Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.0% → 96.7%

Time: 22.3s

Alternatives: 12

Speedup: 1.3×

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: 76.0% 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: 96.7% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- n m) (+ l (pow (- (* 0.5 (+ n m)) M) 2.0))))))

C

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

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(m_1) * exp(((n - m) - (l + (((0.5d0 * (n + m)) - m_1) ** 2.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(((n - m) - (l + (((0.5 * (n + m)) - M) ^ 2.0))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n - m), $MachinePrecision] - N[(l + N[Power[N[(N[(0.5 * N[(n + m), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Simplified 79.2%

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

   1. add-sqr-sqrt 79.2%

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

   2. sqrt-unprod 79.3%

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

   3. pow-prod-up 79.3%

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

   4. div-inv 79.3%

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

   5. fma-neg 79.3%

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

   6. metadata-eval 79.3%

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

   7. metadata-eval 79.3%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - \sqrt{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{\color{blue}{4}}}} \]

6. Applied egg-rr 79.3%

   \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - \color{blue}{\sqrt{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{4}}}} \]
7. Step-by-step derivation

   1. sqrt-pow1 79.2%

      \[\leadsto \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot e^{\left(\left|n - m\right| - \ell\right) - \color{blue}{{\left(\mathsf{fma}\left(m + n, 0.5, -M\right)\right)}^{\left(\frac{4}{2}\right)}}} \]

   2. metadata-eval 79.2%

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

   3. fma-neg 79.2%

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

   4. metadata-eval 79.2%

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

   5. div-inv 79.2%

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

   6. unpow2 79.2%

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

   7. div-inv 79.2%

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

   8. metadata-eval 79.2%

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

   9. +-commutative 79.2%

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

   10. div-inv 79.2%

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

   11. metadata-eval 79.2%

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

   12. +-commutative 79.2%

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

8. Applied egg-rr 79.2%

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

9. Taylor expanded in K around 0 97.3%

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

    1. cos-neg 97.3%

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

11. Simplified 97.3%

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

    1. associate--l- 97.3%

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

    2. add-sqr-sqrt 46.1%

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

    3. fabs-sqr 46.1%

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

    4. add-sqr-sqrt 97.3%

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

    5. pow2 97.3%

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

    6. fma-neg 97.3%

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

13. Applied egg-rr 97.3%

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

    1. +-commutative 97.3%

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

    2. fma-neg 97.3%

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

    3. *-commutative 97.3%

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

    4. +-commutative 97.3%

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

15. Simplified 97.3%

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

Alternative 2: 69.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1660000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) - \left(\ell - \left|n - m\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1660000.0)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m 3.8e-164)
     (* (cos M) (exp (- (* M (- n M)) (- l (fabs (- n m))))))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1660000.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= 3.8e-164) {
		tmp = cos(M) * exp(((M * (n - M)) - (l - fabs((n - m)))));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= (-1660000.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= 3.8d-164) then
        tmp = cos(m_1) * exp(((m_1 * (n - m_1)) - (l - abs((n - m)))))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1660000.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= 3.8e-164) {
		tmp = Math.cos(M) * Math.exp(((M * (n - M)) - (l - Math.abs((n - m)))));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1660000.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= 3.8e-164:
		tmp = math.cos(M) * math.exp(((M * (n - M)) - (l - math.fabs((n - m)))))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1660000.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= 3.8e-164)
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(n - M)) - Float64(l - abs(Float64(n - m))))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1660000.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= 3.8e-164)
		tmp = cos(M) * exp(((M * (n - M)) - (l - abs((n - m)))));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1660000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 3.8e-164], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision] - N[(l - N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.66e6

   1. Initial program 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 100.0%

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

   if -1.66e6 < m < 3.79999999999999989e-164

   1. Initial program 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in K around 0 94.3%

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

      1. cos-neg 94.3%

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

   7. Simplified 94.3%

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

   8. Taylor expanded in n around 0 75.0%

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

      1. +-commutative 75.0%

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

      2. unpow2 75.0%

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

      3. distribute-rgt-out 79.6%

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

   10. Simplified 79.6%

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

   11. Taylor expanded in m around 0 79.6%

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

       1. associate--r+ 79.6%

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

       2. fabs-sub 79.6%

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

       3. associate-*r* 79.6%

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

       4. neg-mul-1 79.6%

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

       5. cancel-sign-sub 79.6%

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

   13. Simplified 79.6%

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

   if 3.79999999999999989e-164 < m

   1. Initial program 72.7%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in n around inf 59.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1660000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 3.8 \cdot 10^{-164}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right) - \left(\ell - \left|n - m\right|\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1660000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 1.18 \cdot 10^{-222}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1660000.0)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m 1.18e-222)
     (* (cos M) (exp (- (pow M 2.0))))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1660000.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= 1.18e-222) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= (-1660000.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= 1.18d-222) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1660000.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= 1.18e-222) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1660000.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= 1.18e-222:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1660000.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= 1.18e-222)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1660000.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= 1.18e-222)
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1660000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.18e-222], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.66e6

   1. Initial program 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 100.0%

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

   if -1.66e6 < m < 1.18000000000000007e-222

   1. Initial program 92.5%

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

   3. Simplified 92.5%

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

   5. Taylor expanded in K around 0 94.8%

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

      1. cos-neg 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in M around inf 71.0%

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

      1. mul-1-neg 71.0%

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

   10. Simplified 71.0%

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

   if 1.18000000000000007e-222 < m

   1. Initial program 72.4%

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

   3. Simplified 72.4%

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

   5. Taylor expanded in K around 0 97.7%

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

      1. cos-neg 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in n around inf 60.0%

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

Alternative 4: 69.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -6.2 \lor \neg \left(M \leq 2.35 \cdot 10^{-32}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -6.2) (not (<= M 2.35e-32)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -6.2) || !(M <= 2.35e-32)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((m_1 <= (-6.2d0)) .or. (.not. (m_1 <= 2.35d-32))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -6.2) || !(M <= 2.35e-32)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -6.2) or not (M <= 2.35e-32):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -6.2) || !(M <= 2.35e-32))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -6.2) || ~((M <= 2.35e-32)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -6.2], N[Not[LessEqual[M, 2.35e-32]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -6.20000000000000018 or 2.3500000000000001e-32 < M

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in M around inf 94.8%

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

      1. mul-1-neg 94.8%

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

   10. Simplified 94.8%

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

   if -6.20000000000000018 < M < 2.3500000000000001e-32

   1. Initial program 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in K around 0 94.6%

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

      1. cos-neg 94.6%

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

   7. Simplified 94.6%

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

   8. Taylor expanded in l around inf 49.5%

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

      1. neg-mul-1 49.5%

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

   10. Simplified 49.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -6.2 \lor \neg \left(M \leq 2.35 \cdot 10^{-32}\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 58.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1660000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-181}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(m \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1660000.0)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= m 1.05e-181)
     (* (cos M) (exp (- (pow M 2.0))))
     (* (cos M) (exp (* n (* m -0.5)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1660000.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (m <= 1.05e-181) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp((n * (m * -0.5)));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (m <= (-1660000.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (m <= 1.05d-181) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp((n * (m * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1660000.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (m <= 1.05e-181) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp((n * (m * -0.5)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1660000.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif m <= 1.05e-181:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp((n * (m * -0.5)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1660000.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (m <= 1.05e-181)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(n * Float64(m * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1660000.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (m <= 1.05e-181)
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp((n * (m * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1660000.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 1.05e-181], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(m * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -1.66e6

   1. Initial program 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 100.0%

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

   if -1.66e6 < m < 1.05000000000000002e-181

   1. Initial program 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

   7. Simplified 94.2%

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

   8. Taylor expanded in M around inf 69.1%

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

      1. mul-1-neg 69.1%

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

   10. Simplified 69.1%

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

   if 1.05000000000000002e-181 < m

   1. Initial program 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in n around 0 72.0%

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

      1. +-commutative 72.0%

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

      2. unpow2 72.0%

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

      3. distribute-rgt-out 80.9%

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

   10. Simplified 80.9%

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

   11. Taylor expanded in n around inf 38.7%

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

   12. Taylor expanded in M around 0 34.4%

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

       1. associate-*r* 34.4%

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

       2. *-commutative 34.4%

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

   14. Simplified 34.4%

       \[\leadsto \cos M \cdot e^{\color{blue}{\left(m \cdot -0.5\right) \cdot n}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1660000:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 1.05 \cdot 10^{-181}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(m \cdot -0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -260:\\ \;\;\;\;e^{\ell} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right) - M\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-20}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(n \cdot \frac{M}{m} - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -260.0)
   (* (exp l) (cos (- (* 0.5 (* (+ n m) K)) M)))
   (if (<= l 1.5e-20)
     (* (cos M) (exp (* m (- (* n (/ M m)) (* n 0.5)))))
     (* (cos M) (exp (- l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -260.0) {
		tmp = exp(l) * cos(((0.5 * ((n + m) * K)) - M));
	} else if (l <= 1.5e-20) {
		tmp = cos(M) * exp((m * ((n * (M / m)) - (n * 0.5))));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-260.0d0)) then
        tmp = exp(l) * cos(((0.5d0 * ((n + m) * k)) - m_1))
    else if (l <= 1.5d-20) then
        tmp = cos(m_1) * exp((m * ((n * (m_1 / m)) - (n * 0.5d0))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -260.0) {
		tmp = Math.exp(l) * Math.cos(((0.5 * ((n + m) * K)) - M));
	} else if (l <= 1.5e-20) {
		tmp = Math.cos(M) * Math.exp((m * ((n * (M / m)) - (n * 0.5))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -260.0:
		tmp = math.exp(l) * math.cos(((0.5 * ((n + m) * K)) - M))
	elif l <= 1.5e-20:
		tmp = math.cos(M) * math.exp((m * ((n * (M / m)) - (n * 0.5))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -260.0)
		tmp = Float64(exp(l) * cos(Float64(Float64(0.5 * Float64(Float64(n + m) * K)) - M)));
	elseif (l <= 1.5e-20)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(Float64(n * Float64(M / m)) - Float64(n * 0.5)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -260.0)
		tmp = exp(l) * cos(((0.5 * ((n + m) * K)) - M));
	elseif (l <= 1.5e-20)
		tmp = cos(M) * exp((m * ((n * (M / m)) - (n * 0.5))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -260.0], N[(N[Exp[l], $MachinePrecision] * N[Cos[N[(N[(0.5 * N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e-20], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(N[(n * N[(M / m), $MachinePrecision]), $MachinePrecision] - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -260

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

      1. add-cbrt-cube 73.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)\right) \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      2. pow3 73.2%

         \[\leadsto \sqrt[3]{\color{blue}{{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)}^{3}}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      3. *-commutative 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(\color{blue}{\frac{K}{2} \cdot \left(m + n\right)} - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      4. div-inv 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(\color{blue}{\left(K \cdot \frac{1}{2}\right)} \cdot \left(m + n\right) - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      5. associate-*l* 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(\color{blue}{K \cdot \left(\frac{1}{2} \cdot \left(m + n\right)\right)} - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      6. *-commutative 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(K \cdot \color{blue}{\left(\left(m + n\right) \cdot \frac{1}{2}\right)} - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      7. metadata-eval 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(K \cdot \left(\left(m + n\right) \cdot \color{blue}{0.5}\right) - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}^{3}}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   7. Taylor expanded in l around inf 18.7%

      \[\leadsto \sqrt[3]{{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}^{3}} \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. neg-mul-1 22.6%

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

   9. Simplified 18.7%

      \[\leadsto \sqrt[3]{{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}^{3}} \cdot e^{\color{blue}{-\ell}} \]
   10. Step-by-step derivation

       1. rem-cbrt-cube 18.7%

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

       2. *-commutative 18.7%

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

       3. add-sqr-sqrt 18.7%

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

       4. sqrt-unprod 18.7%

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

       5. sqr-neg 18.7%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 54.0%

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

       8. associate-*r* 54.0%

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

       9. +-commutative 54.0%

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

   11. Applied egg-rr 54.0%

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

   if -260 < l < 1.50000000000000014e-20

   1. Initial program 82.6%

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

   3. Simplified 82.6%

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

   5. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in n around 0 71.2%

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

      1. +-commutative 71.2%

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

      2. unpow2 71.2%

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

      3. distribute-rgt-out 79.2%

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

   10. Simplified 79.2%

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

   11. Taylor expanded in n around inf 40.8%

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

   12. Taylor expanded in m around -inf 40.8%

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

       1. mul-1-neg 40.8%

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

       2. distribute-rgt-neg-in 40.8%

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

       3. +-commutative 40.8%

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

       4. mul-1-neg 40.8%

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

       5. unsub-neg 40.8%

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

       6. *-commutative 40.8%

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

       7. *-commutative 40.8%

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

       8. associate-/l* 44.6%

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

   14. Simplified 44.6%

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

   if 1.50000000000000014e-20 < l

   1. Initial program 78.1%

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

   3. Simplified 78.1%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 95.4%

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

      1. neg-mul-1 95.4%

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

   10. Simplified 95.4%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -260:\\ \;\;\;\;e^{\ell} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right) - M\right)\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-20}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(n \cdot \frac{M}{m} - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -260:\\ \;\;\;\;e^{\ell} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right) - M\right)\\ \mathbf{elif}\;\ell \leq 3.45 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -260.0)
   (* (exp l) (cos (- (* 0.5 (* (+ n m) K)) M)))
   (if (<= l 3.45e-15)
     (* (cos M) (exp (* n (- M (* m 0.5)))))
     (* (cos M) (exp (- l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -260.0) {
		tmp = exp(l) * cos(((0.5 * ((n + m) * K)) - M));
	} else if (l <= 3.45e-15) {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= (-260.0d0)) then
        tmp = exp(l) * cos(((0.5d0 * ((n + m) * k)) - m_1))
    else if (l <= 3.45d-15) then
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -260.0) {
		tmp = Math.exp(l) * Math.cos(((0.5 * ((n + m) * K)) - M));
	} else if (l <= 3.45e-15) {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -260.0:
		tmp = math.exp(l) * math.cos(((0.5 * ((n + m) * K)) - M))
	elif l <= 3.45e-15:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -260.0)
		tmp = Float64(exp(l) * cos(Float64(Float64(0.5 * Float64(Float64(n + m) * K)) - M)));
	elseif (l <= 3.45e-15)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -260.0)
		tmp = exp(l) * cos(((0.5 * ((n + m) * K)) - M));
	elseif (l <= 3.45e-15)
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -260.0], N[(N[Exp[l], $MachinePrecision] * N[Cos[N[(N[(0.5 * N[(N[(n + m), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.45e-15], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -260

   1. Initial program 73.2%

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

   3. Simplified 73.2%

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

      1. add-cbrt-cube 73.2%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right) \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)\right) \cdot \cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      2. pow3 73.2%

         \[\leadsto \sqrt[3]{\color{blue}{{\cos \left(\left(m + n\right) \cdot \frac{K}{2} - M\right)}^{3}}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      3. *-commutative 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(\color{blue}{\frac{K}{2} \cdot \left(m + n\right)} - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      4. div-inv 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(\color{blue}{\left(K \cdot \frac{1}{2}\right)} \cdot \left(m + n\right) - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      5. associate-*l* 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(\color{blue}{K \cdot \left(\frac{1}{2} \cdot \left(m + n\right)\right)} - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      6. *-commutative 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(K \cdot \color{blue}{\left(\left(m + n\right) \cdot \frac{1}{2}\right)} - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

      7. metadata-eval 73.2%

         \[\leadsto \sqrt[3]{{\cos \left(K \cdot \left(\left(m + n\right) \cdot \color{blue}{0.5}\right) - M\right)}^{3}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   6. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\sqrt[3]{{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}^{3}}} \cdot e^{\left(\left|n - m\right| - \ell\right) - {\left(\frac{m + n}{2} - M\right)}^{2}} \]

   7. Taylor expanded in l around inf 18.7%

      \[\leadsto \sqrt[3]{{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}^{3}} \cdot e^{\color{blue}{-1 \cdot \ell}} \]
   8. Step-by-step derivation

      1. neg-mul-1 22.6%

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

   9. Simplified 18.7%

      \[\leadsto \sqrt[3]{{\cos \left(K \cdot \left(\left(m + n\right) \cdot 0.5\right) - M\right)}^{3}} \cdot e^{\color{blue}{-\ell}} \]
   10. Step-by-step derivation

       1. rem-cbrt-cube 18.7%

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

       2. *-commutative 18.7%

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

       3. add-sqr-sqrt 18.7%

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

       4. sqrt-unprod 18.7%

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

       5. sqr-neg 18.7%

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

       6. sqrt-unprod 0.0%

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

       7. add-sqr-sqrt 54.0%

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

       8. associate-*r* 54.0%

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

       9. +-commutative 54.0%

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

   11. Applied egg-rr 54.0%

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

   if -260 < l < 3.45000000000000005e-15

   1. Initial program 82.7%

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

   3. Simplified 82.7%

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

   5. Taylor expanded in K around 0 97.0%

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

      1. cos-neg 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in n around 0 71.4%

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

      1. +-commutative 71.4%

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

      2. unpow2 71.4%

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

      3. distribute-rgt-out 79.3%

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

   10. Simplified 79.3%

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

   11. Taylor expanded in n around inf 40.9%

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

   if 3.45000000000000005e-15 < l

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 96.0%

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

      1. neg-mul-1 96.0%

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

   10. Simplified 96.0%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -260:\\ \;\;\;\;e^{\ell} \cdot \cos \left(0.5 \cdot \left(\left(n + m\right) \cdot K\right) - M\right)\\ \mathbf{elif}\;\ell \leq 3.45 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{-145}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{elif}\;\ell \leq 3.45 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(m \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 2.4e-145)
   (* (cos M) (exp (* M n)))
   (if (<= l 3.45e-15)
     (* (cos M) (exp (* n (* m -0.5))))
     (* (cos M) (exp (- l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 2.4e-145) {
		tmp = cos(M) * exp((M * n));
	} else if (l <= 3.45e-15) {
		tmp = cos(M) * exp((n * (m * -0.5)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 2.4d-145) then
        tmp = cos(m_1) * exp((m_1 * n))
    else if (l <= 3.45d-15) then
        tmp = cos(m_1) * exp((n * (m * (-0.5d0))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 2.4e-145) {
		tmp = Math.cos(M) * Math.exp((M * n));
	} else if (l <= 3.45e-15) {
		tmp = Math.cos(M) * Math.exp((n * (m * -0.5)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 2.4e-145:
		tmp = math.cos(M) * math.exp((M * n))
	elif l <= 3.45e-15:
		tmp = math.cos(M) * math.exp((n * (m * -0.5)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 2.4e-145)
		tmp = Float64(cos(M) * exp(Float64(M * n)));
	elseif (l <= 3.45e-15)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(m * -0.5))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 2.4e-145)
		tmp = cos(M) * exp((M * n));
	elseif (l <= 3.45e-15)
		tmp = cos(M) * exp((n * (m * -0.5)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 2.4e-145], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.45e-15], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(m * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < 2.40000000000000015e-145

   1. Initial program 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in K around 0 96.3%

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

      1. cos-neg 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in n around 0 74.8%

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

      1. +-commutative 74.8%

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

      2. unpow2 74.8%

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

      3. distribute-rgt-out 81.3%

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

   10. Simplified 81.3%

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

   11. Taylor expanded in n around inf 37.2%

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

   12. Taylor expanded in m around 0 31.2%

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

       1. *-commutative 31.2%

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

       2. *-commutative 31.2%

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

   14. Simplified 31.2%

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

   if 2.40000000000000015e-145 < l < 3.45000000000000005e-15

   1. Initial program 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in K around 0 96.4%

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

      1. cos-neg 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in n around 0 66.9%

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

      1. +-commutative 66.9%

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

      2. unpow2 66.9%

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

      3. distribute-rgt-out 74.4%

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

   10. Simplified 74.4%

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

   11. Taylor expanded in n around inf 44.1%

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

   12. Taylor expanded in M around 0 47.8%

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

       1. associate-*r* 47.8%

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

       2. *-commutative 47.8%

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

   14. Simplified 47.8%

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

   if 3.45000000000000005e-15 < l

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 96.0%

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

      1. neg-mul-1 96.0%

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

   10. Simplified 96.0%

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

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 2.4 \cdot 10^{-145}:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{elif}\;\ell \leq 3.45 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(m \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 3.45 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 3.45e-15)
   (* (cos M) (exp (* n (- M (* m 0.5)))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 3.45e-15) {
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 3.45d-15) then
        tmp = cos(m_1) * exp((n * (m_1 - (m * 0.5d0))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 3.45e-15) {
		tmp = Math.cos(M) * Math.exp((n * (M - (m * 0.5))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 3.45e-15:
		tmp = math.cos(M) * math.exp((n * (M - (m * 0.5))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 3.45e-15)
		tmp = Float64(cos(M) * exp(Float64(n * Float64(M - Float64(m * 0.5)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 3.45e-15)
		tmp = cos(M) * exp((n * (M - (m * 0.5))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 3.45e-15], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 3.45000000000000005e-15

   1. Initial program 79.8%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in K around 0 96.3%

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

      1. cos-neg 96.3%

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

   7. Simplified 96.3%

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

   8. Taylor expanded in n around 0 73.7%

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

      1. +-commutative 73.7%

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

      2. unpow2 73.7%

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

      3. distribute-rgt-out 80.2%

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

   10. Simplified 80.2%

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

   11. Taylor expanded in n around inf 38.2%

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

   if 3.45000000000000005e-15 < l

   1. Initial program 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 96.0%

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

      1. neg-mul-1 96.0%

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

   10. Simplified 96.0%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 3.45 \cdot 10^{-15}:\\ \;\;\;\;\cos M \cdot e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.0135:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 0.0135) (* (cos M) (exp (* M n))) (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.0135) {
		tmp = cos(M) * exp((M * n));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (l <= 0.0135d0) then
        tmp = cos(m_1) * exp((m_1 * n))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 0.0135) {
		tmp = Math.cos(M) * Math.exp((M * n));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 0.0135:
		tmp = math.cos(M) * math.exp((M * n))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 0.0135)
		tmp = Float64(cos(M) * exp(Float64(M * n)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 0.0135)
		tmp = cos(M) * exp((M * n));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 0.0135], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.0134999999999999998

   1. Initial program 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in K around 0 96.4%

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

      1. cos-neg 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in n around 0 73.5%

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

      1. +-commutative 73.5%

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

      2. unpow2 73.5%

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

      3. distribute-rgt-out 80.0%

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

   10. Simplified 80.0%

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

   11. Taylor expanded in n around inf 37.9%

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

   12. Taylor expanded in m around 0 31.8%

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

       1. *-commutative 31.8%

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

       2. *-commutative 31.8%

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

   14. Simplified 31.8%

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

   if 0.0134999999999999998 < l

   1. Initial program 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 98.6%

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

      1. neg-mul-1 98.6%

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

   10. Simplified 98.6%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.0135:\\ \;\;\;\;\cos M \cdot e^{M \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \cos M \cdot e^{-\ell} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (* (cos M) (exp (- l))))

C

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

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(m_1) * exp(-l)
end function

Java

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

Python

def code(K, m, n, M, l):
	return math.cos(M) * math.exp(-l)

Julia

function code(K, m, n, M, l)
	return Float64(cos(M) * exp(Float64(-l)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(-l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

7. Simplified 97.3%

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

8. Taylor expanded in l around inf 39.4%

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

   1. neg-mul-1 39.4%

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

10. Simplified 39.4%

    \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]
11. Add Preprocessing

Alternative 12: 6.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \cos M \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (cos M))

C

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

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(m_1)
end function

Java

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

Python

def code(K, m, n, M, l):
	return math.cos(M)

Julia

function code(K, m, n, M, l)
	return cos(M)
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Cos[M], $MachinePrecision]

Derivation

1. Initial program 79.2%

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

3. Simplified 79.2%

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

5. Taylor expanded in K around 0 97.3%

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

   1. cos-neg 97.3%

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

7. Simplified 97.3%

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

8. Taylor expanded in n around 0 76.4%

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

   1. +-commutative 76.4%

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

   2. unpow2 76.4%

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

   3. distribute-rgt-out 82.7%

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

10. Simplified 82.7%

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

11. Taylor expanded in n around inf 36.7%

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

12. Taylor expanded in n around 0 9.2%

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

Reproduce

herbie shell --seed 2024096 
(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)))))))