Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.4% → 96.6%

Time: 24.4s

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: 75.4% 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: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return cos(M) * exp((fabs((n - m)) - (l + pow((((n + m) * 0.5) - 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((abs((n - m)) - (l + ((((n + m) * 0.5d0) - 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((Math.abs((n - m)) - (l + Math.pow((((n + m) * 0.5) - M), 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

   1. associate-/l* 72.9%

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

   2. +-commutative 72.9%

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

   3. associate-/l* 72.5%

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

   4. associate-/l* 72.9%

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

   5. +-commutative 72.9%

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

   6. exp-diff 24.8%

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

   7. sub-neg 24.8%

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

   8. exp-sum 20.1%

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

   9. associate-/r* 20.1%

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

   10. exp-diff 25.6%

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

3. Simplified 72.9%

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

5. Taylor expanded in K around 0 96.5%

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

7. Simplified 96.5%

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

8. Final simplification 96.5%

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

Alternative 2: 87.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -9:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) + \left(n \cdot 0.5 - M\right) \cdot \left(\left(M - m\right) - n \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -9.0)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (*
    (cos M)
    (exp (+ (- (fabs (- n m)) l) (* (- (* n 0.5) M) (- (- M m) (* n 0.5))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -9.0) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp(((fabs((n - m)) - l) + (((n * 0.5) - M) * ((M - m) - (n * 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 <= (-9.0d0)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((abs((n - m)) - l) + (((n * 0.5d0) - m_1) * ((m_1 - m) - (n * 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 <= -9.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((Math.abs((n - m)) - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -9.0:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((math.fabs((n - m)) - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -9.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(abs(Float64(n - m)) - l) + Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - m) - Float64(n * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -9.0)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp(((abs((n - m)) - l) + (((n * 0.5) - M) * ((M - m) - (n * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -9.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] + N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - m), $MachinePrecision] - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -9

   1. Initial program 63.3%

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

      1. associate-/l* 63.3%

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

      2. +-commutative 63.3%

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

      3. associate-/l* 63.3%

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

      4. associate-/l* 63.3%

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

      5. +-commutative 63.3%

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

      6. exp-diff 3.3%

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

      7. sub-neg 3.3%

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

      8. exp-sum 0.0%

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

      9. associate-/r* 0.0%

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

      10. exp-diff 0.0%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in K around 0 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 88.5%

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

   9. Taylor expanded in m around inf 98.4%

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

   if -9 < m

   1. Initial program 75.3%

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

      1. associate-/l* 75.8%

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

      2. +-commutative 75.8%

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

      3. associate-/l* 75.3%

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

      4. associate-/l* 75.8%

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

      5. +-commutative 75.8%

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

      6. exp-diff 31.4%

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

      7. sub-neg 31.4%

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

      8. exp-sum 26.3%

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

      9. associate-/r* 26.3%

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

      10. exp-diff 33.4%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in m around 0 67.2%

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

      1. +-commutative 67.2%

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

      2. unpow2 67.2%

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

      3. distribute-rgt-out 68.7%

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

      4. *-commutative 68.7%

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

      5. *-commutative 68.7%

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

   7. Simplified 68.7%

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

   8. Taylor expanded in K around 0 82.4%

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

   10. Simplified 82.4%

       \[\leadsto \color{blue}{\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) - \left(0.5 \cdot n - M\right) \cdot \left(\left(m - M\right) + 0.5 \cdot n\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -9:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|n - m\right| - \ell\right) + \left(n \cdot 0.5 - M\right) \cdot \left(\left(M - m\right) - n \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 4.5 \cdot 10^{-81}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + {m}^{2} \cdot 0.25\right)}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 4.5e-81)
   (exp (- (fabs (- n m)) (+ l (* (pow m 2.0) 0.25))))
   (if (<= n 53.0)
     (* (cos (- (* K (/ (+ n m) 2.0)) 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 (n <= 4.5e-81) {
		tmp = exp((fabs((n - m)) - (l + (pow(m, 2.0) * 0.25))));
	} else if (n <= 53.0) {
		tmp = cos(((K * ((n + m) / 2.0)) - 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 (n <= 4.5d-81) then
        tmp = exp((abs((n - m)) - (l + ((m ** 2.0d0) * 0.25d0))))
    else if (n <= 53.0d0) then
        tmp = cos(((k * ((n + m) / 2.0d0)) - 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 (n <= 4.5e-81) {
		tmp = Math.exp((Math.abs((n - m)) - (l + (Math.pow(m, 2.0) * 0.25))));
	} else if (n <= 53.0) {
		tmp = Math.cos(((K * ((n + m) / 2.0)) - 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 n <= 4.5e-81:
		tmp = math.exp((math.fabs((n - m)) - (l + (math.pow(m, 2.0) * 0.25))))
	elif n <= 53.0:
		tmp = math.cos(((K * ((n + m) / 2.0)) - 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 (n <= 4.5e-81)
		tmp = exp(Float64(abs(Float64(n - m)) - Float64(l + Float64((m ^ 2.0) * 0.25))));
	elseif (n <= 53.0)
		tmp = Float64(cos(Float64(Float64(K * Float64(Float64(n + m) / 2.0)) - 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 (n <= 4.5e-81)
		tmp = exp((abs((n - m)) - (l + ((m ^ 2.0) * 0.25))));
	elseif (n <= 53.0)
		tmp = cos(((K * ((n + m) / 2.0)) - 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[n, 4.5e-81], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[(N[Power[m, 2.0], $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 53.0], N[(N[Cos[N[(N[(K * N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $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 n < 4.5e-81

   1. Initial program 74.3%

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

      1. associate-/l* 74.9%

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

      2. +-commutative 74.9%

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

      3. associate-/l* 74.3%

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

      4. associate-/l* 74.9%

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

      5. +-commutative 74.9%

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

      6. exp-diff 28.3%

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

      7. sub-neg 28.3%

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

      8. exp-sum 23.7%

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

      9. associate-/r* 23.7%

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

      10. exp-diff 30.1%

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

   3. Simplified 74.9%

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

   5. Taylor expanded in K around 0 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in m around inf 65.2%

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

   9. Taylor expanded in M around 0 64.6%

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

   if 4.5e-81 < n < 53

   1. Initial program 81.9%

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

      1. associate-/l* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-/l* 81.9%

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

      4. associate-/l* 81.9%

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

      5. +-commutative 81.9%

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

      6. exp-diff 48.5%

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

      7. sub-neg 48.5%

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

      8. exp-sum 48.5%

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

      9. associate-/r* 48.5%

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

      10. exp-diff 58.0%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in m around 0 67.8%

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

      1. +-commutative 67.8%

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

      2. unpow2 67.8%

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

      3. distribute-rgt-out 72.6%

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

      4. *-commutative 72.6%

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

      5. *-commutative 72.6%

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

   7. Simplified 72.6%

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

   8. Taylor expanded in M around inf 60.0%

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

      1. mul-1-neg 60.0%

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

   10. Simplified 60.0%

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

   if 53 < n

   1. Initial program 63.9%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. associate-/l* 63.9%

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

      4. associate-/l* 63.9%

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

      5. +-commutative 63.9%

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

      6. exp-diff 6.6%

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

      7. sub-neg 6.6%

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

      8. exp-sum 0.0%

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

      9. associate-/r* 0.0%

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

      10. exp-diff 1.6%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in m around 0 60.7%

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

      1. +-commutative 60.7%

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

      2. unpow2 60.7%

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

      3. distribute-rgt-out 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

   7. Simplified 62.3%

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

   8. Taylor expanded in n around inf 62.3%

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

      1. *-commutative 62.3%

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

   10. Simplified 62.3%

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

   11. Taylor expanded in K around 0 98.4%

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

       1. cos-neg 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 4.5 \cdot 10^{-81}:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + {m}^{2} \cdot 0.25\right)}\\ \mathbf{elif}\;n \leq 53:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{if}\;m \leq -0.0014:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 7.2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;m \leq 0.052:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* -0.25 (pow m 2.0))))))
   (if (<= m -0.0014)
     t_0
     (if (<= m 7.2e-259)
       (/ (cos M) (exp l))
       (if (<= m 0.052)
         (* (cos (- (* K (/ (+ n m) 2.0)) M)) (exp (* -0.25 (* n n))))
         t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((-0.25 * pow(m, 2.0)));
	double tmp;
	if (m <= -0.0014) {
		tmp = t_0;
	} else if (m <= 7.2e-259) {
		tmp = cos(M) / exp(l);
	} else if (m <= 0.052) {
		tmp = cos(((K * ((n + m) / 2.0)) - M)) * exp((-0.25 * (n * n)));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    if (m <= (-0.0014d0)) then
        tmp = t_0
    else if (m <= 7.2d-259) then
        tmp = cos(m_1) / exp(l)
    else if (m <= 0.052d0) then
        tmp = cos(((k * ((n + m) / 2.0d0)) - m_1)) * exp(((-0.25d0) * (n * n)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	double tmp;
	if (m <= -0.0014) {
		tmp = t_0;
	} else if (m <= 7.2e-259) {
		tmp = Math.cos(M) / Math.exp(l);
	} else if (m <= 0.052) {
		tmp = Math.cos(((K * ((n + m) / 2.0)) - M)) * Math.exp((-0.25 * (n * n)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	tmp = 0
	if m <= -0.0014:
		tmp = t_0
	elif m <= 7.2e-259:
		tmp = math.cos(M) / math.exp(l)
	elif m <= 0.052:
		tmp = math.cos(((K * ((n + m) / 2.0)) - M)) * math.exp((-0.25 * (n * n)))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))))
	tmp = 0.0
	if (m <= -0.0014)
		tmp = t_0;
	elseif (m <= 7.2e-259)
		tmp = Float64(cos(M) / exp(l));
	elseif (m <= 0.052)
		tmp = Float64(cos(Float64(Float64(K * Float64(Float64(n + m) / 2.0)) - M)) * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((-0.25 * (m ^ 2.0)));
	tmp = 0.0;
	if (m <= -0.0014)
		tmp = t_0;
	elseif (m <= 7.2e-259)
		tmp = cos(M) / exp(l);
	elseif (m <= 0.052)
		tmp = cos(((K * ((n + m) / 2.0)) - M)) * exp((-0.25 * (n * n)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -0.0014], t$95$0, If[LessEqual[m, 7.2e-259], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[m, 0.052], N[(N[Cos[N[(N[(K * N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.00139999999999999999 or 0.0519999999999999976 < m

   1. Initial program 62.6%

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

      1. associate-/l* 63.4%

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

      2. +-commutative 63.4%

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

      3. associate-/l* 62.6%

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

      4. associate-/l* 63.4%

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

      5. +-commutative 63.4%

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

      6. exp-diff 4.9%

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

      7. sub-neg 4.9%

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

      8. exp-sum 0.0%

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

      9. associate-/r* 0.0%

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

      10. exp-diff 0.8%

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

   3. Simplified 63.4%

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

   5. Taylor expanded in K around 0 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf 84.8%

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

   9. Taylor expanded in m around inf 98.4%

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

   if -0.00139999999999999999 < m < 7.1999999999999996e-259

   1. Initial program 81.4%

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

      1. associate-/l* 81.4%

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

      2. +-commutative 81.4%

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

      3. associate-/l* 81.4%

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

      4. associate-/l* 81.4%

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

      5. +-commutative 81.4%

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

      6. exp-diff 42.9%

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

      7. sub-neg 42.9%

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

      8. exp-sum 37.8%

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

      9. associate-/r* 37.8%

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

      10. exp-diff 49.3%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in m around 0 81.4%

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

      1. +-commutative 81.4%

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

      2. unpow2 81.4%

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

      3. distribute-rgt-out 81.4%

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

      4. *-commutative 81.4%

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

      5. *-commutative 81.4%

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

   7. Simplified 81.4%

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

   8. Taylor expanded in l around inf 44.5%

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

      1. mul-1-neg 44.5%

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

   10. Simplified 44.5%

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

       1. exp-neg 44.5%

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

   12. Applied egg-rr 44.5%

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

   13. Taylor expanded in K around 0 44.9%

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

       1. cos-neg 44.9%

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

   15. Simplified 44.9%

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

   if 7.1999999999999996e-259 < m < 0.0519999999999999976

   1. Initial program 81.9%

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

      1. associate-/l* 81.9%

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

      2. +-commutative 81.9%

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

      3. associate-/l* 81.9%

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

      4. associate-/l* 81.9%

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

      5. +-commutative 81.9%

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

      6. exp-diff 43.7%

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

      7. sub-neg 43.7%

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

      8. exp-sum 40.1%

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

      9. associate-/r* 40.1%

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

      10. exp-diff 47.3%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in m around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. unpow2 81.9%

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

      3. distribute-rgt-out 81.9%

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

      4. *-commutative 81.9%

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

      5. *-commutative 81.9%

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

   7. Simplified 81.9%

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

   8. Taylor expanded in n around inf 41.9%

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

      1. *-commutative 41.9%

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

   10. Simplified 41.9%

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

       1. unpow2 41.9%

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

   12. Applied egg-rr 41.9%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.0014:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;m \leq 7.2 \cdot 10^{-259}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{elif}\;m \leq 0.052:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 54:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + {m}^{2} \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 54.0)
   (exp (- (fabs (- n m)) (+ l (* (pow m 2.0) 0.25))))
   (* (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 (n <= 54.0) {
		tmp = exp((fabs((n - m)) - (l + (pow(m, 2.0) * 0.25))));
	} 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 (n <= 54.0d0) then
        tmp = exp((abs((n - m)) - (l + ((m ** 2.0d0) * 0.25d0))))
    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 (n <= 54.0) {
		tmp = Math.exp((Math.abs((n - m)) - (l + (Math.pow(m, 2.0) * 0.25))));
	} 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 n <= 54.0:
		tmp = math.exp((math.fabs((n - m)) - (l + (math.pow(m, 2.0) * 0.25))))
	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 (n <= 54.0)
		tmp = exp(Float64(abs(Float64(n - m)) - Float64(l + Float64((m ^ 2.0) * 0.25))));
	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 (n <= 54.0)
		tmp = exp((abs((n - m)) - (l + ((m ^ 2.0) * 0.25))));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 54.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[(N[Power[m, 2.0], $MachinePrecision] * 0.25), $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 2 regimes

2. if n < 54

   1. Initial program 75.1%

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

      1. associate-/l* 75.6%

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

      2. +-commutative 75.6%

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

      3. associate-/l* 75.1%

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

      4. associate-/l* 75.6%

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

      5. +-commutative 75.6%

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

      6. exp-diff 30.5%

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

      7. sub-neg 30.5%

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

      8. exp-sum 26.4%

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

      9. associate-/r* 26.4%

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

      10. exp-diff 33.1%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in K around 0 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in m around inf 63.8%

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

   9. Taylor expanded in M around 0 63.3%

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

   if 54 < n

   1. Initial program 63.9%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. associate-/l* 63.9%

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

      4. associate-/l* 63.9%

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

      5. +-commutative 63.9%

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

      6. exp-diff 6.6%

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

      7. sub-neg 6.6%

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

      8. exp-sum 0.0%

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

      9. associate-/r* 0.0%

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

      10. exp-diff 1.6%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in m around 0 60.7%

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

      1. +-commutative 60.7%

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

      2. unpow2 60.7%

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

      3. distribute-rgt-out 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

   7. Simplified 62.3%

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

   8. Taylor expanded in n around inf 62.3%

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

      1. *-commutative 62.3%

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

   10. Simplified 62.3%

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

   11. Taylor expanded in K around 0 98.4%

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

       1. cos-neg 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 54:\\ \;\;\;\;e^{\left|n - m\right| - \left(\ell + {m}^{2} \cdot 0.25\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 54.0)
   (* (cos M) (exp (* -0.25 (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 (n <= 54.0) {
		tmp = cos(M) * exp((-0.25 * 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 (n <= 54.0d0) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 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 (n <= 54.0) {
		tmp = Math.cos(M) * Math.exp((-0.25 * 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 n <= 54.0:
		tmp = math.cos(M) * math.exp((-0.25 * 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 (n <= 54.0)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (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 (n <= 54.0)
		tmp = cos(M) * exp((-0.25 * (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[n, 54.0], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $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 2 regimes

2. if n < 54

   1. Initial program 75.1%

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

      1. associate-/l* 75.6%

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

      2. +-commutative 75.6%

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

      3. associate-/l* 75.1%

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

      4. associate-/l* 75.6%

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

      5. +-commutative 75.6%

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

      6. exp-diff 30.5%

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

      7. sub-neg 30.5%

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

      8. exp-sum 26.4%

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

      9. associate-/r* 26.4%

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

      10. exp-diff 33.1%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in K around 0 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in m around inf 63.8%

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

   9. Taylor expanded in m around inf 54.6%

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

   if 54 < n

   1. Initial program 63.9%

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

      1. associate-/l* 63.9%

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

      2. +-commutative 63.9%

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

      3. associate-/l* 63.9%

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

      4. associate-/l* 63.9%

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

      5. +-commutative 63.9%

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

      6. exp-diff 6.6%

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

      7. sub-neg 6.6%

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

      8. exp-sum 0.0%

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

      9. associate-/r* 0.0%

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

      10. exp-diff 1.6%

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

   3. Simplified 63.9%

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

   5. Taylor expanded in m around 0 60.7%

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

      1. +-commutative 60.7%

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

      2. unpow2 60.7%

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

      3. distribute-rgt-out 62.3%

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

      4. *-commutative 62.3%

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

      5. *-commutative 62.3%

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

   7. Simplified 62.3%

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

   8. Taylor expanded in n around inf 62.3%

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

      1. *-commutative 62.3%

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

   10. Simplified 62.3%

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

   11. Taylor expanded in K around 0 98.4%

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

       1. cos-neg 98.4%

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

   13. Simplified 98.4%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 54:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot \frac{n + m}{2} - M\right)\\ t_1 := t\_0 \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)}{e^{\ell}}\\ \mathbf{elif}\;\ell \leq -4.3 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{+137}:\\ \;\;\;\;t\_0 \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;\ell \leq -16200000000:\\ \;\;\;\;t\_0 \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 0.00125:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (cos (- (* K (/ (+ n m) 2.0)) M)))
        (t_1 (* t_0 (exp (* -0.25 (* n n))))))
   (if (<= l -7.8e+253)
     (/ (cos (- (* 0.5 (* n K)) M)) (exp l))
     (if (<= l -4.3e+186)
       t_1
       (if (<= l -2.4e+137)
         (*
          t_0
          (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0))))
         (if (<= l -16200000000.0)
           (* t_0 (exp (* m (- M (* n 0.5)))))
           (if (<= l 0.00125) t_1 (/ (cos M) (exp l)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(((K * ((n + m) / 2.0)) - M));
	double t_1 = t_0 * exp((-0.25 * (n * n)));
	double tmp;
	if (l <= -7.8e+253) {
		tmp = cos(((0.5 * (n * K)) - M)) / exp(l);
	} else if (l <= -4.3e+186) {
		tmp = t_1;
	} else if (l <= -2.4e+137) {
		tmp = t_0 * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	} else if (l <= -16200000000.0) {
		tmp = t_0 * exp((m * (M - (n * 0.5))));
	} else if (l <= 0.00125) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(((k * ((n + m) / 2.0d0)) - m_1))
    t_1 = t_0 * exp(((-0.25d0) * (n * n)))
    if (l <= (-7.8d+253)) then
        tmp = cos(((0.5d0 * (n * k)) - m_1)) / exp(l)
    else if (l <= (-4.3d+186)) then
        tmp = t_1
    else if (l <= (-2.4d+137)) then
        tmp = t_0 * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
    else if (l <= (-16200000000.0d0)) then
        tmp = t_0 * exp((m * (m_1 - (n * 0.5d0))))
    else if (l <= 0.00125d0) then
        tmp = t_1
    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 t_0 = Math.cos(((K * ((n + m) / 2.0)) - M));
	double t_1 = t_0 * Math.exp((-0.25 * (n * n)));
	double tmp;
	if (l <= -7.8e+253) {
		tmp = Math.cos(((0.5 * (n * K)) - M)) / Math.exp(l);
	} else if (l <= -4.3e+186) {
		tmp = t_1;
	} else if (l <= -2.4e+137) {
		tmp = t_0 * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	} else if (l <= -16200000000.0) {
		tmp = t_0 * Math.exp((m * (M - (n * 0.5))));
	} else if (l <= 0.00125) {
		tmp = t_1;
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(((K * ((n + m) / 2.0)) - M))
	t_1 = t_0 * math.exp((-0.25 * (n * n)))
	tmp = 0
	if l <= -7.8e+253:
		tmp = math.cos(((0.5 * (n * K)) - M)) / math.exp(l)
	elif l <= -4.3e+186:
		tmp = t_1
	elif l <= -2.4e+137:
		tmp = t_0 * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))
	elif l <= -16200000000.0:
		tmp = t_0 * math.exp((m * (M - (n * 0.5))))
	elif l <= 0.00125:
		tmp = t_1
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = cos(Float64(Float64(K * Float64(Float64(n + m) / 2.0)) - M))
	t_1 = Float64(t_0 * exp(Float64(-0.25 * Float64(n * n))))
	tmp = 0.0
	if (l <= -7.8e+253)
		tmp = Float64(cos(Float64(Float64(0.5 * Float64(n * K)) - M)) / exp(l));
	elseif (l <= -4.3e+186)
		tmp = t_1;
	elseif (l <= -2.4e+137)
		tmp = Float64(t_0 * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))));
	elseif (l <= -16200000000.0)
		tmp = Float64(t_0 * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	elseif (l <= 0.00125)
		tmp = t_1;
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(((K * ((n + m) / 2.0)) - M));
	t_1 = t_0 * exp((-0.25 * (n * n)));
	tmp = 0.0;
	if (l <= -7.8e+253)
		tmp = cos(((0.5 * (n * K)) - M)) / exp(l);
	elseif (l <= -4.3e+186)
		tmp = t_1;
	elseif (l <= -2.4e+137)
		tmp = t_0 * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
	elseif (l <= -16200000000.0)
		tmp = t_0 * exp((m * (M - (n * 0.5))));
	elseif (l <= 0.00125)
		tmp = t_1;
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Cos[N[(N[(K * N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -7.8e+253], N[(N[Cos[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -4.3e+186], t$95$1, If[LessEqual[l, -2.4e+137], N[(t$95$0 * N[(1.0 + N[(l * N[(N[(l * N[(0.5 + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -16200000000.0], N[(t$95$0 * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.00125], t$95$1, N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if l < -7.8000000000000003e253

   1. Initial program 58.8%

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

      1. associate-/l* 58.8%

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

      2. +-commutative 58.8%

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

      3. associate-/l* 58.8%

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

      4. associate-/l* 58.8%

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

      5. +-commutative 58.8%

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

      6. exp-diff 17.6%

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

      7. sub-neg 17.6%

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

      8. exp-sum 17.6%

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

      9. associate-/r* 17.6%

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

      10. exp-diff 41.2%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in m around 0 58.8%

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

      1. +-commutative 58.8%

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

      2. unpow2 58.8%

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

      3. distribute-rgt-out 58.8%

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

      4. *-commutative 58.8%

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

      5. *-commutative 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in l around inf 35.7%

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

      1. mul-1-neg 35.7%

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

   10. Simplified 35.7%

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

       1. exp-neg 35.7%

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

   12. Applied egg-rr 35.7%

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

   13. Taylor expanded in m around 0 35.9%

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

   if -7.8000000000000003e253 < l < -4.3e186 or -1.62e10 < l < 0.00125000000000000003

   1. Initial program 72.8%

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

      1. associate-/l* 73.4%

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

      2. +-commutative 73.4%

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

      3. associate-/l* 72.8%

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

      4. associate-/l* 73.4%

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

      5. +-commutative 73.4%

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

      6. exp-diff 19.6%

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

      7. sub-neg 19.6%

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

      8. exp-sum 19.6%

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

      9. associate-/r* 19.6%

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

      10. exp-diff 21.7%

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

   3. Simplified 73.4%

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

   5. Taylor expanded in m around 0 55.1%

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

      1. +-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. distribute-rgt-out 57.2%

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

      4. *-commutative 57.2%

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

      5. *-commutative 57.2%

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

   7. Simplified 57.2%

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

   8. Taylor expanded in n around inf 38.6%

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

      1. *-commutative 38.6%

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

   10. Simplified 38.6%

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

       1. unpow2 38.6%

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

   12. Applied egg-rr 38.6%

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

   if -4.3e186 < l < -2.39999999999999983e137

   1. Initial program 85.7%

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

      1. associate-/l* 85.7%

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

      2. +-commutative 85.7%

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

      3. associate-/l* 85.7%

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

      4. associate-/l* 85.7%

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

      5. +-commutative 85.7%

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

      6. exp-diff 42.9%

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

      7. sub-neg 42.9%

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

      8. exp-sum 42.9%

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

      9. associate-/r* 42.9%

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

      10. exp-diff 71.4%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in m around 0 85.7%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. distribute-rgt-out 85.7%

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

      4. *-commutative 85.7%

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

      5. *-commutative 85.7%

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

   7. Simplified 85.7%

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

   8. Taylor expanded in l around inf 57.6%

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

      1. mul-1-neg 57.6%

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

   10. Simplified 57.6%

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

   11. Taylor expanded in l around 0 57.6%

       \[\leadsto \cos \left(K \cdot \frac{m + n}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) - 1\right)\right)} \]

   if -2.39999999999999983e137 < l < -1.62e10

   1. Initial program 72.7%

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

      1. associate-/l* 72.7%

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

      2. +-commutative 72.7%

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

      3. associate-/l* 72.7%

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

      4. associate-/l* 72.7%

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

      5. +-commutative 72.7%

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

      6. exp-diff 9.1%

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

      7. sub-neg 9.1%

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

      8. exp-sum 9.1%

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

      9. associate-/r* 9.1%

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

      10. exp-diff 31.8%

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

   3. Simplified 72.7%

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

   5. Taylor expanded in m around 0 68.3%

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

      1. +-commutative 68.3%

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

      2. unpow2 68.3%

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

      3. distribute-rgt-out 68.3%

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

      4. *-commutative 68.3%

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

      5. *-commutative 68.3%

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

   7. Simplified 68.3%

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

   8. Taylor expanded in m around inf 23.8%

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

   if 0.00125000000000000003 < l

   1. Initial program 73.8%

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

      1. associate-/l* 73.8%

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

      2. +-commutative 73.8%

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

      3. associate-/l* 73.8%

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

      4. associate-/l* 73.8%

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

      5. +-commutative 73.8%

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

      6. exp-diff 41.5%

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

      7. sub-neg 41.5%

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

      8. exp-sum 23.1%

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

      9. associate-/r* 23.1%

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

      10. exp-diff 23.1%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in m around 0 57.1%

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

      1. +-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. distribute-rgt-out 61.7%

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

      4. *-commutative 61.7%

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

      5. *-commutative 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in l around inf 67.9%

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

      1. mul-1-neg 67.9%

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

   10. Simplified 67.9%

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

       1. exp-neg 67.9%

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

   12. Applied egg-rr 67.9%

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

   13. Taylor expanded in K around 0 94.0%

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

       1. cos-neg 94.0%

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

   15. Simplified 94.0%

       \[\leadsto \color{blue}{\frac{\cos M}{e^{\ell}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -7.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)}{e^{\ell}}\\ \mathbf{elif}\;\ell \leq -4.3 \cdot 10^{+186}:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{elif}\;\ell \leq -2.4 \cdot 10^{+137}:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{elif}\;\ell \leq -16200000000:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq 0.00125:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)}{e^{\ell}}\\ \mathbf{elif}\;\ell \leq 0.00195:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -3.8e+253)
   (/ (cos (- (* 0.5 (* n K)) M)) (exp l))
   (if (<= l 0.00195)
     (* (cos (- (* K (/ (+ n m) 2.0)) M)) (exp (* -0.25 (* n n))))
     (/ (cos M) (exp l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -3.8e+253) {
		tmp = cos(((0.5 * (n * K)) - M)) / exp(l);
	} else if (l <= 0.00195) {
		tmp = cos(((K * ((n + m) / 2.0)) - M)) * exp((-0.25 * (n * 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 <= (-3.8d+253)) then
        tmp = cos(((0.5d0 * (n * k)) - m_1)) / exp(l)
    else if (l <= 0.00195d0) then
        tmp = cos(((k * ((n + m) / 2.0d0)) - m_1)) * exp(((-0.25d0) * (n * 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 <= -3.8e+253) {
		tmp = Math.cos(((0.5 * (n * K)) - M)) / Math.exp(l);
	} else if (l <= 0.00195) {
		tmp = Math.cos(((K * ((n + m) / 2.0)) - M)) * Math.exp((-0.25 * (n * n)));
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -3.8e+253:
		tmp = math.cos(((0.5 * (n * K)) - M)) / math.exp(l)
	elif l <= 0.00195:
		tmp = math.cos(((K * ((n + m) / 2.0)) - M)) * math.exp((-0.25 * (n * n)))
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -3.8e+253)
		tmp = Float64(cos(Float64(Float64(0.5 * Float64(n * K)) - M)) / exp(l));
	elseif (l <= 0.00195)
		tmp = Float64(cos(Float64(Float64(K * Float64(Float64(n + m) / 2.0)) - M)) * exp(Float64(-0.25 * Float64(n * n))));
	else
		tmp = Float64(cos(M) / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -3.8e+253)
		tmp = cos(((0.5 * (n * K)) - M)) / exp(l);
	elseif (l <= 0.00195)
		tmp = cos(((K * ((n + m) / 2.0)) - M)) * exp((-0.25 * (n * n)));
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -3.8e+253], N[(N[Cos[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 0.00195], N[(N[Cos[N[(N[(K * N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.79999999999999989e253

   1. Initial program 58.8%

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

      1. associate-/l* 58.8%

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

      2. +-commutative 58.8%

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

      3. associate-/l* 58.8%

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

      4. associate-/l* 58.8%

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

      5. +-commutative 58.8%

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

      6. exp-diff 17.6%

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

      7. sub-neg 17.6%

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

      8. exp-sum 17.6%

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

      9. associate-/r* 17.6%

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

      10. exp-diff 41.2%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in m around 0 58.8%

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

      1. +-commutative 58.8%

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

      2. unpow2 58.8%

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

      3. distribute-rgt-out 58.8%

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

      4. *-commutative 58.8%

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

      5. *-commutative 58.8%

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

   7. Simplified 58.8%

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

   8. Taylor expanded in l around inf 35.7%

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

      1. mul-1-neg 35.7%

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

   10. Simplified 35.7%

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

       1. exp-neg 35.7%

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

   12. Applied egg-rr 35.7%

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

   13. Taylor expanded in m around 0 35.9%

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

   if -3.79999999999999989e253 < l < 0.0019499999999999999

   1. Initial program 73.3%

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

      1. associate-/l* 73.8%

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

      2. +-commutative 73.8%

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

      3. associate-/l* 73.3%

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

      4. associate-/l* 73.8%

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

      5. +-commutative 73.8%

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

      6. exp-diff 19.2%

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

      7. sub-neg 19.2%

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

      8. exp-sum 19.2%

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

      9. associate-/r* 19.2%

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

      10. exp-diff 25.0%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in m around 0 58.0%

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

      1. +-commutative 58.0%

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

      2. unpow2 58.0%

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

      3. distribute-rgt-out 59.7%

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

      4. *-commutative 59.7%

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

      5. *-commutative 59.7%

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

   7. Simplified 59.7%

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

   8. Taylor expanded in n around inf 37.5%

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

      1. *-commutative 37.5%

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

   10. Simplified 37.5%

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

       1. unpow2 37.5%

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

   12. Applied egg-rr 37.5%

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

   if 0.0019499999999999999 < l

   1. Initial program 73.8%

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

      1. associate-/l* 73.8%

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

      2. +-commutative 73.8%

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

      3. associate-/l* 73.8%

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

      4. associate-/l* 73.8%

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

      5. +-commutative 73.8%

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

      6. exp-diff 41.5%

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

      7. sub-neg 41.5%

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

      8. exp-sum 23.1%

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

      9. associate-/r* 23.1%

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

      10. exp-diff 23.1%

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

   3. Simplified 73.8%

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

   5. Taylor expanded in m around 0 57.1%

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

      1. +-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. distribute-rgt-out 61.7%

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

      4. *-commutative 61.7%

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

      5. *-commutative 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in l around inf 67.9%

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

      1. mul-1-neg 67.9%

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

   10. Simplified 67.9%

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

       1. exp-neg 67.9%

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

   12. Applied egg-rr 67.9%

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

   13. Taylor expanded in K around 0 94.0%

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

       1. cos-neg 94.0%

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

   15. Simplified 94.0%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.8 \cdot 10^{+253}:\\ \;\;\;\;\frac{\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right)}{e^{\ell}}\\ \mathbf{elif}\;\ell \leq 0.00195:\\ \;\;\;\;\cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot e^{-0.25 \cdot \left(n \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.6% accurate, 2.1× speedup

Math

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

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(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 72.5%

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

   1. associate-/l* 72.9%

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

   2. +-commutative 72.9%

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

   3. associate-/l* 72.5%

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

   4. associate-/l* 72.9%

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

   5. +-commutative 72.9%

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

   6. exp-diff 24.8%

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

   7. sub-neg 24.8%

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

   8. exp-sum 20.1%

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

   9. associate-/r* 20.1%

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

   10. exp-diff 25.6%

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

3. Simplified 72.9%

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

5. Taylor expanded in m around 0 57.8%

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

   1. +-commutative 57.8%

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

   2. unpow2 57.8%

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

   3. distribute-rgt-out 60.2%

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

   4. *-commutative 60.2%

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

   5. *-commutative 60.2%

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

7. Simplified 60.2%

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

8. Taylor expanded in l around inf 28.3%

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

   1. mul-1-neg 28.3%

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

10. Simplified 28.3%

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

    1. exp-neg 28.3%

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

12. Applied egg-rr 28.3%

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

13. Taylor expanded in K around 0 35.2%

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

    1. cos-neg 35.2%

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

15. Simplified 35.2%

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

16. Final simplification 35.2%

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

Alternative 10: 9.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (*
  (cos (- (* K (/ (+ n m) 2.0)) M))
  (+ 1.0 (* l (+ (* l (+ 0.5 (* l -0.16666666666666666))) -1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(((K * ((n + m) / 2.0)) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
}

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(((k * ((n + m) / 2.0d0)) - m_1)) * (1.0d0 + (l * ((l * (0.5d0 + (l * (-0.16666666666666666d0)))) + (-1.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(((K * ((n + m) / 2.0)) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
}

Python

def code(K, m, n, M, l):
	return math.cos(((K * ((n + m) / 2.0)) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(K * Float64(Float64(n + m) / 2.0)) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * Float64(0.5 + Float64(l * -0.16666666666666666))) + -1.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(((K * ((n + m) / 2.0)) - M)) * (1.0 + (l * ((l * (0.5 + (l * -0.16666666666666666))) + -1.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(K * N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * N[(0.5 + N[(l * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

   1. associate-/l* 72.9%

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

   2. +-commutative 72.9%

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

   3. associate-/l* 72.5%

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

   4. associate-/l* 72.9%

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

   5. +-commutative 72.9%

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

   6. exp-diff 24.8%

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

   7. sub-neg 24.8%

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

   8. exp-sum 20.1%

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

   9. associate-/r* 20.1%

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

   10. exp-diff 25.6%

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

3. Simplified 72.9%

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

5. Taylor expanded in m around 0 57.8%

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

   1. +-commutative 57.8%

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

   2. unpow2 57.8%

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

   3. distribute-rgt-out 60.2%

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

   4. *-commutative 60.2%

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

   5. *-commutative 60.2%

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

7. Simplified 60.2%

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

8. Taylor expanded in l around inf 28.3%

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

   1. mul-1-neg 28.3%

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

10. Simplified 28.3%

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

11. Taylor expanded in l around 0 10.7%

    \[\leadsto \cos \left(K \cdot \frac{m + n}{2} - M\right) \cdot \color{blue}{\left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + -0.16666666666666666 \cdot \ell\right) - 1\right)\right)} \]

12. Final simplification 10.7%

    \[\leadsto \cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot \left(0.5 + \ell \cdot -0.16666666666666666\right) + -1\right)\right) \]
13. Add Preprocessing

Alternative 11: 9.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right) \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos (- (* K (/ (+ n m) 2.0)) M)) (+ 1.0 (* l (+ (* l 0.5) -1.0)))))

C

double code(double K, double m, double n, double M, double l) {
	return cos(((K * ((n + m) / 2.0)) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
}

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(((k * ((n + m) / 2.0d0)) - m_1)) * (1.0d0 + (l * ((l * 0.5d0) + (-1.0d0))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(((K * ((n + m) / 2.0)) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
}

Python

def code(K, m, n, M, l):
	return math.cos(((K * ((n + m) / 2.0)) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)))

Julia

function code(K, m, n, M, l)
	return Float64(cos(Float64(Float64(K * Float64(Float64(n + m) / 2.0)) - M)) * Float64(1.0 + Float64(l * Float64(Float64(l * 0.5) + -1.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(((K * ((n + m) / 2.0)) - M)) * (1.0 + (l * ((l * 0.5) + -1.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(N[Cos[N[(N[(K * N[(N[(n + m), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(l * N[(N[(l * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

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

   1. associate-/l* 72.9%

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

   2. +-commutative 72.9%

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

   3. associate-/l* 72.5%

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

   4. associate-/l* 72.9%

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

   5. +-commutative 72.9%

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

   6. exp-diff 24.8%

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

   7. sub-neg 24.8%

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

   8. exp-sum 20.1%

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

   9. associate-/r* 20.1%

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

   10. exp-diff 25.6%

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

3. Simplified 72.9%

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

5. Taylor expanded in m around 0 57.8%

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

   1. +-commutative 57.8%

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

   2. unpow2 57.8%

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

   3. distribute-rgt-out 60.2%

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

   4. *-commutative 60.2%

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

   5. *-commutative 60.2%

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

7. Simplified 60.2%

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

8. Taylor expanded in l around inf 28.3%

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

   1. mul-1-neg 28.3%

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

10. Simplified 28.3%

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

11. Taylor expanded in l around 0 9.8%

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

12. Final simplification 9.8%

    \[\leadsto \cos \left(K \cdot \frac{n + m}{2} - M\right) \cdot \left(1 + \ell \cdot \left(\ell \cdot 0.5 + -1\right)\right) \]
13. Add Preprocessing

Alternative 12: 7.2% 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 72.5%

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

   1. associate-/l* 72.9%

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

   2. +-commutative 72.9%

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

   3. associate-/l* 72.5%

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

   4. associate-/l* 72.9%

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

   5. +-commutative 72.9%

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

   6. exp-diff 24.8%

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

   7. sub-neg 24.8%

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

   8. exp-sum 20.1%

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

   9. associate-/r* 20.1%

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

   10. exp-diff 25.6%

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

3. Simplified 72.9%

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

5. Taylor expanded in m around 0 57.8%

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

   1. +-commutative 57.8%

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

   2. unpow2 57.8%

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

   3. distribute-rgt-out 60.2%

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

   4. *-commutative 60.2%

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

   5. *-commutative 60.2%

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

7. Simplified 60.2%

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

8. Taylor expanded in n around inf 34.2%

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

   1. *-commutative 34.2%

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

10. Simplified 34.2%

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

11. Taylor expanded in n around 0 7.5%

    \[\leadsto \color{blue}{\cos \left(0.5 \cdot \left(K \cdot m\right) - M\right)} \]
12. Step-by-step derivation

    1. associate-*r* 7.5%

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

13. Simplified 7.5%

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

14. Taylor expanded in K around 0 7.9%

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

    1. cos-neg 7.9%

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

16. Simplified 7.9%

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

17. Final simplification 7.9%

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

Reproduce

herbie shell --seed 2024067 
(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)))))))