Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.2% → 96.4%

Time: 16.1s

Alternatives: 8

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: 75.2% 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: 96.4% accurate, 1.3× speedup

Math

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

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
 (* (exp (- (- (- n m) l) (pow (- (* 0.5 (+ n m)) M) 2.0))) (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 exp((((n - m) - l) - pow(((0.5 * (n + m)) - M), 2.0))) * 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 = exp((((n - m) - l) - (((0.5d0 * (n + m)) - m_1) ** 2.0d0))) * 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.exp((((n - m) - l) - Math.pow(((0.5 * (n + m)) - M), 2.0))) * Math.cos(M);
}

Python

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

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	return Float64(exp(Float64(Float64(Float64(n - m) - l) - (Float64(Float64(0.5 * Float64(n + m)) - M) ^ 2.0))) * 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 = exp((((n - m) - l) - (((0.5 * (n + m)) - M) ^ 2.0))) * 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[(N[Exp[N[(N[(N[(n - m), $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[M], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in K around 0 96.7%

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

5. Simplified 96.7%

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

   1. unpow2 96.7%

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

   2. +-commutative 96.7%

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

   3. +-commutative 96.7%

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

7. Applied egg-rr 96.7%

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

8. Taylor expanded in n around -inf 96.7%

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

   1. fabs-neg 96.7%

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

   2. neg-mul-1 96.7%

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

   3. sub-neg 96.7%

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

   4. rem-square-sqrt 51.3%

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

   5. fabs-sqr 51.3%

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

   6. rem-square-sqrt 96.6%

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

10. Simplified 96.6%

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

    1. sub-neg 96.6%

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

    2. add-sqr-sqrt 51.3%

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

    3. fabs-sqr 51.3%

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

    4. add-sqr-sqrt 96.7%

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

    5. fabs-sub 96.7%

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

    6. add-sqr-sqrt 45.4%

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

    7. fabs-sqr 45.4%

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

    8. add-sqr-sqrt 96.3%

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

    9. pow2 96.3%

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

    10. *-commutative 96.3%

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

12. Applied egg-rr 96.3%

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

    1. sub-neg 96.3%

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

    2. associate--r+ 96.3%

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

    3. *-commutative 96.3%

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

    4. +-commutative 96.3%

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

14. Simplified 96.3%

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

15. Final simplification 96.3%

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

Alternative 2: 96.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in K around 0 96.7%

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

5. Simplified 96.7%

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

   1. unpow2 96.7%

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

   2. +-commutative 96.7%

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

   3. +-commutative 96.7%

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

7. Applied egg-rr 96.7%

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

8. Taylor expanded in n around -inf 96.7%

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

   1. fabs-neg 96.7%

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

   2. neg-mul-1 96.7%

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

   3. sub-neg 96.7%

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

   4. rem-square-sqrt 51.3%

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

   5. fabs-sqr 51.3%

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

   6. rem-square-sqrt 96.6%

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

10. Simplified 96.6%

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

11. Final simplification 96.6%

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

Alternative 3: 88.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(0.5 \cdot \left(n + m\right) - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) - \ell}\\ \end{array} \end{array} \]

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 (<= n 2e+138)
   (* (cos M) (exp (+ (- m n) (- (* M (- (* 0.5 (+ n m)) M)) l))))
   (* (cos M) (exp (- (- m n) 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 <= 2e+138) {
		tmp = cos(M) * exp(((m - n) + ((M * ((0.5 * (n + m)) - M)) - l)));
	} else {
		tmp = cos(M) * exp(((m - n) - 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 <= 2d+138) then
        tmp = cos(m_1) * exp(((m - n) + ((m_1 * ((0.5d0 * (n + m)) - m_1)) - l)))
    else
        tmp = cos(m_1) * exp(((m - n) - 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 <= 2e+138) {
		tmp = Math.cos(M) * Math.exp(((m - n) + ((M * ((0.5 * (n + m)) - M)) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp(((m - n) - 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 <= 2e+138:
		tmp = math.cos(M) * math.exp(((m - n) + ((M * ((0.5 * (n + m)) - M)) - l)))
	else:
		tmp = math.cos(M) * math.exp(((m - n) - 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 <= 2e+138)
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(Float64(M * Float64(Float64(0.5 * Float64(n + m)) - M)) - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) - 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 <= 2e+138)
		tmp = cos(M) * exp(((m - n) + ((M * ((0.5 * (n + m)) - M)) - l)));
	else
		tmp = cos(M) * exp(((m - n) - 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[LessEqual[n, 2e+138], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(M * N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 2.0000000000000001e138

   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 96.2%

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

   5. Simplified 96.2%

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

      1. unpow2 96.2%

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

      2. +-commutative 96.2%

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

      3. +-commutative 96.2%

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

   7. Applied egg-rr 96.2%

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

   8. Taylor expanded in n around -inf 96.2%

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

      1. fabs-neg 96.2%

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

      2. neg-mul-1 96.2%

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

      3. sub-neg 96.2%

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

      4. rem-square-sqrt 59.1%

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

      5. fabs-sqr 59.1%

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

      6. rem-square-sqrt 96.0%

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

   10. Simplified 96.0%

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

   11. Taylor expanded in M around inf 67.7%

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

       1. neg-mul-1 67.7%

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

   13. Simplified 67.7%

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

   if 2.0000000000000001e138 < n

   1. Initial program 56.8%

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

   3. Taylor expanded in K around 0 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 6.9%

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

   7. Taylor expanded in n around -inf 6.9%

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

      1. fabs-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. sub-neg 100.0%

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

      4. rem-square-sqrt 5.4%

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

      5. fabs-sqr 5.4%

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

      6. rem-square-sqrt 100.0%

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

   9. Simplified 89.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(0.5 \cdot \left(n + m\right) - M\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+59}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) - \ell}\\ \end{array} \end{array} \]

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 (<= l -5e+59) (* (cos M) (exp l)) (* (cos M) (exp (- (- m n) 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 (l <= -5e+59) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = cos(M) * exp(((m - n) - 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 (l <= (-5d+59)) then
        tmp = cos(m_1) * exp(l)
    else
        tmp = cos(m_1) * exp(((m - n) - 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 (l <= -5e+59) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.cos(M) * Math.exp(((m - n) - 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 l <= -5e+59:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.cos(M) * math.exp(((m - n) - 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 (l <= -5e+59)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) - 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 (l <= -5e+59)
		tmp = cos(M) * exp(l);
	else
		tmp = cos(M) * exp(((m - n) - 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[LessEqual[l, -5e+59], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] - l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.9999999999999997e59

   1. Initial program 75.0%

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

   3. Taylor expanded in l around inf 17.0%

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

      1. mul-1-neg 17.0%

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

   5. Simplified 17.0%

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

   6. Taylor expanded in K around 0 12.0%

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

      1. cos-neg 12.0%

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

   8. Simplified 12.0%

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

      1. pow1 12.0%

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

      2. add-sqr-sqrt 12.0%

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

      3. sqrt-unprod 12.0%

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

      4. sqr-neg 12.0%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 82.4%

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

   10. Applied egg-rr 82.4%

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

       1. unpow1 82.4%

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

   12. Simplified 82.4%

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

   if -4.9999999999999997e59 < l

   1. Initial program 78.1%

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

   3. Taylor expanded in K around 0 97.8%

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

   5. Simplified 97.8%

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

   6. Taylor expanded in l around inf 29.5%

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

   7. Taylor expanded in n around -inf 29.5%

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

      1. fabs-neg 97.8%

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

      2. neg-mul-1 97.8%

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

      3. sub-neg 97.8%

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

      4. rem-square-sqrt 51.2%

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

      5. fabs-sqr 51.2%

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

      6. rem-square-sqrt 97.6%

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

   9. Simplified 58.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+59}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) - \ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-34}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

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 (<= l -1e-34) (* (cos M) (exp l)) (* (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 (l <= -1e-34) {
		tmp = cos(M) * exp(l);
	} 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 (l <= (-1d-34)) then
        tmp = cos(m_1) * exp(l)
    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 (l <= -1e-34) {
		tmp = Math.cos(M) * Math.exp(l);
	} 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 l <= -1e-34:
		tmp = math.cos(M) * math.exp(l)
	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 (l <= -1e-34)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -1e-34)
		tmp = cos(M) * exp(l);
	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[LessEqual[l, -1e-34], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.99999999999999928e-35

   1. Initial program 78.3%

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

   3. Taylor expanded in l around inf 18.4%

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

      1. mul-1-neg 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in K around 0 14.4%

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

      1. cos-neg 14.4%

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

   8. Simplified 14.4%

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

      1. pow1 14.4%

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

      2. add-sqr-sqrt 14.4%

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

      3. sqrt-unprod 14.4%

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

      4. sqr-neg 14.4%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 74.4%

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

   10. Applied egg-rr 74.4%

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

       1. unpow1 74.4%

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

   12. Simplified 74.4%

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

   if -9.99999999999999928e-35 < l

   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 l around inf 39.1%

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

      1. mul-1-neg 39.1%

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

   5. Simplified 39.1%

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

   6. Taylor expanded in K around 0 44.6%

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

      1. cos-neg 44.6%

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

   8. Simplified 44.6%

      \[\leadsto \color{blue}{\cos M} \cdot e^{-\ell} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 49.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [K, m, n, M, l] = \mathsf{sort}([K, m, n, M, l])\\ \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.12:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
(FPCore (K m n M l)
 :precision binary64
 (if (<= l 0.12) (* (cos M) (exp l)) (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 (l <= 0.12) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 0.12d0) then
        tmp = cos(m_1) * exp(l)
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

assert K < m && m < n && n < M && M < l;
public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.12) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

[K, m, n, M, l] = sort([K, m, n, M, l])
def code(K, m, n, M, l):
	tmp = 0
	if l <= 0.12:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.exp(-l)
	return tmp

Julia

K, m, n, M, l = sort([K, m, n, M, l])
function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 0.12)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

K, m, n, M, l = num2cell(sort([K, m, n, M, l])){:}
function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 0.12)
		tmp = cos(M) * exp(l);
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: K, m, n, M, and l should be sorted in increasing order before calling this function.
code[K_, m_, n_, M_, l_] := If[LessEqual[l, 0.12], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.12

   1. Initial program 74.9%

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

   3. Taylor expanded in l around inf 15.7%

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

      1. mul-1-neg 15.7%

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

   5. Simplified 15.7%

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

   6. Taylor expanded in K around 0 14.4%

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

      1. cos-neg 14.4%

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

   8. Simplified 14.4%

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

      1. pow1 14.4%

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

      2. add-sqr-sqrt 8.9%

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

      3. sqrt-unprod 14.4%

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

      4. sqr-neg 14.4%

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

      5. sqrt-unprod 5.6%

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

      6. add-sqr-sqrt 36.2%

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

   10. Applied egg-rr 36.2%

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

       1. unpow1 36.2%

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

   12. Simplified 36.2%

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

   if 0.12 < l

   1. Initial program 84.8%

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

   3. Taylor expanded in l around inf 84.8%

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

      1. mul-1-neg 84.8%

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

   5. Simplified 84.8%

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

   6. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in M around 0 100.0%

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

Alternative 7: 35.4% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in l around inf 33.5%

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

   1. mul-1-neg 33.5%

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

5. Simplified 33.5%

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

6. Taylor expanded in K around 0 36.5%

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

   1. cos-neg 36.5%

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

8. Simplified 36.5%

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

9. Taylor expanded in M around 0 36.5%

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

Alternative 8: 7.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.4%

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

3. Taylor expanded in l around inf 33.5%

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

   1. mul-1-neg 33.5%

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

5. Simplified 33.5%

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

6. Taylor expanded in l around 0 8.0%

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

7. Taylor expanded in K around 0 8.4%

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

   1. cos-neg 8.4%

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

9. Simplified 8.4%

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

Reproduce

herbie shell --seed 2024144 
(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)))))))