93.5%

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

      8. sub-neg 93.5%

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

      9. rem-square-sqrt 47.6%

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

      10. fabs-sqr 47.6%

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

      11. rem-square-sqrt 93.0%

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

      12. associate--r+ 93.0%

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

      13. metadata-eval 93.0%

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

      14. +-commutative 93.0%

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

   8. Simplified 93.0%

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

   9. Taylor expanded in M around 0 93.0%

      \[\leadsto \color{blue}{1} \cdot e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot {\left(n + m\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 93.0%

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

   11. Applied egg-rr 93.0%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2050 \lor \neg \left(M \leq 38\right):\\ \;\;\;\;\cos \left(-M\right) \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := n - \left(m + \ell\right)\\ \mathbf{if}\;n \leq 7.2 \cdot 10^{-209} \lor \neg \left(n \leq 1.2 \cdot 10^{-80}\right):\\ \;\;\;\;e^{t\_0 + -0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{t\_0 + M \cdot \left(m - M\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- n (+ m l))))
   (if (or (<= n 7.2e-209) (not (<= n 1.2e-80)))
     (exp (+ t_0 (* -0.25 (* (+ n m) (+ n m)))))
     (* (cos (- (* K (/ (+ n m) 2.0)) M)) (exp (+ t_0 (* M (- m M))))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = n - (m + l);
	double tmp;
	if ((n <= 7.2e-209) || !(n <= 1.2e-80)) {
		tmp = exp((t_0 + (-0.25 * ((n + m) * (n + m)))));
	} else {
		tmp = cos(((K * ((n + m) / 2.0)) - M)) * exp((t_0 + (M * (m - M))));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = n - (m + l)
    if ((n <= 7.2d-209) .or. (.not. (n <= 1.2d-80))) then
        tmp = exp((t_0 + ((-0.25d0) * ((n + m) * (n + m)))))
    else
        tmp = cos(((k * ((n + m) / 2.0d0)) - m_1)) * exp((t_0 + (m_1 * (m - m_1))))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double t_0 = n - (m + l);
	double tmp;
	if ((n <= 7.2e-209) || !(n <= 1.2e-80)) {
		tmp = Math.exp((t_0 + (-0.25 * ((n + m) * (n + m)))));
	} else {
		tmp = Math.cos(((K * ((n + m) / 2.0)) - M)) * Math.exp((t_0 + (M * (m - M))));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	t_0 = n - (m + l)
	tmp = 0
	if (n <= 7.2e-209) or not (n <= 1.2e-80):
		tmp = math.exp((t_0 + (-0.25 * ((n + m) * (n + m)))))
	else:
		tmp = math.cos(((K * ((n + m) / 2.0)) - M)) * math.exp((t_0 + (M * (m - M))))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(n - Float64(m + l))
	tmp = 0.0
	if ((n <= 7.2e-209) || !(n <= 1.2e-80))
		tmp = exp(Float64(t_0 + Float64(-0.25 * Float64(Float64(n + m) * Float64(n + m)))));
	else
		tmp = Float64(cos(Float64(Float64(K * Float64(Float64(n + m) / 2.0)) - M)) * exp(Float64(t_0 + Float64(M * Float64(m - M)))));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	t_0 = n - (m + l);
	tmp = 0.0;
	if ((n <= 7.2e-209) || ~((n <= 1.2e-80)))
		tmp = exp((t_0 + (-0.25 * ((n + m) * (n + m)))));
	else
		tmp = cos(((K * ((n + m) / 2.0)) - M)) * exp((t_0 + (M * (m - M))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(n - N[(m + l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[n, 7.2e-209], N[Not[LessEqual[n, 1.2e-80]], $MachinePrecision]], N[Exp[N[(t$95$0 + N[(-0.25 * N[(N[(n + m), $MachinePrecision] * N[(n + m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(N[(K * N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$0 + N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 7.20000000000000032e-209 or 1.2e-80 < n

   1. Initial program 76.9%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in K around 0 96.3%

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

   6. Taylor expanded in M around 0 85.8%

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

      1. fabs-sub 85.8%

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

      2. associate--r+ 85.8%

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

      3. cancel-sign-sub-inv 85.8%

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

      4. fabs-sub 85.8%

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

      5. sub-neg 85.8%

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

      6. neg-mul-1 85.8%

         \[\leadsto \cos \left(-M\right) \cdot e^{\left(\left|n + \color{blue}{-1 \cdot m}\right| - \ell\right) + \left(-0.25\right) \cdot {\left(m + n\right)}^{2}} \]

      7. neg-mul-1 85.8%

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

      8. sub-neg 85.8%

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

      9. rem-square-sqrt 44.2%

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

      10. fabs-sqr 44.2%

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

      11. rem-square-sqrt 85.6%

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

      12. associate--r+ 85.6%

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

      13. metadata-eval 85.6%

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

      14. +-commutative 85.6%

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

   8. Simplified 85.6%

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

   9. Taylor expanded in M around 0 85.6%

      \[\leadsto \color{blue}{1} \cdot e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot {\left(n + m\right)}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 85.6%

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

   11. Applied egg-rr 85.6%

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

   if 7.20000000000000032e-209 < n < 1.2e-80

   1. Initial program 95.8%

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

      1. associate-/l* 95.8%

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

      2. +-commutative 95.8%

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

      3. associate-/l* 95.8%

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

      4. associate-/l* 95.8%

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

      5. +-commutative 95.8%

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

      6. exp-diff 62.5%

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

      7. sub-neg 62.5%

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

      8. exp-sum 50.0%

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

      9. associate-/r* 50.0%

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

      10. exp-diff 54.2%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in m around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. unpow2 71.1%

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

      3. distribute-rgt-out 75.3%

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

      4. *-commutative 75.3%

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

      5. *-commutative 75.3%

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

   7. Simplified 75.3%

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

   8. Taylor expanded in n around 0 75.3%

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

      1. fabs-sub 75.3%

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

      2. associate--r+ 75.3%

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

      3. associate-*r* 75.3%

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

      4. neg-mul-1 75.3%

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

      5. cancel-sign-sub 75.3%

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

      6. fabs-sub 75.3%

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

      7. sub-neg 75.3%

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

      8. neg-mul-1 75.3%

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

      9. neg-mul-1 75.3%

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

      10. sub-neg 75.3%

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

      11. rem-square-sqrt 54.2%

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

      12. fabs-sqr 54.2%

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

      13. rem-square-sqrt 79.4%

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

      14. associate--r+ 79.4%

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

   10. Simplified 79.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 7.2 \cdot 10^{-209} \lor \neg \left(n \leq 1.2 \cdot 10^{-80}\right):\\ \;\;\;\;e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{\left(n - \left(m + \ell\right)\right) + M \cdot \left(m - M\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)} \end{array} \]

FPCore

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

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return exp(((n - (m + l)) + (-0.25 * ((n + m) * (n + m)))));
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = exp(((n - (m + l)) + ((-0.25d0) * ((n + m) * (n + m)))))
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return Math.exp(((n - (m + l)) + (-0.25 * ((n + m) * (n + m)))));
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return math.exp(((n - (m + l)) + (-0.25 * ((n + m) * (n + m)))))

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return exp(Float64(Float64(n - Float64(m + l)) + Float64(-0.25 * Float64(Float64(n + m) * Float64(n + m)))))
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = exp(((n - (m + l)) + (-0.25 * ((n + m) * (n + m)))));
end

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Simplified 79.1%

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

5. Taylor expanded in K around 0 96.7%

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

6. Taylor expanded in M around 0 84.5%

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

   1. fabs-sub 84.5%

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

   2. associate--r+ 84.5%

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

   3. cancel-sign-sub-inv 84.5%

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

   4. fabs-sub 84.5%

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

   5. sub-neg 84.5%

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

   6. neg-mul-1 84.5%

      \[\leadsto \cos \left(-M\right) \cdot e^{\left(\left|n + \color{blue}{-1 \cdot m}\right| - \ell\right) + \left(-0.25\right) \cdot {\left(m + n\right)}^{2}} \]

   7. neg-mul-1 84.5%

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

   8. sub-neg 84.5%

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

   9. rem-square-sqrt 43.6%

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

   10. fabs-sqr 43.6%

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

   11. rem-square-sqrt 84.3%

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

   12. associate--r+ 84.3%

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

   13. metadata-eval 84.3%

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

   14. +-commutative 84.3%

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

8. Simplified 84.3%

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

9. Taylor expanded in M around 0 84.3%

   \[\leadsto \color{blue}{1} \cdot e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot {\left(n + m\right)}^{2}} \]
10. Step-by-step derivation

    1. unpow2 84.3%

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

11. Applied egg-rr 84.3%

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

12. Final simplification 84.3%

    \[\leadsto e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot \left(\left(n + m\right) \cdot \left(n + m\right)\right)} \]
13. Add Preprocessing

Alternative 6: 34.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

3. Simplified 79.1%

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

5. Taylor expanded in K around 0 96.7%

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

6. Taylor expanded in M around 0 84.5%

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

   1. fabs-sub 84.5%

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

   2. associate--r+ 84.5%

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

   3. cancel-sign-sub-inv 84.5%

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

   4. fabs-sub 84.5%

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

   5. sub-neg 84.5%

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

   6. neg-mul-1 84.5%

      \[\leadsto \cos \left(-M\right) \cdot e^{\left(\left|n + \color{blue}{-1 \cdot m}\right| - \ell\right) + \left(-0.25\right) \cdot {\left(m + n\right)}^{2}} \]

   7. neg-mul-1 84.5%

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

   8. sub-neg 84.5%

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

   9. rem-square-sqrt 43.6%

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

   10. fabs-sqr 43.6%

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

   11. rem-square-sqrt 84.3%

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

   12. associate--r+ 84.3%

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

   13. metadata-eval 84.3%

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

   14. +-commutative 84.3%

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

8. Simplified 84.3%

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

9. Taylor expanded in M around 0 84.3%

   \[\leadsto \color{blue}{1} \cdot e^{\left(n - \left(m + \ell\right)\right) + -0.25 \cdot {\left(n + m\right)}^{2}} \]

10. Taylor expanded in l around inf 35.2%

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

    1. neg-mul-1 35.2%

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

12. Simplified 35.2%

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

13. Final simplification 35.2%

    \[\leadsto e^{-\ell} \]
14. Add Preprocessing

Alternative 7: 7.0% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.7%

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

   1. associate-/l* 79.1%

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

   2. +-commutative 79.1%

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

   3. associate-/l* 78.7%

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

   4. associate-/l* 79.1%

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

   5. +-commutative 79.1%

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

   6. exp-diff 29.5%

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

   7. sub-neg 29.5%

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

   8. exp-sum 21.7%

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

   9. associate-/r* 21.7%

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

   10. exp-diff 24.8%

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

3. Simplified 79.1%

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

5. Taylor expanded in l around inf 29.4%

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

   1. mul-1-neg 29.4%

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

7. Simplified 29.4%

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

8. Taylor expanded in l around 0 5.9%

   \[\leadsto \cos \left(K \cdot \frac{m + n}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 5.9%

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

   2. unsub-neg 5.9%

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

10. Simplified 5.9%

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

11. Taylor expanded in K around 0 6.1%

    \[\leadsto \color{blue}{\cos \left(-M\right) \cdot \left(1 - \ell\right)} \]
12. Step-by-step derivation

    1. cos-neg 6.1%

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

13. Simplified 6.1%

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

14. Taylor expanded in l around 0 6.4%

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

Alternative 8: 6.7% accurate, 141.7× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ 1 - \ell \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l) :precision binary64 (- 1.0 l))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return 1.0 - l;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 - l
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	return 1.0 - l;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	return 1.0 - l

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(1.0 - l)
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp = code(K, m, n, M, l)
	tmp = 1.0 - l;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := N[(1.0 - l), $MachinePrecision]

Derivation

1. Initial program 78.7%

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

   1. associate-/l* 79.1%

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

   2. +-commutative 79.1%

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

   3. associate-/l* 78.7%

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

   4. associate-/l* 79.1%

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

   5. +-commutative 79.1%

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

   6. exp-diff 29.5%

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

   7. sub-neg 29.5%

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

   8. exp-sum 21.7%

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

   9. associate-/r* 21.7%

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

   10. exp-diff 24.8%

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

3. Simplified 79.1%

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

5. Taylor expanded in l around inf 29.4%

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

   1. mul-1-neg 29.4%

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

7. Simplified 29.4%

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

8. Taylor expanded in l around 0 5.9%

   \[\leadsto \cos \left(K \cdot \frac{m + n}{2} - M\right) \cdot \color{blue}{\left(1 + -1 \cdot \ell\right)} \]
9. Step-by-step derivation

   1. neg-mul-1 5.9%

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

   2. unsub-neg 5.9%

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

10. Simplified 5.9%

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

11. Taylor expanded in K around 0 6.1%

    \[\leadsto \color{blue}{\cos \left(-M\right) \cdot \left(1 - \ell\right)} \]
12. Step-by-step derivation

    1. cos-neg 6.1%

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

13. Simplified 6.1%

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

14. Taylor expanded in M around 0 6.1%

    \[\leadsto \color{blue}{1 - \ell} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(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)))))))