Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.6% → 96.5%

Time: 14.2s

Alternatives: 6

Speedup: 3.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r/ 77.2%

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

   3. associate--r- 77.2%

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

   4. +-commutative 77.2%

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

   5. associate-+r- 77.2%

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

   6. unsub-neg 77.2%

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

   7. associate--r+ 77.2%

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

   8. +-commutative 77.2%

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

   9. associate--r+ 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

6. Simplified 98.1%

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

   1. associate--l- 98.1%

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

   2. sub-neg 98.1%

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

   3. add-sqr-sqrt 53.4%

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

   4. fabs-sqr 53.4%

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

   5. add-sqr-sqrt 98.1%

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

   6. div-inv 98.1%

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

   7. metadata-eval 98.1%

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

8. Applied egg-rr 98.1%

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

   1. sub-neg 98.1%

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

   2. associate--l- 98.1%

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

   3. associate-+r+ 98.1%

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

   4. *-commutative 98.1%

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

   5. +-commutative 98.1%

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

10. Simplified 98.1%

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

11. Final simplification 98.1%

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

Alternative 2: 93.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r/ 77.2%

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

   3. associate--r- 77.2%

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

   4. +-commutative 77.2%

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

   5. associate-+r- 77.2%

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

   6. unsub-neg 77.2%

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

   7. associate--r+ 77.2%

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

   8. +-commutative 77.2%

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

   9. associate--r+ 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

6. Simplified 98.1%

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

   1. associate--l- 98.1%

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

   2. sub-neg 98.1%

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

   3. add-sqr-sqrt 53.4%

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

   4. fabs-sqr 53.4%

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

   5. add-sqr-sqrt 98.1%

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

   6. div-inv 98.1%

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

   7. metadata-eval 98.1%

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

8. Applied egg-rr 98.1%

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

   1. sub-neg 98.1%

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

   2. associate--l- 98.1%

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

   3. associate-+r+ 98.1%

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

   4. *-commutative 98.1%

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

   5. +-commutative 98.1%

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

10. Simplified 98.1%

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

11. Taylor expanded in M around inf 73.1%

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

    1. unpow2 73.1%

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

13. Simplified 73.1%

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

14. Final simplification 73.1%

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

Alternative 3: 93.4% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r/ 77.2%

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

   3. associate--r- 77.2%

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

   4. +-commutative 77.2%

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

   5. associate-+r- 77.2%

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

   6. unsub-neg 77.2%

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

   7. associate--r+ 77.2%

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

   8. +-commutative 77.2%

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

   9. associate--r+ 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in K around 0 98.1%

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

   1. cos-neg 98.1%

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

6. Simplified 98.1%

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

   1. associate--l- 98.1%

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

   2. sub-neg 98.1%

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

   3. add-sqr-sqrt 53.4%

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

   4. fabs-sqr 53.4%

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

   5. add-sqr-sqrt 98.1%

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

   6. div-inv 98.1%

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

   7. metadata-eval 98.1%

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

8. Applied egg-rr 98.1%

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

   1. sub-neg 98.1%

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

   2. associate--l- 98.1%

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

   3. associate-+r+ 98.1%

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

   4. *-commutative 98.1%

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

   5. +-commutative 98.1%

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

10. Simplified 98.1%

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

11. Taylor expanded in M around inf 73.1%

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

    1. unpow2 73.1%

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

13. Simplified 73.1%

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

14. Taylor expanded in M around 0 72.7%

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

15. Final simplification 72.7%

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

Alternative 4: 35.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r/ 77.2%

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

   3. associate--r- 77.2%

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

   4. +-commutative 77.2%

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

   5. associate-+r- 77.2%

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

   6. unsub-neg 77.2%

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

   7. associate--r+ 77.2%

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

   8. +-commutative 77.2%

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

   9. associate--r+ 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in l around inf 25.2%

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

   1. neg-mul-1 25.2%

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

6. Simplified 25.2%

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

7. Taylor expanded in K around 0 31.6%

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

   1. cos-neg 31.6%

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

9. Simplified 31.6%

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

10. Taylor expanded in M around 0 31.2%

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

11. Final simplification 31.2%

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

Alternative 5: 6.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r/ 77.2%

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

   3. associate--r- 77.2%

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

   4. +-commutative 77.2%

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

   5. associate-+r- 77.2%

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

   6. unsub-neg 77.2%

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

   7. associate--r+ 77.2%

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

   8. +-commutative 77.2%

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

   9. associate--r+ 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in m around -inf 37.4%

   \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \left(\left(0.5 \cdot n - M\right) \cdot m\right) + -0.25 \cdot {m}^{2}}} \]
5. Step-by-step derivation

   1. +-commutative 37.4%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{-0.25 \cdot {m}^{2} + -1 \cdot \left(\left(0.5 \cdot n - M\right) \cdot m\right)}} \]

   2. mul-1-neg 37.4%

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

   3. unsub-neg 37.4%

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

   4. *-commutative 37.4%

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

   5. unpow2 37.4%

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

   6. associate-*l* 37.4%

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

   7. *-commutative 37.4%

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

6. Simplified 37.4%

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

7. Taylor expanded in m around 0 6.1%

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

8. Taylor expanded in n around 0 6.7%

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

   1. cos-neg 6.7%

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

10. Simplified 6.7%

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

11. Final simplification 6.7%

    \[\leadsto \cos M \]

Alternative 6: 6.9% accurate, 425.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: m and n should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := 1.0

Derivation

1. Initial program 77.2%

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

   1. *-commutative 77.2%

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

   2. associate-*r/ 77.2%

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

   3. associate--r- 77.2%

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

   4. +-commutative 77.2%

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

   5. associate-+r- 77.2%

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

   6. unsub-neg 77.2%

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

   7. associate--r+ 77.2%

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

   8. +-commutative 77.2%

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

   9. associate--r+ 77.2%

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

3. Simplified 77.2%

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

4. Taylor expanded in m around -inf 37.4%

   \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{-1 \cdot \left(\left(0.5 \cdot n - M\right) \cdot m\right) + -0.25 \cdot {m}^{2}}} \]
5. Step-by-step derivation

   1. +-commutative 37.4%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\color{blue}{-0.25 \cdot {m}^{2} + -1 \cdot \left(\left(0.5 \cdot n - M\right) \cdot m\right)}} \]

   2. mul-1-neg 37.4%

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

   3. unsub-neg 37.4%

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

   4. *-commutative 37.4%

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

   5. unpow2 37.4%

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

   6. associate-*l* 37.4%

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

   7. *-commutative 37.4%

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

6. Simplified 37.4%

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

7. Taylor expanded in m around 0 6.1%

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

8. Taylor expanded in n around 0 6.7%

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

   1. cos-neg 6.7%

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

10. Simplified 6.7%

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

11. Taylor expanded in M around 0 6.7%

    \[\leadsto \color{blue}{1} \]

12. Final simplification 6.7%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023215 
(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)))))))