Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.9% → 97.0%

Time: 21.9s

Alternatives: 15

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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: 97.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- (fabs (- n m)) l) (pow (- (/ (+ m n) 2.0) 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((((m + n) / 2.0) - 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) - ((((m + n) / 2.0d0) - 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((((m + n) / 2.0) - M), 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 76.6%

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

   2. +-commutative 76.6%

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

   3. fabs-sub 76.6%

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

   4. +-commutative 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

7. Simplified 97.5%

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

8. Final simplification 97.5%

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

Alternative 2: 86.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{+30}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|n - m\right| - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -3e+30)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (*
    (cos M)
    (exp (+ (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (- (fabs (- n m)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3e+30) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (fabs((n - m)) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-3d+30)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) + (abs((n - m)) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3e+30) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (Math.abs((n - m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -3e+30:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (math.fabs((n - m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -3e+30)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) + Float64(abs(Float64(n - m)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -3e+30)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) + (abs((n - m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -3e+30], 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[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -2.99999999999999978e30

   1. Initial program 61.1%

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

      1. associate-/l* 61.1%

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

      2. +-commutative 61.1%

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

      3. fabs-sub 61.1%

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

      4. +-commutative 61.1%

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

   3. Simplified 61.1%

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

   5. Taylor expanded in K around 0 98.1%

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

      1. cos-neg 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in m around inf 96.4%

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

   if -2.99999999999999978e30 < m

   1. Initial program 79.8%

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

      1. associate-/l* 80.8%

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

      2. +-commutative 80.8%

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

      3. fabs-sub 80.8%

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

      4. +-commutative 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in K around 0 97.4%

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

      1. cos-neg 97.4%

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

   7. Simplified 97.4%

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

   8. Taylor expanded in m around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. distribute-rgt-out 82.7%

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

      4. *-commutative 82.7%

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

      5. *-commutative 82.7%

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

   10. Simplified 82.7%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3 \cdot 10^{+30}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) + \left(\left|n - m\right| - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -27 \lor \neg \left(M \leq 30\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -27.0) (not (<= M 30.0)))
   (* (cos M) (exp (- (pow M 2.0))))
   (* (cos M) (exp (+ (- m n) (- (* M (+ M m)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 30.0)) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else {
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-27.0d0)) .or. (.not. (m_1 <= 30.0d0))) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else
        tmp = cos(m_1) * exp(((m - n) + ((m_1 * (m_1 + m)) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -27.0) || !(M <= 30.0)) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else {
		tmp = Math.cos(M) * Math.exp(((m - n) + ((M * (M + m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -27.0) or not (M <= 30.0):
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	else:
		tmp = math.cos(M) * math.exp(((m - n) + ((M * (M + m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -27.0) || !(M <= 30.0))
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(Float64(M * Float64(M + m)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -27.0) || ~((M <= 30.0)))
		tmp = cos(M) * exp(-(M ^ 2.0));
	else
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -27.0], N[Not[LessEqual[M, 30.0]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(M * N[(M + m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -27 or 30 < M

   1. Initial program 82.2%

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

      1. associate-/l* 83.0%

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

      2. +-commutative 83.0%

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

      3. fabs-sub 83.0%

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

      4. +-commutative 83.0%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in K around 0 99.3%

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

      1. cos-neg 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in M around inf 97.8%

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

      1. mul-1-neg 97.8%

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

   10. Simplified 97.8%

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

   if -27 < M < 30

   1. Initial program 68.7%

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

      1. associate-/l* 69.6%

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

      2. +-commutative 69.6%

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

      3. fabs-sub 69.6%

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

      4. +-commutative 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in K around 0 95.6%

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

      1. cos-neg 95.6%

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

   7. Simplified 95.6%

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

   8. Taylor expanded in m around 0 66.2%

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

      1. +-commutative 66.2%

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

      2. unpow2 66.2%

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

      3. distribute-rgt-out 68.7%

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

      4. *-commutative 68.7%

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

      5. *-commutative 68.7%

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

   10. Simplified 68.7%

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

   11. Taylor expanded in n around 0 32.9%

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

       1. associate--r+ 32.9%

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

       2. sub-neg 32.9%

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

       3. neg-mul-1 32.9%

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

       4. associate-*r* 32.9%

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

       5. neg-mul-1 32.9%

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

       6. cancel-sign-sub 32.9%

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

       7. neg-mul-1 32.9%

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

       8. sub-neg 32.9%

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

       9. unpow1 32.9%

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

       10. sqr-pow 18.8%

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

       11. fabs-sqr 18.8%

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

       12. sqr-pow 68.7%

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

       13. unpow1 68.7%

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

   13. Simplified 68.7%

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

       1. associate-+l- 68.7%

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

       2. add-sqr-sqrt 18.8%

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

       3. fabs-sqr 18.8%

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

       4. add-sqr-sqrt 32.9%

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

       5. fabs-sub 32.9%

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

       6. add-sqr-sqrt 14.1%

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

       7. fabs-sqr 14.1%

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

       8. add-sqr-sqrt 56.5%

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

       9. sub-neg 56.5%

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

       10. add-sqr-sqrt 26.0%

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

       11. sqrt-unprod 56.5%

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

       12. sqr-neg 56.5%

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

       13. sqrt-unprod 30.5%

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

       14. add-sqr-sqrt 56.5%

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

   15. Applied egg-rr 56.5%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -27 \lor \neg \left(M \leq 30\right):\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-\ell}\\ t_1 := \cos M \cdot e^{\ell}\\ \mathbf{if}\;\ell \leq -1.36 \cdot 10^{+202}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{+177}:\\ \;\;\;\;\left({M}^{2} \cdot -0.5 + 1\right) \cdot t\_0\\ \mathbf{elif}\;\ell \leq -2.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\ell \leq 3.6:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (- l))) (t_1 (* (cos M) (exp l))))
   (if (<= l -1.36e+202)
     t_1
     (if (<= l -2.1e+177)
       (* (+ (* (pow M 2.0) -0.5) 1.0) t_0)
       (if (<= l -2.2)
         t_1
         (if (<= l 3.6) (* (cos M) (exp (* m (- M (* n 0.5))))) t_0))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(-l);
	double t_1 = cos(M) * exp(l);
	double tmp;
	if (l <= -1.36e+202) {
		tmp = t_1;
	} else if (l <= -2.1e+177) {
		tmp = ((pow(M, 2.0) * -0.5) + 1.0) * t_0;
	} else if (l <= -2.2) {
		tmp = t_1;
	} else if (l <= 3.6) {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	} 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) :: t_1
    real(8) :: tmp
    t_0 = exp(-l)
    t_1 = cos(m_1) * exp(l)
    if (l <= (-1.36d+202)) then
        tmp = t_1
    else if (l <= (-2.1d+177)) then
        tmp = (((m_1 ** 2.0d0) * (-0.5d0)) + 1.0d0) * t_0
    else if (l <= (-2.2d0)) then
        tmp = t_1
    else if (l <= 3.6d0) then
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    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.exp(-l);
	double t_1 = Math.cos(M) * Math.exp(l);
	double tmp;
	if (l <= -1.36e+202) {
		tmp = t_1;
	} else if (l <= -2.1e+177) {
		tmp = ((Math.pow(M, 2.0) * -0.5) + 1.0) * t_0;
	} else if (l <= -2.2) {
		tmp = t_1;
	} else if (l <= 3.6) {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(-l)
	t_1 = math.cos(M) * math.exp(l)
	tmp = 0
	if l <= -1.36e+202:
		tmp = t_1
	elif l <= -2.1e+177:
		tmp = ((math.pow(M, 2.0) * -0.5) + 1.0) * t_0
	elif l <= -2.2:
		tmp = t_1
	elif l <= 3.6:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(-l))
	t_1 = Float64(cos(M) * exp(l))
	tmp = 0.0
	if (l <= -1.36e+202)
		tmp = t_1;
	elseif (l <= -2.1e+177)
		tmp = Float64(Float64(Float64((M ^ 2.0) * -0.5) + 1.0) * t_0);
	elseif (l <= -2.2)
		tmp = t_1;
	elseif (l <= 3.6)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(-l);
	t_1 = cos(M) * exp(l);
	tmp = 0.0;
	if (l <= -1.36e+202)
		tmp = t_1;
	elseif (l <= -2.1e+177)
		tmp = (((M ^ 2.0) * -0.5) + 1.0) * t_0;
	elseif (l <= -2.2)
		tmp = t_1;
	elseif (l <= 3.6)
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[(-l)], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.36e+202], t$95$1, If[LessEqual[l, -2.1e+177], N[(N[(N[(N[Power[M, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[l, -2.2], t$95$1, If[LessEqual[l, 3.6], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.36e202 or -2.10000000000000013e177 < l < -2.2000000000000002

   1. Initial program 83.1%

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

      1. associate-/l* 84.6%

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

      2. +-commutative 84.6%

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

      3. fabs-sub 84.6%

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

      4. +-commutative 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in K around 0 96.9%

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

      1. cos-neg 96.9%

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

   7. Simplified 96.9%

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

   8. Taylor expanded in l around inf 21.2%

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

      1. mul-1-neg 21.2%

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

   10. Simplified 21.2%

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

       1. expm1-log1p-u 20.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

       2. expm1-udef 20.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

       3. add-sqr-sqrt 20.8%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

       4. sqrt-unprod 20.8%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

       5. sqr-neg 20.8%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

       6. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

       7. add-sqr-sqrt 75.8%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   12. Applied egg-rr 75.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 75.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 75.8%

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

   14. Simplified 75.8%

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

   if -1.36e202 < l < -2.10000000000000013e177

   1. Initial program 50.0%

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

      1. associate-/l* 66.7%

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

      2. +-commutative 66.7%

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

      3. fabs-sub 66.7%

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

      4. +-commutative 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in K around 0 66.7%

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

      1. cos-neg 66.7%

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

   7. Simplified 66.7%

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

   8. Taylor expanded in l around inf 66.7%

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

      1. mul-1-neg 66.7%

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

   10. Simplified 66.7%

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

   11. Taylor expanded in M around 0 0.0%

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

       1. associate-*r* 0.0%

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

       2. distribute-rgt1-in 83.3%

          \[\leadsto \color{blue}{\left(-0.5 \cdot {M}^{2} + 1\right) \cdot e^{-\ell}} \]

   13. Simplified 83.3%

       \[\leadsto \color{blue}{\left(-0.5 \cdot {M}^{2} + 1\right) \cdot e^{-\ell}} \]

   if -2.2000000000000002 < l < 3.60000000000000009

   1. Initial program 71.3%

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

      1. associate-/l* 71.4%

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

      2. +-commutative 71.4%

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

      3. fabs-sub 71.4%

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

      4. +-commutative 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in m around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. unpow2 71.1%

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

      3. distribute-rgt-out 78.8%

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

      4. *-commutative 78.8%

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

      5. *-commutative 78.8%

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

   10. Simplified 78.8%

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

   11. Taylor expanded in m around inf 43.8%

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

       1. *-commutative 43.8%

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

   13. Simplified 43.8%

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

   if 3.60000000000000009 < l

   1. Initial program 79.1%

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

      1. associate-/l* 79.1%

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

      2. +-commutative 79.1%

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

      3. fabs-sub 79.1%

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

      4. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 97.1%

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

      1. mul-1-neg 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in M around 0 97.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.36 \cdot 10^{+202}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq -2.1 \cdot 10^{+177}:\\ \;\;\;\;\left({M}^{2} \cdot -0.5 + 1\right) \cdot e^{-\ell}\\ \mathbf{elif}\;\ell \leq -2.2:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 3.6:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \left(M + m\right)\\ \mathbf{if}\;M \leq -9 \cdot 10^{+108}:\\ \;\;\;\;\cos M \cdot e^{\left(n - \left(m + \ell\right)\right) - t\_0}\\ \mathbf{elif}\;M \leq -0.26:\\ \;\;\;\;\cos \left(\frac{n \cdot K}{2} - M\right) \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{elif}\;M \leq 0.0095:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(t\_0 - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) - \left(m + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* M (+ M m))))
   (if (<= M -9e+108)
     (* (cos M) (exp (- (- n (+ m l)) t_0)))
     (if (<= M -0.26)
       (* (cos (- (/ (* n K) 2.0) M)) (exp (- m (+ n l))))
       (if (<= M 0.0095)
         (* (cos M) (exp (+ (- m n) (- t_0 l))))
         (* (cos M) (exp (- (* M (- m M)) (+ m l)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M * (M + m);
	double tmp;
	if (M <= -9e+108) {
		tmp = cos(M) * exp(((n - (m + l)) - t_0));
	} else if (M <= -0.26) {
		tmp = cos((((n * K) / 2.0) - M)) * exp((m - (n + l)));
	} else if (M <= 0.0095) {
		tmp = cos(M) * exp(((m - n) + (t_0 - l)));
	} else {
		tmp = cos(M) * exp(((M * (m - M)) - (m + 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) :: tmp
    t_0 = m_1 * (m_1 + m)
    if (m_1 <= (-9d+108)) then
        tmp = cos(m_1) * exp(((n - (m + l)) - t_0))
    else if (m_1 <= (-0.26d0)) then
        tmp = cos((((n * k) / 2.0d0) - m_1)) * exp((m - (n + l)))
    else if (m_1 <= 0.0095d0) then
        tmp = cos(m_1) * exp(((m - n) + (t_0 - l)))
    else
        tmp = cos(m_1) * exp(((m_1 * (m - m_1)) - (m + 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 = M * (M + m);
	double tmp;
	if (M <= -9e+108) {
		tmp = Math.cos(M) * Math.exp(((n - (m + l)) - t_0));
	} else if (M <= -0.26) {
		tmp = Math.cos((((n * K) / 2.0) - M)) * Math.exp((m - (n + l)));
	} else if (M <= 0.0095) {
		tmp = Math.cos(M) * Math.exp(((m - n) + (t_0 - l)));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * (m - M)) - (m + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M * (M + m)
	tmp = 0
	if M <= -9e+108:
		tmp = math.cos(M) * math.exp(((n - (m + l)) - t_0))
	elif M <= -0.26:
		tmp = math.cos((((n * K) / 2.0) - M)) * math.exp((m - (n + l)))
	elif M <= 0.0095:
		tmp = math.cos(M) * math.exp(((m - n) + (t_0 - l)))
	else:
		tmp = math.cos(M) * math.exp(((M * (m - M)) - (m + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M * Float64(M + m))
	tmp = 0.0
	if (M <= -9e+108)
		tmp = Float64(cos(M) * exp(Float64(Float64(n - Float64(m + l)) - t_0)));
	elseif (M <= -0.26)
		tmp = Float64(cos(Float64(Float64(Float64(n * K) / 2.0) - M)) * exp(Float64(m - Float64(n + l))));
	elseif (M <= 0.0095)
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(t_0 - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(m - M)) - Float64(m + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M * (M + m);
	tmp = 0.0;
	if (M <= -9e+108)
		tmp = cos(M) * exp(((n - (m + l)) - t_0));
	elseif (M <= -0.26)
		tmp = cos((((n * K) / 2.0) - M)) * exp((m - (n + l)));
	elseif (M <= 0.0095)
		tmp = cos(M) * exp(((m - n) + (t_0 - l)));
	else
		tmp = cos(M) * exp(((M * (m - M)) - (m + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M * N[(M + m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -9e+108], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n - N[(m + l), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, -0.26], N[(N[Cos[N[(N[(N[(n * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(m - N[(n + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 0.0095], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(t$95$0 - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if M < -9e108

   1. Initial program 78.6%

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

      1. associate-/l* 78.6%

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

      2. +-commutative 78.6%

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

      3. fabs-sub 78.6%

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

      4. +-commutative 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 81.0%

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

      1. +-commutative 81.0%

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

      2. unpow2 81.0%

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

      3. distribute-rgt-out 97.7%

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

      4. *-commutative 97.7%

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

      5. *-commutative 97.7%

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

   10. Simplified 97.7%

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

   11. Taylor expanded in n around 0 95.3%

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

       1. associate--r+ 95.3%

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

       2. sub-neg 95.3%

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

       3. neg-mul-1 95.3%

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

       4. associate-*r* 95.3%

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

       5. neg-mul-1 95.3%

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

       6. cancel-sign-sub 95.3%

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

       7. neg-mul-1 95.3%

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

       8. sub-neg 95.3%

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

       9. unpow1 95.3%

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

       10. sqr-pow 54.8%

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

       11. fabs-sqr 54.8%

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

       12. sqr-pow 95.3%

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

       13. unpow1 95.3%

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

   13. Simplified 95.3%

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

       1. add-sqr-sqrt 52.5%

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

       2. fma-def 52.5%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 5.6%

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

       5. sqr-neg 5.6%

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

       6. sqrt-unprod 5.6%

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

       7. add-sqr-sqrt 5.6%

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

       8. distribute-lft-neg-out 5.6%

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

       9. fma-neg 5.6%

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

       10. add-sqr-sqrt 6.3%

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

       11. associate--l- 6.3%

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

       12. sub-neg 6.3%

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

       13. add-sqr-sqrt 6.3%

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

       14. sqrt-unprod 6.3%

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

       15. sqr-neg 6.3%

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

       16. sqrt-unprod 0.0%

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

       17. add-sqr-sqrt 90.6%

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

   15. Applied egg-rr 90.6%

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

   if -9e108 < M < -0.26000000000000001

   1. Initial program 76.7%

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

   3. Taylor expanded in m around 0 83.3%

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

   4. Taylor expanded in m around 0 70.1%

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

      1. +-commutative 76.8%

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

      2. unpow2 76.8%

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

      3. distribute-rgt-out 86.8%

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

      4. *-commutative 86.8%

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

      5. *-commutative 86.8%

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

   6. Simplified 73.5%

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

   7. Taylor expanded in n around 0 50.5%

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

      1. associate-*r* 50.5%

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

      2. neg-mul-1 50.5%

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

   9. Simplified 50.5%

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

   10. Taylor expanded in M around 0 8.0%

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

       1. rem-square-sqrt 4.1%

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

       2. fabs-sqr 4.1%

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

       3. rem-square-sqrt 34.2%

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

       4. associate--l- 34.2%

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

   12. Simplified 34.2%

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

   if -0.26000000000000001 < M < 0.00949999999999999976

   1. Initial program 68.5%

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

      1. associate-/l* 69.3%

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

      2. +-commutative 69.3%

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

      3. fabs-sub 69.3%

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

      4. +-commutative 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in m around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. unpow2 66.0%

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

      3. distribute-rgt-out 68.5%

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

      4. *-commutative 68.5%

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

      5. *-commutative 68.5%

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

   10. Simplified 68.5%

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

   11. Taylor expanded in n around 0 32.4%

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

       1. associate--r+ 32.4%

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

       2. sub-neg 32.4%

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

       3. neg-mul-1 32.4%

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

       4. associate-*r* 32.4%

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

       5. neg-mul-1 32.4%

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

       6. cancel-sign-sub 32.4%

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

       7. neg-mul-1 32.4%

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

       8. sub-neg 32.4%

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

       9. unpow1 32.4%

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

       10. sqr-pow 18.2%

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

       11. fabs-sqr 18.2%

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

       12. sqr-pow 68.5%

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

       13. unpow1 68.5%

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

   13. Simplified 68.5%

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

       1. associate-+l- 68.5%

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

       2. add-sqr-sqrt 18.2%

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

       3. fabs-sqr 18.2%

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

       4. add-sqr-sqrt 32.4%

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

       5. fabs-sub 32.4%

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

       6. add-sqr-sqrt 14.2%

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

       7. fabs-sqr 14.2%

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

       8. add-sqr-sqrt 56.2%

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

       9. sub-neg 56.2%

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

       10. add-sqr-sqrt 26.2%

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

       11. sqrt-unprod 56.2%

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

       12. sqr-neg 56.2%

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

       13. sqrt-unprod 30.0%

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

       14. add-sqr-sqrt 56.2%

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

   15. Applied egg-rr 56.2%

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

   if 0.00949999999999999976 < M

   1. Initial program 87.5%

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

      1. associate-/l* 87.5%

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

      2. +-commutative 87.5%

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

      3. fabs-sub 87.5%

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

      4. +-commutative 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. distribute-rgt-out 86.2%

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

      4. *-commutative 86.2%

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

      5. *-commutative 86.2%

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

   10. Simplified 86.2%

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

   11. Taylor expanded in n around 0 78.5%

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

       1. associate--r+ 78.5%

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

       2. sub-neg 78.5%

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

       3. neg-mul-1 78.5%

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

       4. associate-*r* 78.5%

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

       5. neg-mul-1 78.5%

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

       6. cancel-sign-sub 78.5%

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

       7. neg-mul-1 78.5%

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

       8. sub-neg 78.5%

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

       9. unpow1 78.5%

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

       10. sqr-pow 43.8%

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

       11. fabs-sqr 43.8%

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

       12. sqr-pow 81.5%

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

       13. unpow1 81.5%

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

   13. Simplified 81.5%

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

   14. Taylor expanded in n around 0 86.2%

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

       1. +-commutative 86.2%

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

   16. Simplified 86.2%

       \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot \left(m - M\right) - \left(m + \ell\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -9 \cdot 10^{+108}:\\ \;\;\;\;\cos M \cdot e^{\left(n - \left(m + \ell\right)\right) - M \cdot \left(M + m\right)}\\ \mathbf{elif}\;M \leq -0.26:\\ \;\;\;\;\cos \left(\frac{n \cdot K}{2} - M\right) \cdot e^{m - \left(n + \ell\right)}\\ \mathbf{elif}\;M \leq 0.0095:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) - \left(m + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -2.05 \cdot 10^{+65} \lor \neg \left(M \leq 0.0095\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) - \left(m + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= M -2.05e+65) (not (<= M 0.0095)))
   (* (cos M) (exp (- (* M (- m M)) (+ m l))))
   (* (cos M) (exp (+ (- m n) (- (* M (+ M m)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.05e+65) || !(M <= 0.0095)) {
		tmp = cos(M) * exp(((M * (m - M)) - (m + l)));
	} else {
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m_1 <= (-2.05d+65)) .or. (.not. (m_1 <= 0.0095d0))) then
        tmp = cos(m_1) * exp(((m_1 * (m - m_1)) - (m + l)))
    else
        tmp = cos(m_1) * exp(((m - n) + ((m_1 * (m_1 + m)) - l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((M <= -2.05e+65) || !(M <= 0.0095)) {
		tmp = Math.cos(M) * Math.exp(((M * (m - M)) - (m + l)));
	} else {
		tmp = Math.cos(M) * Math.exp(((m - n) + ((M * (M + m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (M <= -2.05e+65) or not (M <= 0.0095):
		tmp = math.cos(M) * math.exp(((M * (m - M)) - (m + l)))
	else:
		tmp = math.cos(M) * math.exp(((m - n) + ((M * (M + m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((M <= -2.05e+65) || !(M <= 0.0095))
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(m - M)) - Float64(m + l))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(Float64(M * Float64(M + m)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((M <= -2.05e+65) || ~((M <= 0.0095)))
		tmp = cos(M) * exp(((M * (m - M)) - (m + l)));
	else
		tmp = cos(M) * exp(((m - n) + ((M * (M + m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[M, -2.05e+65], N[Not[LessEqual[M, 0.0095]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(M * N[(M + m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < -2.0500000000000001e65 or 0.00949999999999999976 < M

   1. Initial program 84.7%

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

      1. associate-/l* 84.7%

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

      2. +-commutative 84.7%

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

      3. fabs-sub 84.7%

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

      4. +-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 78.9%

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

      1. +-commutative 78.9%

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

      2. unpow2 78.9%

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

      3. distribute-rgt-out 90.8%

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

      4. *-commutative 90.8%

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

      5. *-commutative 90.8%

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

   10. Simplified 90.8%

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

   11. Taylor expanded in n around 0 84.2%

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

       1. associate--r+ 84.2%

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

       2. sub-neg 84.2%

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

       3. neg-mul-1 84.2%

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

       4. associate-*r* 84.2%

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

       5. neg-mul-1 84.2%

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

       6. cancel-sign-sub 84.2%

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

       7. neg-mul-1 84.2%

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

       8. sub-neg 84.2%

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

       9. unpow1 84.2%

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

       10. sqr-pow 46.7%

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

       11. fabs-sqr 46.7%

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

       12. sqr-pow 86.7%

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

       13. unpow1 86.7%

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

   13. Simplified 86.7%

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

   14. Taylor expanded in n around 0 90.0%

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

       1. +-commutative 90.0%

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

   16. Simplified 90.0%

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

   if -2.0500000000000001e65 < M < 0.00949999999999999976

   1. Initial program 68.2%

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

      1. associate-/l* 69.7%

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

      2. +-commutative 69.7%

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

      3. fabs-sub 69.7%

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

      4. +-commutative 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in K around 0 95.4%

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

      1. cos-neg 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in m around 0 67.5%

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

      1. +-commutative 67.5%

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

      2. unpow2 67.5%

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

      3. distribute-rgt-out 70.4%

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

      4. *-commutative 70.4%

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

      5. *-commutative 70.4%

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

   10. Simplified 70.4%

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

   11. Taylor expanded in n around 0 32.6%

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

       1. associate--r+ 32.6%

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

       2. sub-neg 32.6%

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

       3. neg-mul-1 32.6%

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

       4. associate-*r* 32.6%

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

       5. neg-mul-1 32.6%

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

       6. cancel-sign-sub 32.6%

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

       7. neg-mul-1 32.6%

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

       8. sub-neg 32.6%

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

       9. unpow1 32.6%

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

       10. sqr-pow 15.9%

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

       11. fabs-sqr 15.9%

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

       12. sqr-pow 67.6%

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

       13. unpow1 67.6%

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

   13. Simplified 67.6%

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

       1. associate-+l- 67.6%

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

       2. add-sqr-sqrt 15.9%

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

       3. fabs-sqr 15.9%

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

       4. add-sqr-sqrt 32.6%

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

       5. fabs-sub 32.6%

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

       6. add-sqr-sqrt 16.8%

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

       7. fabs-sqr 16.8%

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

       8. add-sqr-sqrt 56.2%

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

       9. sub-neg 56.2%

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

       10. add-sqr-sqrt 30.1%

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

       11. sqrt-unprod 56.2%

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

       12. sqr-neg 56.2%

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

       13. sqrt-unprod 26.0%

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

       14. add-sqr-sqrt 55.4%

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

   15. Applied egg-rr 55.4%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.05 \cdot 10^{+65} \lor \neg \left(M \leq 0.0095\right):\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) - \left(m + \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M \cdot \left(M + m\right)\\ \mathbf{if}\;M \leq -1.15 \cdot 10^{+63}:\\ \;\;\;\;\cos M \cdot e^{\left(n - \left(m + \ell\right)\right) - t\_0}\\ \mathbf{elif}\;M \leq 0.0095:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(t\_0 - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) - \left(m + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* M (+ M m))))
   (if (<= M -1.15e+63)
     (* (cos M) (exp (- (- n (+ m l)) t_0)))
     (if (<= M 0.0095)
       (* (cos M) (exp (+ (- m n) (- t_0 l))))
       (* (cos M) (exp (- (* M (- m M)) (+ m l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M * (M + m);
	double tmp;
	if (M <= -1.15e+63) {
		tmp = cos(M) * exp(((n - (m + l)) - t_0));
	} else if (M <= 0.0095) {
		tmp = cos(M) * exp(((m - n) + (t_0 - l)));
	} else {
		tmp = cos(M) * exp(((M * (m - M)) - (m + 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) :: tmp
    t_0 = m_1 * (m_1 + m)
    if (m_1 <= (-1.15d+63)) then
        tmp = cos(m_1) * exp(((n - (m + l)) - t_0))
    else if (m_1 <= 0.0095d0) then
        tmp = cos(m_1) * exp(((m - n) + (t_0 - l)))
    else
        tmp = cos(m_1) * exp(((m_1 * (m - m_1)) - (m + 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 = M * (M + m);
	double tmp;
	if (M <= -1.15e+63) {
		tmp = Math.cos(M) * Math.exp(((n - (m + l)) - t_0));
	} else if (M <= 0.0095) {
		tmp = Math.cos(M) * Math.exp(((m - n) + (t_0 - l)));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * (m - M)) - (m + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M * (M + m)
	tmp = 0
	if M <= -1.15e+63:
		tmp = math.cos(M) * math.exp(((n - (m + l)) - t_0))
	elif M <= 0.0095:
		tmp = math.cos(M) * math.exp(((m - n) + (t_0 - l)))
	else:
		tmp = math.cos(M) * math.exp(((M * (m - M)) - (m + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M * Float64(M + m))
	tmp = 0.0
	if (M <= -1.15e+63)
		tmp = Float64(cos(M) * exp(Float64(Float64(n - Float64(m + l)) - t_0)));
	elseif (M <= 0.0095)
		tmp = Float64(cos(M) * exp(Float64(Float64(m - n) + Float64(t_0 - l))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(m - M)) - Float64(m + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M * (M + m);
	tmp = 0.0;
	if (M <= -1.15e+63)
		tmp = cos(M) * exp(((n - (m + l)) - t_0));
	elseif (M <= 0.0095)
		tmp = cos(M) * exp(((m - n) + (t_0 - l)));
	else
		tmp = cos(M) * exp(((M * (m - M)) - (m + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M * N[(M + m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -1.15e+63], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(n - N[(m + l), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 0.0095], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(t$95$0 - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if M < -1.14999999999999997e63

   1. Initial program 81.5%

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

      1. associate-/l* 81.5%

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

      2. +-commutative 81.5%

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

      3. fabs-sub 81.5%

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

      4. +-commutative 81.5%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 79.7%

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

      1. +-commutative 79.7%

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

      2. unpow2 79.7%

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

      3. distribute-rgt-out 96.4%

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

      4. *-commutative 96.4%

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

      5. *-commutative 96.4%

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

   10. Simplified 96.4%

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

   11. Taylor expanded in n around 0 90.9%

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

       1. associate--r+ 90.9%

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

       2. sub-neg 90.9%

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

       3. neg-mul-1 90.9%

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

       4. associate-*r* 90.9%

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

       5. neg-mul-1 90.9%

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

       6. cancel-sign-sub 90.9%

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

       7. neg-mul-1 90.9%

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

       8. sub-neg 90.9%

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

       9. unpow1 90.9%

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

       10. sqr-pow 50.1%

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

       11. fabs-sqr 50.1%

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

       12. sqr-pow 92.7%

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

       13. unpow1 92.7%

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

   13. Simplified 92.7%

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

       1. add-sqr-sqrt 48.3%

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

       2. fma-def 48.3%

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

       3. add-sqr-sqrt 0.0%

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

       4. sqrt-unprod 6.3%

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

       5. sqr-neg 6.3%

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

       6. sqrt-unprod 6.3%

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

       7. add-sqr-sqrt 6.3%

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

       8. distribute-lft-neg-out 6.3%

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

       9. fma-neg 6.3%

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

       10. add-sqr-sqrt 8.9%

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

       11. associate--l- 8.9%

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

       12. sub-neg 8.9%

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

       13. add-sqr-sqrt 8.9%

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

       14. sqrt-unprod 8.9%

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

       15. sqr-neg 8.9%

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

       16. sqrt-unprod 0.0%

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

       17. add-sqr-sqrt 83.6%

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

   15. Applied egg-rr 83.6%

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

   if -1.14999999999999997e63 < M < 0.00949999999999999976

   1. Initial program 68.2%

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

      1. associate-/l* 69.7%

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

      2. +-commutative 69.7%

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

      3. fabs-sub 69.7%

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

      4. +-commutative 69.7%

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

   3. Simplified 69.7%

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

   5. Taylor expanded in K around 0 95.4%

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

      1. cos-neg 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in m around 0 67.5%

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

      1. +-commutative 67.5%

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

      2. unpow2 67.5%

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

      3. distribute-rgt-out 70.4%

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

      4. *-commutative 70.4%

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

      5. *-commutative 70.4%

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

   10. Simplified 70.4%

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

   11. Taylor expanded in n around 0 32.6%

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

       1. associate--r+ 32.6%

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

       2. sub-neg 32.6%

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

       3. neg-mul-1 32.6%

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

       4. associate-*r* 32.6%

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

       5. neg-mul-1 32.6%

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

       6. cancel-sign-sub 32.6%

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

       7. neg-mul-1 32.6%

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

       8. sub-neg 32.6%

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

       9. unpow1 32.6%

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

       10. sqr-pow 15.9%

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

       11. fabs-sqr 15.9%

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

       12. sqr-pow 67.6%

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

       13. unpow1 67.6%

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

   13. Simplified 67.6%

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

       1. associate-+l- 67.6%

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

       2. add-sqr-sqrt 15.9%

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

       3. fabs-sqr 15.9%

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

       4. add-sqr-sqrt 32.6%

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

       5. fabs-sub 32.6%

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

       6. add-sqr-sqrt 16.8%

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

       7. fabs-sqr 16.8%

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

       8. add-sqr-sqrt 56.2%

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

       9. sub-neg 56.2%

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

       10. add-sqr-sqrt 30.1%

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

       11. sqrt-unprod 56.2%

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

       12. sqr-neg 56.2%

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

       13. sqrt-unprod 26.0%

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

       14. add-sqr-sqrt 55.4%

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

   15. Applied egg-rr 55.4%

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

   if 0.00949999999999999976 < M

   1. Initial program 87.5%

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

      1. associate-/l* 87.5%

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

      2. +-commutative 87.5%

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

      3. fabs-sub 87.5%

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

      4. +-commutative 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 78.2%

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

      1. +-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. distribute-rgt-out 86.2%

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

      4. *-commutative 86.2%

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

      5. *-commutative 86.2%

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

   10. Simplified 86.2%

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

   11. Taylor expanded in n around 0 78.5%

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

       1. associate--r+ 78.5%

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

       2. sub-neg 78.5%

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

       3. neg-mul-1 78.5%

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

       4. associate-*r* 78.5%

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

       5. neg-mul-1 78.5%

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

       6. cancel-sign-sub 78.5%

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

       7. neg-mul-1 78.5%

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

       8. sub-neg 78.5%

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

       9. unpow1 78.5%

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

       10. sqr-pow 43.8%

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

       11. fabs-sqr 43.8%

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

       12. sqr-pow 81.5%

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

       13. unpow1 81.5%

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

   13. Simplified 81.5%

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

   14. Taylor expanded in n around 0 86.2%

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

       1. +-commutative 86.2%

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

   16. Simplified 86.2%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -1.15 \cdot 10^{+63}:\\ \;\;\;\;\cos M \cdot e^{\left(n - \left(m + \ell\right)\right) - M \cdot \left(M + m\right)}\\ \mathbf{elif}\;M \leq 0.0095:\\ \;\;\;\;\cos M \cdot e^{\left(m - n\right) + \left(M \cdot \left(M + m\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) - \left(m + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 3.6:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -2.2)
   (* (cos M) (exp l))
   (if (<= l 3.6) (* (cos M) (exp (* m (- M (* n 0.5))))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -2.2) {
		tmp = cos(M) * exp(l);
	} else if (l <= 3.6) {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2.2d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 3.6d0) then
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -2.2) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 3.6) {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -2.2:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 3.6:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -2.2)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 3.6)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -2.2)
		tmp = cos(M) * exp(l);
	elseif (l <= 3.6)
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -2.2], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.6], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.2000000000000002

   1. Initial program 80.3%

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

      1. associate-/l* 83.1%

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

      2. +-commutative 83.1%

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

      3. fabs-sub 83.1%

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

      4. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in K around 0 94.4%

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

      1. cos-neg 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in l around inf 25.1%

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

      1. mul-1-neg 25.1%

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

   10. Simplified 25.1%

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

       1. expm1-log1p-u 24.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

       2. expm1-udef 24.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

       3. add-sqr-sqrt 24.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

       4. sqrt-unprod 24.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

       5. sqr-neg 24.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

       6. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

       7. add-sqr-sqrt 69.5%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   12. Applied egg-rr 69.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 69.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 69.5%

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

   14. Simplified 69.5%

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

   if -2.2000000000000002 < l < 3.60000000000000009

   1. Initial program 71.3%

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

      1. associate-/l* 71.4%

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

      2. +-commutative 71.4%

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

      3. fabs-sub 71.4%

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

      4. +-commutative 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in m around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. unpow2 71.1%

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

      3. distribute-rgt-out 78.8%

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

      4. *-commutative 78.8%

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

      5. *-commutative 78.8%

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

   10. Simplified 78.8%

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

   11. Taylor expanded in m around inf 43.8%

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

       1. *-commutative 43.8%

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

   13. Simplified 43.8%

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

   if 3.60000000000000009 < l

   1. Initial program 79.1%

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

      1. associate-/l* 79.1%

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

      2. +-commutative 79.1%

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

      3. fabs-sub 79.1%

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

      4. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 97.1%

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

      1. mul-1-neg 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in M around 0 97.1%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 3.6:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.2:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 3.6:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -2.2)
   (* (cos M) (exp l))
   (if (<= l 3.6) (* (cos M) (exp (* m (+ M -1.0)))) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -2.2) {
		tmp = cos(M) * exp(l);
	} else if (l <= 3.6) {
		tmp = cos(M) * exp((m * (M + -1.0)));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2.2d0)) then
        tmp = cos(m_1) * exp(l)
    else if (l <= 3.6d0) then
        tmp = cos(m_1) * exp((m * (m_1 + (-1.0d0))))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -2.2) {
		tmp = Math.cos(M) * Math.exp(l);
	} else if (l <= 3.6) {
		tmp = Math.cos(M) * Math.exp((m * (M + -1.0)));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -2.2:
		tmp = math.cos(M) * math.exp(l)
	elif l <= 3.6:
		tmp = math.cos(M) * math.exp((m * (M + -1.0)))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -2.2)
		tmp = Float64(cos(M) * exp(l));
	elseif (l <= 3.6)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M + -1.0))));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -2.2)
		tmp = cos(M) * exp(l);
	elseif (l <= 3.6)
		tmp = cos(M) * exp((m * (M + -1.0)));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -2.2], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 3.6], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -2.2000000000000002

   1. Initial program 80.3%

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

      1. associate-/l* 83.1%

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

      2. +-commutative 83.1%

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

      3. fabs-sub 83.1%

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

      4. +-commutative 83.1%

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

   3. Simplified 83.1%

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

   5. Taylor expanded in K around 0 94.4%

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

      1. cos-neg 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in l around inf 25.1%

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

      1. mul-1-neg 25.1%

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

   10. Simplified 25.1%

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

       1. expm1-log1p-u 24.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

       2. expm1-udef 24.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

       3. add-sqr-sqrt 24.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

       4. sqrt-unprod 24.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

       5. sqr-neg 24.7%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

       6. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

       7. add-sqr-sqrt 69.5%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   12. Applied egg-rr 69.5%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 69.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 69.5%

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

   14. Simplified 69.5%

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

   if -2.2000000000000002 < l < 3.60000000000000009

   1. Initial program 71.3%

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

      1. associate-/l* 71.4%

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

      2. +-commutative 71.4%

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

      3. fabs-sub 71.4%

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

      4. +-commutative 71.4%

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

   3. Simplified 71.4%

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

   5. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in m around 0 71.1%

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

      1. +-commutative 71.1%

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

      2. unpow2 71.1%

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

      3. distribute-rgt-out 78.8%

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

      4. *-commutative 78.8%

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

      5. *-commutative 78.8%

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

   10. Simplified 78.8%

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

   11. Taylor expanded in n around 0 48.8%

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

       1. associate--r+ 48.8%

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

       2. sub-neg 48.8%

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

       3. neg-mul-1 48.8%

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

       4. associate-*r* 48.8%

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

       5. neg-mul-1 48.8%

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

       6. cancel-sign-sub 48.8%

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

       7. neg-mul-1 48.8%

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

       8. sub-neg 48.8%

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

       9. unpow1 48.8%

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

       10. sqr-pow 23.5%

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

       11. fabs-sqr 23.5%

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

       12. sqr-pow 72.2%

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

       13. unpow1 72.2%

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

   13. Simplified 72.2%

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

   14. Taylor expanded in m around inf 39.7%

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

   if 3.60000000000000009 < l

   1. Initial program 79.1%

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

      1. associate-/l* 79.1%

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

      2. +-commutative 79.1%

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

      3. fabs-sub 79.1%

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

      4. +-commutative 79.1%

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

   3. Simplified 79.1%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in l around inf 97.1%

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

      1. mul-1-neg 97.1%

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

   10. Simplified 97.1%

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

   11. Taylor expanded in M around 0 97.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.2:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 3.6:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -7 \cdot 10^{+138}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) - \left(m + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -7e+138)
   (* (cos M) (exp (* m (- M (* n 0.5)))))
   (* (cos M) (exp (- (* M (- m M)) (+ m l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -7e+138) {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	} else {
		tmp = cos(M) * exp(((M * (m - M)) - (m + l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-7d+138)) then
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    else
        tmp = cos(m_1) * exp(((m_1 * (m - m_1)) - (m + l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -7e+138) {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	} else {
		tmp = Math.cos(M) * Math.exp(((M * (m - M)) - (m + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -7e+138:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	else:
		tmp = math.cos(M) * math.exp(((M * (m - M)) - (m + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -7e+138)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(M * Float64(m - M)) - Float64(m + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -7e+138)
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	else
		tmp = cos(M) * exp(((M * (m - M)) - (m + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -7e+138], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision] - N[(m + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -6.9999999999999996e138

   1. Initial program 60.0%

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

      1. associate-/l* 60.0%

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

      2. +-commutative 60.0%

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

      3. fabs-sub 60.0%

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

      4. +-commutative 60.0%

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

   3. Simplified 60.0%

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

   5. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in m around 0 40.6%

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

      1. +-commutative 40.6%

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

      2. unpow2 40.6%

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

      3. distribute-rgt-out 60.6%

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

      4. *-commutative 60.6%

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

      5. *-commutative 60.6%

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

   10. Simplified 60.6%

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

   11. Taylor expanded in m around inf 47.5%

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

       1. *-commutative 47.5%

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

   13. Simplified 47.5%

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

   if -6.9999999999999996e138 < m

   1. Initial program 78.0%

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

      1. associate-/l* 78.8%

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

      2. +-commutative 78.8%

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

      3. fabs-sub 78.8%

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

      4. +-commutative 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in K around 0 97.2%

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

      1. cos-neg 97.2%

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

   7. Simplified 97.2%

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

   8. Taylor expanded in m around 0 77.0%

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

      1. +-commutative 77.0%

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

      2. unpow2 77.0%

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

      3. distribute-rgt-out 82.4%

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

      4. *-commutative 82.4%

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

      5. *-commutative 82.4%

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

   10. Simplified 82.4%

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

   11. Taylor expanded in n around 0 58.4%

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

       1. associate--r+ 58.4%

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

       2. sub-neg 58.4%

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

       3. neg-mul-1 58.4%

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

       4. associate-*r* 58.4%

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

       5. neg-mul-1 58.4%

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

       6. cancel-sign-sub 58.4%

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

       7. neg-mul-1 58.4%

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

       8. sub-neg 58.4%

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

       9. unpow1 58.4%

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

       10. sqr-pow 28.7%

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

       11. fabs-sqr 28.7%

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

       12. sqr-pow 80.2%

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

       13. unpow1 80.2%

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

   13. Simplified 80.2%

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

   14. Taylor expanded in n around 0 79.0%

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

       1. +-commutative 79.0%

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

   16. Simplified 79.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -7 \cdot 10^{+138}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right) - \left(m + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 49.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.00052:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -0.00052) (* (cos M) (exp l)) (/ 1.0 (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -0.00052) {
		tmp = cos(M) * exp(l);
	} else {
		tmp = 1.0 / exp(l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-0.00052d0)) then
        tmp = cos(m_1) * exp(l)
    else
        tmp = 1.0d0 / exp(l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -0.00052) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = 1.0 / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -0.00052:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = 1.0 / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -0.00052)
		tmp = Float64(cos(M) * exp(l));
	else
		tmp = Float64(1.0 / exp(l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -0.00052)
		tmp = cos(M) * exp(l);
	else
		tmp = 1.0 / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -0.00052], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -5.19999999999999954e-4

   1. Initial program 80.6%

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

      1. associate-/l* 83.3%

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

      2. +-commutative 83.3%

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

      3. fabs-sub 83.3%

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

      4. +-commutative 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in K around 0 94.4%

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

      1. cos-neg 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in l around inf 24.8%

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

      1. mul-1-neg 24.8%

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

   10. Simplified 24.8%

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

       1. expm1-log1p-u 24.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

       2. expm1-udef 24.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

       3. add-sqr-sqrt 24.4%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

       4. sqrt-unprod 24.4%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

       5. sqr-neg 24.4%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

       6. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

       7. add-sqr-sqrt 68.6%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   12. Applied egg-rr 68.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 68.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 68.6%

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

   14. Simplified 68.6%

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

   if -5.19999999999999954e-4 < l

   1. Initial program 74.0%

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

      1. associate-/l* 74.0%

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

      2. +-commutative 74.0%

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

      3. fabs-sub 74.0%

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

      4. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in l around inf 40.8%

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

      1. mul-1-neg 40.8%

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

   10. Simplified 40.8%

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

       1. exp-neg 40.8%

          \[\leadsto \cos M \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

       2. un-div-inv 40.8%

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

   12. Applied egg-rr 40.8%

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

   13. Taylor expanded in M around 0 40.8%

       \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.00052:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{-6}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -9.2e-6) (* (cos M) (exp l)) (/ (cos M) (exp l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -9.2e-6) {
		tmp = cos(M) * exp(l);
	} 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 <= (-9.2d-6)) then
        tmp = cos(m_1) * exp(l)
    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 <= -9.2e-6) {
		tmp = Math.cos(M) * Math.exp(l);
	} else {
		tmp = Math.cos(M) / Math.exp(l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -9.2e-6:
		tmp = math.cos(M) * math.exp(l)
	else:
		tmp = math.cos(M) / math.exp(l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -9.2e-6)
		tmp = Float64(cos(M) * exp(l));
	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 <= -9.2e-6)
		tmp = cos(M) * exp(l);
	else
		tmp = cos(M) / exp(l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -9.2e-6], N[(N[Cos[M], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -9.2e-6

   1. Initial program 80.6%

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

      1. associate-/l* 83.3%

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

      2. +-commutative 83.3%

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

      3. fabs-sub 83.3%

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

      4. +-commutative 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in K around 0 94.4%

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

      1. cos-neg 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in l around inf 24.8%

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

      1. mul-1-neg 24.8%

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

   10. Simplified 24.8%

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

       1. expm1-log1p-u 24.4%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)\right)} \]

       2. expm1-udef 24.4%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{-\ell}\right)} - 1} \]

       3. add-sqr-sqrt 24.4%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{-\ell} \cdot \sqrt{-\ell}}}\right)} - 1 \]

       4. sqrt-unprod 24.4%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\left(-\ell\right) \cdot \left(-\ell\right)}}}\right)} - 1 \]

       5. sqr-neg 24.4%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\sqrt{\color{blue}{\ell \cdot \ell}}}\right)} - 1 \]

       6. sqrt-unprod 0.0%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)} - 1 \]

       7. add-sqr-sqrt 68.6%

          \[\leadsto e^{\mathsf{log1p}\left(\cos M \cdot e^{\color{blue}{\ell}}\right)} - 1 \]

   12. Applied egg-rr 68.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 68.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos M \cdot e^{\ell}\right)\right)} \]

       2. expm1-log1p 68.6%

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

   14. Simplified 68.6%

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

   if -9.2e-6 < l

   1. Initial program 74.0%

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

      1. associate-/l* 74.0%

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

      2. +-commutative 74.0%

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

      3. fabs-sub 74.0%

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

      4. +-commutative 74.0%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in l around inf 40.8%

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

      1. mul-1-neg 40.8%

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

   10. Simplified 40.8%

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

       1. exp-neg 40.8%

          \[\leadsto \cos M \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

       2. un-div-inv 40.8%

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

   12. Applied egg-rr 40.8%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -9.2 \cdot 10^{-6}:\\ \;\;\;\;\cos M \cdot e^{\ell}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{e^{\ell}} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (/ 1.0 (exp l)))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 / 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 = 1.0d0 / exp(l)
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 / Math.exp(l);
}

Python

def code(K, m, n, M, l):
	return 1.0 / math.exp(l)

Julia

function code(K, m, n, M, l)
	return Float64(1.0 / exp(l))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 / exp(l);
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 / N[Exp[l], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 76.6%

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

   2. +-commutative 76.6%

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

   3. fabs-sub 76.6%

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

   4. +-commutative 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in l around inf 36.3%

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

   1. mul-1-neg 36.3%

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

10. Simplified 36.3%

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

    1. exp-neg 36.3%

       \[\leadsto \cos M \cdot \color{blue}{\frac{1}{e^{\ell}}} \]

    2. un-div-inv 36.3%

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

12. Applied egg-rr 36.3%

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

13. Taylor expanded in M around 0 36.3%

    \[\leadsto \color{blue}{\frac{1}{e^{\ell}}} \]

14. Final simplification 36.3%

    \[\leadsto \frac{1}{e^{\ell}} \]
15. Add Preprocessing

Alternative 14: 34.7% accurate, 4.2× speedup

Math

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

FPCore

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

C

double code(double K, double m, double n, double M, double l) {
	return 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 = exp(-l)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.9%

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

   1. associate-/l* 76.6%

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

   2. +-commutative 76.6%

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

   3. fabs-sub 76.6%

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

   4. +-commutative 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in l around inf 36.3%

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

   1. mul-1-neg 36.3%

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

10. Simplified 36.3%

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

11. Taylor expanded in M around 0 36.3%

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

12. Final simplification 36.3%

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

Alternative 15: 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 75.9%

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

   1. associate-/l* 76.6%

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

   2. +-commutative 76.6%

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

   3. fabs-sub 76.6%

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

   4. +-commutative 76.6%

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

3. Simplified 76.6%

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

5. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

7. Simplified 97.5%

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

8. Taylor expanded in l around inf 36.3%

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

   1. mul-1-neg 36.3%

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

10. Simplified 36.3%

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

11. Taylor expanded in l around 0 5.4%

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

12. Final simplification 5.4%

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

Reproduce

herbie shell --seed 2024040 
(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)))))))