Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.5% → 95.3%

Time: 18.7s

Alternatives: 12

Speedup: 2.0×

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: 76.5% 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: 95.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := m \cdot 0.5 - M\\ \mathbf{if}\;n \leq 8.6 \cdot 10^{+42}:\\ \;\;\;\;\cos M \cdot e^{-\mathsf{fma}\left(t\_0, n + t\_0, m - \left(n - \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (* m 0.5) M)))
   (if (<= n 8.6e+42)
     (* (cos M) (exp (- (fma t_0 (+ n t_0) (- m (- n l))))))
     (exp (* -0.25 (pow n 2.0))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double t_0 = (m * 0.5) - M;
	double tmp;
	if (n <= 8.6e+42) {
		tmp = cos(M) * exp(-fma(t_0, (n + t_0), (m - (n - l))));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(Float64(m * 0.5) - M)
	tmp = 0.0
	if (n <= 8.6e+42)
		tmp = Float64(cos(M) * exp(Float64(-fma(t_0, Float64(n + t_0), Float64(m - Float64(n - l))))));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return 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[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]}, If[LessEqual[n, 8.6e+42], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[(t$95$0 * N[(n + t$95$0), $MachinePrecision] + N[(m - N[(n - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 8.5999999999999996e42

   1. Initial program 71.9%

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

   3. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in n around 0 79.1%

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

      1. +-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. distribute-rgt-out 85.3%

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

      4. *-commutative 85.3%

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

      5. *-commutative 85.3%

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

   8. Simplified 85.3%

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

      1. sub-neg 85.3%

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

      2. exp-sum 32.6%

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

      3. distribute-rgt-neg-in 32.6%

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

      4. exp-prod 33.2%

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

      5. associate-+l- 33.2%

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

      6. fabs-sub 33.2%

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

   10. Applied egg-rr 33.2%

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

   12. Applied egg-rr 90.9%

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

       1. rec-exp 90.9%

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

       2. fma-define 90.9%

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

       3. *-commutative 90.9%

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

       4. associate-+l- 90.9%

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

       5. +-commutative 90.9%

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

       6. *-commutative 90.9%

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

       7. associate--r- 90.9%

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

   14. Simplified 90.9%

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

   if 8.5999999999999996e42 < n

   1. Initial program 66.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around inf 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 8.6 \cdot 10^{+42}:\\ \;\;\;\;\cos M \cdot e^{-\mathsf{fma}\left(m \cdot 0.5 - M, n + \left(m \cdot 0.5 - M\right), m - \left(n - \ell\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

3. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

5. Simplified 97.4%

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

6. Final simplification 97.4%

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

Alternative 3: 95.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\cos M \cdot \frac{1}{e^{\left(m \cdot 0.5 - M\right) \cdot \left(m \cdot 0.5 + \left(n - M\right)\right) + \left(\ell + \left(m - n\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= n 5e+41)
   (*
    (cos M)
    (/ 1.0 (exp (+ (* (- (* m 0.5) M) (+ (* m 0.5) (- n M))) (+ l (- m n))))))
   (exp (* -0.25 (pow n 2.0)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5e+41) {
		tmp = cos(M) * (1.0 / exp(((((m * 0.5) - M) * ((m * 0.5) + (n - M))) + (l + (m - n)))));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 5d+41) then
        tmp = cos(m_1) * (1.0d0 / exp(((((m * 0.5d0) - m_1) * ((m * 0.5d0) + (n - m_1))) + (l + (m - n)))))
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 5e+41) {
		tmp = Math.cos(M) * (1.0 / Math.exp(((((m * 0.5) - M) * ((m * 0.5) + (n - M))) + (l + (m - n)))));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if n <= 5e+41:
		tmp = math.cos(M) * (1.0 / math.exp(((((m * 0.5) - M) * ((m * 0.5) + (n - M))) + (l + (m - n)))))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 5e+41)
		tmp = Float64(cos(M) * Float64(1.0 / exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(m * 0.5) + Float64(n - M))) + Float64(l + Float64(m - n))))));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 5e+41)
		tmp = cos(M) * (1.0 / exp(((((m * 0.5) - M) * ((m * 0.5) + (n - M))) + (l + (m - n)))));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[n, 5e+41], N[(N[Cos[M], $MachinePrecision] * N[(1.0 / N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(m * 0.5), $MachinePrecision] + N[(n - M), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(l + N[(m - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 5.00000000000000022e41

   1. Initial program 71.9%

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

   3. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in n around 0 79.1%

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

      1. +-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. distribute-rgt-out 85.3%

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

      4. *-commutative 85.3%

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

      5. *-commutative 85.3%

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

   8. Simplified 85.3%

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

      1. sub-neg 85.3%

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

      2. exp-sum 32.6%

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

      3. distribute-rgt-neg-in 32.6%

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

      4. exp-prod 33.2%

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

      5. associate-+l- 33.2%

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

      6. fabs-sub 33.2%

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

   10. Applied egg-rr 33.2%

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

   12. Applied egg-rr 90.9%

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

   if 5.00000000000000022e41 < n

   1. Initial program 66.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around inf 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 5 \cdot 10^{+41}:\\ \;\;\;\;\cos M \cdot \frac{1}{e^{\left(m \cdot 0.5 - M\right) \cdot \left(m \cdot 0.5 + \left(n - M\right)\right) + \left(\ell + \left(m - n\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := \frac{\cos M}{e^{\ell}}\\ t_1 := e^{-0.25 \cdot {n}^{2}}\\ \mathbf{if}\;n \leq -0.0014:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -1.45 \cdot 10^{-159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-157}:\\ \;\;\;\;{n}^{2} \cdot \left(\cos M \cdot -0.25\right)\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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) (exp l))) (t_1 (exp (* -0.25 (pow n 2.0)))))
   (if (<= n -0.0014)
     t_1
     (if (<= n -1.45e-159)
       t_0
       (if (<= n 2e-157)
         (* (pow n 2.0) (* (cos M) -0.25))
         (if (<= n 2.55e-6) t_0 t_1))))))

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) / exp(l);
	double t_1 = exp((-0.25 * pow(n, 2.0)));
	double tmp;
	if (n <= -0.0014) {
		tmp = t_1;
	} else if (n <= -1.45e-159) {
		tmp = t_0;
	} else if (n <= 2e-157) {
		tmp = pow(n, 2.0) * (cos(M) * -0.25);
	} else if (n <= 2.55e-6) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = cos(m_1) / exp(l)
    t_1 = exp(((-0.25d0) * (n ** 2.0d0)))
    if (n <= (-0.0014d0)) then
        tmp = t_1
    else if (n <= (-1.45d-159)) then
        tmp = t_0
    else if (n <= 2d-157) then
        tmp = (n ** 2.0d0) * (cos(m_1) * (-0.25d0))
    else if (n <= 2.55d-6) then
        tmp = t_0
    else
        tmp = t_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 = Math.cos(M) / Math.exp(l);
	double t_1 = Math.exp((-0.25 * Math.pow(n, 2.0)));
	double tmp;
	if (n <= -0.0014) {
		tmp = t_1;
	} else if (n <= -1.45e-159) {
		tmp = t_0;
	} else if (n <= 2e-157) {
		tmp = Math.pow(n, 2.0) * (Math.cos(M) * -0.25);
	} else if (n <= 2.55e-6) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) / math.exp(l)
	t_1 = math.exp((-0.25 * math.pow(n, 2.0)))
	tmp = 0
	if n <= -0.0014:
		tmp = t_1
	elif n <= -1.45e-159:
		tmp = t_0
	elif n <= 2e-157:
		tmp = math.pow(n, 2.0) * (math.cos(M) * -0.25)
	elif n <= 2.55e-6:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = Float64(cos(M) / exp(l))
	t_1 = exp(Float64(-0.25 * (n ^ 2.0)))
	tmp = 0.0
	if (n <= -0.0014)
		tmp = t_1;
	elseif (n <= -1.45e-159)
		tmp = t_0;
	elseif (n <= 2e-157)
		tmp = Float64((n ^ 2.0) * Float64(cos(M) * -0.25));
	elseif (n <= 2.55e-6)
		tmp = t_0;
	else
		tmp = t_1;
	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) / exp(l);
	t_1 = exp((-0.25 * (n ^ 2.0)));
	tmp = 0.0;
	if (n <= -0.0014)
		tmp = t_1;
	elseif (n <= -1.45e-159)
		tmp = t_0;
	elseif (n <= 2e-157)
		tmp = (n ^ 2.0) * (cos(M) * -0.25);
	elseif (n <= 2.55e-6)
		tmp = t_0;
	else
		tmp = t_1;
	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[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -0.0014], t$95$1, If[LessEqual[n, -1.45e-159], t$95$0, If[LessEqual[n, 2e-157], N[(N[Power[n, 2.0], $MachinePrecision] * N[(N[Cos[M], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.55e-6], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -0.00139999999999999999 or 2.5500000000000001e-6 < n

   1. Initial program 64.9%

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

   3. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in n around inf 92.8%

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

   7. Taylor expanded in M around 0 92.8%

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

   if -0.00139999999999999999 < n < -1.44999999999999995e-159 or 1.99999999999999989e-157 < n < 2.5500000000000001e-6

   1. Initial program 72.8%

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

   3. Taylor expanded in K around 0 95.2%

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

      1. cos-neg 95.2%

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

   5. Simplified 95.2%

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

   6. Taylor expanded in l around inf 44.1%

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

      1. neg-mul-1 44.1%

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

   8. Simplified 44.1%

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

   9. Taylor expanded in l around -inf 44.1%

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

       1. neg-mul-1 44.1%

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

       2. rem-exp-log 36.7%

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

       3. exp-sum 36.7%

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

       4. unsub-neg 36.7%

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

       5. exp-diff 36.7%

          \[\leadsto \color{blue}{\frac{e^{\log \cos M}}{e^{\ell}}} \]

       6. rem-exp-log 44.1%

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

   11. Simplified 44.1%

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

   if -1.44999999999999995e-159 < n < 1.99999999999999989e-157

   1. Initial program 80.9%

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

   3. Taylor expanded in K around 0 95.8%

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

      1. cos-neg 95.8%

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

   5. Simplified 95.8%

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

   6. Taylor expanded in n around inf 10.0%

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

   7. Taylor expanded in n around 0 10.0%

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

      1. associate-*r* 10.0%

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

      2. distribute-rgt1-in 10.0%

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

   9. Simplified 10.0%

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

   10. Taylor expanded in n around inf 80.6%

       \[\leadsto \color{blue}{-0.25 \cdot \left({n}^{2} \cdot \cos M\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 80.6%

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

       2. *-commutative 80.6%

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

       3. associate-*l* 80.6%

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

   12. Simplified 80.6%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.0014:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;n \leq -1.45 \cdot 10^{-159}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-157}:\\ \;\;\;\;{n}^{2} \cdot \left(\cos M \cdot -0.25\right)\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} t_0 := e^{-0.25 \cdot {n}^{2}}\\ \mathbf{if}\;n \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-157}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 1.04 \cdot 10^{-158}:\\ \;\;\;\;{n}^{2} \cdot \left(\cos M \cdot -0.25\right)\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 (exp (* -0.25 (pow n 2.0)))))
   (if (<= n -1.15e+21)
     t_0
     (if (<= n -4.2e-157)
       (* (cos M) (exp (* n (- M (* m 0.5)))))
       (if (<= n 1.04e-158)
         (* (pow n 2.0) (* (cos M) -0.25))
         (if (<= n 2.55e-6) (/ (cos M) (exp l)) t_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 = exp((-0.25 * pow(n, 2.0)));
	double tmp;
	if (n <= -1.15e+21) {
		tmp = t_0;
	} else if (n <= -4.2e-157) {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	} else if (n <= 1.04e-158) {
		tmp = pow(n, 2.0) * (cos(M) * -0.25);
	} else if (n <= 2.55e-6) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = t_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 = exp(((-0.25d0) * (n ** 2.0d0)))
    if (n <= (-1.15d+21)) then
        tmp = t_0
    else if (n <= (-4.2d-157)) then
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    else if (n <= 1.04d-158) then
        tmp = (n ** 2.0d0) * (cos(m_1) * (-0.25d0))
    else if (n <= 2.55d-6) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = t_0
    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.exp((-0.25 * Math.pow(n, 2.0)));
	double tmp;
	if (n <= -1.15e+21) {
		tmp = t_0;
	} else if (n <= -4.2e-157) {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	} else if (n <= 1.04e-158) {
		tmp = Math.pow(n, 2.0) * (Math.cos(M) * -0.25);
	} else if (n <= 2.55e-6) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = t_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.exp((-0.25 * math.pow(n, 2.0)))
	tmp = 0
	if n <= -1.15e+21:
		tmp = t_0
	elif n <= -4.2e-157:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	elif n <= 1.04e-158:
		tmp = math.pow(n, 2.0) * (math.cos(M) * -0.25)
	elif n <= 2.55e-6:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = t_0
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	t_0 = exp(Float64(-0.25 * (n ^ 2.0)))
	tmp = 0.0
	if (n <= -1.15e+21)
		tmp = t_0;
	elseif (n <= -4.2e-157)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	elseif (n <= 1.04e-158)
		tmp = Float64((n ^ 2.0) * Float64(cos(M) * -0.25));
	elseif (n <= 2.55e-6)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = t_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 = exp((-0.25 * (n ^ 2.0)));
	tmp = 0.0;
	if (n <= -1.15e+21)
		tmp = t_0;
	elseif (n <= -4.2e-157)
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	elseif (n <= 1.04e-158)
		tmp = (n ^ 2.0) * (cos(M) * -0.25);
	elseif (n <= 2.55e-6)
		tmp = cos(M) / exp(l);
	else
		tmp = t_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[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, -1.15e+21], t$95$0, If[LessEqual[n, -4.2e-157], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.04e-158], N[(N[Power[n, 2.0], $MachinePrecision] * N[(N[Cos[M], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.55e-6], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.15e21 or 2.5500000000000001e-6 < n

   1. Initial program 65.3%

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

   3. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in n around inf 93.5%

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

   7. Taylor expanded in M around 0 93.5%

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

   if -1.15e21 < n < -4.2e-157

   1. Initial program 74.2%

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

   3. Taylor expanded in K around 0 94.7%

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

      1. cos-neg 94.7%

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

   5. Simplified 94.7%

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

   6. Taylor expanded in n around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. unpow2 90.6%

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

      3. distribute-rgt-out 90.6%

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

      4. *-commutative 90.6%

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

      5. *-commutative 90.6%

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

   8. Simplified 90.6%

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

   9. Taylor expanded in n around inf 45.7%

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

   if -4.2e-157 < n < 1.0399999999999999e-158

   1. Initial program 80.9%

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

   3. Taylor expanded in K around 0 95.8%

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

      1. cos-neg 95.8%

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

   5. Simplified 95.8%

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

   6. Taylor expanded in n around inf 10.0%

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

   7. Taylor expanded in n around 0 10.0%

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

      1. associate-*r* 10.0%

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

      2. distribute-rgt1-in 10.0%

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

   9. Simplified 10.0%

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

   10. Taylor expanded in n around inf 80.6%

       \[\leadsto \color{blue}{-0.25 \cdot \left({n}^{2} \cdot \cos M\right)} \]
   11. Step-by-step derivation

       1. associate-*r* 80.6%

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

       2. *-commutative 80.6%

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

       3. associate-*l* 80.6%

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

   12. Simplified 80.6%

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

   if 1.0399999999999999e-158 < n < 2.5500000000000001e-6

   1. Initial program 67.2%

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

   3. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   5. Simplified 97.0%

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

   6. Taylor expanded in l around inf 36.8%

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

      1. neg-mul-1 36.8%

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

   8. Simplified 36.8%

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

   9. Taylor expanded in l around -inf 36.8%

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

       1. neg-mul-1 36.8%

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

       2. rem-exp-log 23.9%

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

       3. exp-sum 23.9%

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

       4. unsub-neg 23.9%

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

       5. exp-diff 23.9%

          \[\leadsto \color{blue}{\frac{e^{\log \cos M}}{e^{\ell}}} \]

       6. rem-exp-log 36.8%

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

   11. Simplified 36.8%

       \[\leadsto \color{blue}{\frac{\cos M}{e^{\ell}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+21}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;n \leq -4.2 \cdot 10^{-157}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;n \leq 1.04 \cdot 10^{-158}:\\ \;\;\;\;{n}^{2} \cdot \left(\cos M \cdot -0.25\right)\\ \mathbf{elif}\;n \leq 2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= n 6.8e+43)
   (* (cos M) (exp (- (* (- (* m 0.5) M) (- (- M (* m 0.5)) n)) l)))
   (exp (* -0.25 (pow n 2.0)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 6.8e+43) {
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 6.8d+43) then
        tmp = cos(m_1) * exp(((((m * 0.5d0) - m_1) * ((m_1 - (m * 0.5d0)) - n)) - l))
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 6.8e+43) {
		tmp = Math.cos(M) * Math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if n <= 6.8e+43:
		tmp = math.cos(M) * math.exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 6.8e+43)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(Float64(M - Float64(m * 0.5)) - n)) - l)));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 6.8e+43)
		tmp = cos(M) * exp(((((m * 0.5) - M) * ((M - (m * 0.5)) - n)) - l));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[n, 6.8e+43], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision] - n), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 6.80000000000000024e43

   1. Initial program 71.9%

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

   3. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in n around 0 79.1%

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

      1. +-commutative 79.1%

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

      2. unpow2 79.1%

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

      3. distribute-rgt-out 85.3%

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

      4. *-commutative 85.3%

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

      5. *-commutative 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in l around inf 87.0%

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

   if 6.80000000000000024e43 < n

   1. Initial program 66.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around inf 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5 - M\right) \cdot \left(\left(M - m \cdot 0.5\right) - n\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-272}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5\right) \cdot \left(\left(-n\right) - m \cdot 0.5\right) - \ell}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+27}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= n -2.65e-272)
   (* (cos M) (exp (- (* (* m 0.5) (- (- n) (* m 0.5))) l)))
   (if (<= n 1.1e+27)
     (* (cos M) (exp (- (* M (- n M)) l)))
     (exp (* -0.25 (pow n 2.0))))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -2.65e-272) {
		tmp = cos(M) * exp((((m * 0.5) * (-n - (m * 0.5))) - l));
	} else if (n <= 1.1e+27) {
		tmp = cos(M) * exp(((M * (n - M)) - l));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= (-2.65d-272)) then
        tmp = cos(m_1) * exp((((m * 0.5d0) * (-n - (m * 0.5d0))) - l))
    else if (n <= 1.1d+27) then
        tmp = cos(m_1) * exp(((m_1 * (n - m_1)) - l))
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -2.65e-272) {
		tmp = Math.cos(M) * Math.exp((((m * 0.5) * (-n - (m * 0.5))) - l));
	} else if (n <= 1.1e+27) {
		tmp = Math.cos(M) * Math.exp(((M * (n - M)) - l));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if n <= -2.65e-272:
		tmp = math.cos(M) * math.exp((((m * 0.5) * (-n - (m * 0.5))) - l))
	elif n <= 1.1e+27:
		tmp = math.cos(M) * math.exp(((M * (n - M)) - l))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -2.65e-272)
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(m * 0.5) * Float64(Float64(-n) - Float64(m * 0.5))) - l)));
	elseif (n <= 1.1e+27)
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(n - M)) - l)));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -2.65e-272)
		tmp = cos(M) * exp((((m * 0.5) * (-n - (m * 0.5))) - l));
	elseif (n <= 1.1e+27)
		tmp = cos(M) * exp(((M * (n - M)) - l));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[n, -2.65e-272], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(m * 0.5), $MachinePrecision] * N[((-n) - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.1e+27], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.65e-272

   1. Initial program 70.9%

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

   3. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in n around 0 69.5%

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

      1. +-commutative 69.5%

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

      2. unpow2 69.5%

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

      3. distribute-rgt-out 79.0%

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

      4. *-commutative 79.0%

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

      5. *-commutative 79.0%

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

   8. Simplified 79.0%

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

   9. Taylor expanded in l around inf 82.0%

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

   10. Taylor expanded in M around 0 69.4%

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

       1. associate-*r* 69.4%

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

       2. *-commutative 69.4%

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

       3. *-commutative 69.4%

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

   12. Simplified 69.4%

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

   if -2.65e-272 < n < 1.0999999999999999e27

   1. Initial program 74.5%

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

   3. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   5. Simplified 95.1%

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

   6. Taylor expanded in n around 0 93.9%

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

      1. +-commutative 93.9%

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

      2. unpow2 93.9%

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

      3. distribute-rgt-out 95.1%

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

      4. *-commutative 95.1%

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

      5. *-commutative 95.1%

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

   8. Simplified 95.1%

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

   9. Taylor expanded in l around inf 94.6%

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

   10. Taylor expanded in m around 0 77.8%

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

       1. neg-sub0 77.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left(\ell + -1 \cdot \left(M \cdot \left(n - M\right)\right)\right)}} \]

       2. +-commutative 77.8%

          \[\leadsto \cos M \cdot e^{0 - \color{blue}{\left(-1 \cdot \left(M \cdot \left(n - M\right)\right) + \ell\right)}} \]

       3. associate--r+ 77.8%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - -1 \cdot \left(M \cdot \left(n - M\right)\right)\right) - \ell}} \]

       4. neg-sub0 77.8%

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

       5. mul-1-neg 77.8%

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

       6. remove-double-neg 77.8%

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

   12. Simplified 77.8%

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

   if 1.0999999999999999e27 < n

   1. Initial program 64.6%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around inf 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.65 \cdot 10^{-272}:\\ \;\;\;\;\cos M \cdot e^{\left(m \cdot 0.5\right) \cdot \left(\left(-n\right) - m \cdot 0.5\right) - \ell}\\ \mathbf{elif}\;n \leq 1.1 \cdot 10^{+27}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq -0.0014 \lor \neg \left(n \leq 2.55 \cdot 10^{-6}\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -0.0014) (not (<= n 2.55e-6)))
   (exp (* -0.25 (pow n 2.0)))
   (/ (cos M) (exp l))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -0.0014) || !(n <= 2.55e-6)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos(M) / exp(l);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((n <= (-0.0014d0)) .or. (.not. (n <= 2.55d-6))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos(m_1) / exp(l)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -0.0014) || !(n <= 2.55e-6)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if (n <= -0.0014) or not (n <= 2.55e-6):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) / math.exp(l)
	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 ((n <= -0.0014) || !(n <= 2.55e-6))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -0.0014) || ~((n <= 2.55e-6)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -0.0014], N[Not[LessEqual[n, 2.55e-6]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -0.00139999999999999999 or 2.5500000000000001e-6 < n

   1. Initial program 64.9%

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

   3. Taylor expanded in K around 0 99.2%

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

      1. cos-neg 99.2%

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

   5. Simplified 99.2%

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

   6. Taylor expanded in n around inf 92.8%

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

   7. Taylor expanded in M around 0 92.8%

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

   if -0.00139999999999999999 < n < 2.5500000000000001e-6

   1. Initial program 77.1%

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

   3. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   5. Simplified 95.5%

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

   6. Taylor expanded in l around inf 42.8%

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

      1. neg-mul-1 42.8%

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

   8. Simplified 42.8%

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

   9. Taylor expanded in l around -inf 42.8%

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

       1. neg-mul-1 42.8%

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

       2. rem-exp-log 31.8%

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

       3. exp-sum 31.8%

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

       4. unsub-neg 31.8%

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

       5. exp-diff 31.8%

          \[\leadsto \color{blue}{\frac{e^{\log \cos M}}{e^{\ell}}} \]

       6. rem-exp-log 42.8%

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

   11. Simplified 42.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -0.0014 \lor \neg \left(n \leq 2.55 \cdot 10^{-6}\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 2.4 \cdot 10^{+29}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= n 2.4e+29)
   (* (cos M) (exp (- (* M (- n M)) l)))
   (exp (* -0.25 (pow n 2.0)))))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.4e+29) {
		tmp = cos(M) * exp(((M * (n - M)) - l));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 2.4d+29) then
        tmp = cos(m_1) * exp(((m_1 * (n - m_1)) - l))
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 2.4e+29) {
		tmp = Math.cos(M) * Math.exp(((M * (n - M)) - l));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if n <= 2.4e+29:
		tmp = math.cos(M) * math.exp(((M * (n - M)) - l))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 2.4e+29)
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(n - M)) - l)));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 2.4e+29)
		tmp = cos(M) * exp(((M * (n - M)) - l));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[n, 2.4e+29], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.4000000000000001e29

   1. Initial program 72.3%

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

   3. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   5. Simplified 96.8%

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

   6. Taylor expanded in n around 0 79.0%

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

      1. +-commutative 79.0%

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

      2. unpow2 79.0%

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

      3. distribute-rgt-out 85.2%

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

      4. *-commutative 85.2%

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

      5. *-commutative 85.2%

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

   8. Simplified 85.2%

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

   9. Taylor expanded in l around inf 86.9%

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

   10. Taylor expanded in m around 0 74.3%

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

       1. neg-sub0 74.3%

          \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left(\ell + -1 \cdot \left(M \cdot \left(n - M\right)\right)\right)}} \]

       2. +-commutative 74.3%

          \[\leadsto \cos M \cdot e^{0 - \color{blue}{\left(-1 \cdot \left(M \cdot \left(n - M\right)\right) + \ell\right)}} \]

       3. associate--r+ 74.3%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - -1 \cdot \left(M \cdot \left(n - M\right)\right)\right) - \ell}} \]

       4. neg-sub0 74.3%

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

       5. mul-1-neg 74.3%

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

       6. remove-double-neg 74.3%

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

   12. Simplified 74.3%

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

   if 2.4000000000000001e29 < n

   1. Initial program 64.6%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in n around inf 100.0%

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

   7. Taylor expanded in M around 0 100.0%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2.4 \cdot 10^{+29}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) - \ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.0% accurate, 2.1× speedup

Math

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

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 (/ (cos M) (exp l)))

C

assert(K < m && m < n && n < M && M < l);
double code(double K, double m, double n, double M, double l) {
	return cos(M) / 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 = cos(m_1) / 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.cos(M) / Math.exp(l);
}

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(l)

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(cos(M) / exp(l))
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(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[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.8%

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

3. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

5. Simplified 97.4%

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

6. Taylor expanded in l around inf 37.5%

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

   1. neg-mul-1 37.5%

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

8. Simplified 37.5%

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

9. Taylor expanded in l around -inf 37.5%

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

    1. neg-mul-1 37.5%

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

    2. rem-exp-log 26.8%

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

    3. exp-sum 26.8%

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

    4. unsub-neg 26.8%

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

    5. exp-diff 26.8%

       \[\leadsto \color{blue}{\frac{e^{\log \cos M}}{e^{\ell}}} \]

    6. rem-exp-log 37.5%

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

11. Simplified 37.5%

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

12. Final simplification 37.5%

    \[\leadsto \frac{\cos M}{e^{\ell}} \]
13. Add Preprocessing

Alternative 11: 35.6% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.8%

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

3. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

5. Simplified 97.4%

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

6. Taylor expanded in l around inf 37.5%

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

   1. neg-mul-1 37.5%

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

8. Simplified 37.5%

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

9. Taylor expanded in M around 0 36.7%

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

10. Final simplification 36.7%

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

Alternative 12: 6.7% 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 70.8%

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

3. Taylor expanded in K around 0 97.4%

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

   1. cos-neg 97.4%

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

5. Simplified 97.4%

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

6. Taylor expanded in l around inf 37.5%

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

   1. neg-mul-1 37.5%

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

8. Simplified 37.5%

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

9. Taylor expanded in l around 0 7.2%

   \[\leadsto \color{blue}{\cos M} \]

10. Final simplification 7.2%

    \[\leadsto \cos M \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024050 
(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)))))))