Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.9%

Time: 23.8s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 1.0× speedup

Math

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

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(abs(Float64(n - m)) - Float64(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[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - N[(l + N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-/l* 76.3%

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

   2. exp-diff 23.1%

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

   3. sub-neg 23.1%

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

   4. exp-sum 16.9%

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

   5. associate-/r* 16.9%

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

   6. exp-diff 24.3%

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

   7. exp-diff 76.3%

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

   8. sub-neg 76.3%

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

   9. remove-double-neg 76.3%

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

   10. fabs-sub 76.3%

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

3. Simplified 76.3%

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

4. Taylor expanded in K around 0 96.7%

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

   1. cos-neg 96.7%

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

6. Simplified 96.7%

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

7. Final simplification 96.7%

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

Alternative 2: 87.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -1e+14)
   (* (cos M) (exp (* (pow m 2.0) -0.25)))
   (*
    (cos M)
    (exp (+ (fabs (- n m)) (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -1e+14) {
		tmp = cos(M) * exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = cos(M) * exp((fabs((n - m)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - 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 <= (-1d+14)) then
        tmp = cos(m_1) * exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = cos(m_1) * exp((abs((n - m)) + ((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - 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 <= -1e+14) {
		tmp = Math.cos(M) * Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.cos(M) * Math.exp((Math.abs((n - m)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -1e+14:
		tmp = math.cos(M) * math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.cos(M) * math.exp((math.fabs((n - m)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -1e+14)
		tmp = Float64(cos(M) * exp(Float64((m ^ 2.0) * -0.25)));
	else
		tmp = Float64(cos(M) * exp(Float64(abs(Float64(n - m)) + Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -1e+14)
		tmp = cos(M) * exp(((m ^ 2.0) * -0.25));
	else
		tmp = cos(M) * exp((abs((n - m)) + ((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -1e+14], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -1e14

   1. Initial program 69.5%

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

      1. associate-/l* 69.5%

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

      2. exp-diff 8.5%

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

      3. sub-neg 8.5%

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

      4. exp-sum 0.0%

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

      5. associate-/r* 0.0%

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

      6. exp-diff 6.8%

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

      7. exp-diff 69.5%

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

      8. sub-neg 69.5%

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

      9. remove-double-neg 69.5%

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

      10. fabs-sub 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

   6. Simplified 98.3%

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

   7. Taylor expanded in n around 0 91.6%

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

      1. +-commutative 91.6%

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

      2. unpow2 91.6%

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

      3. distribute-rgt-out 96.6%

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

      4. *-commutative 96.6%

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

      5. *-commutative 96.6%

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

   9. Simplified 96.6%

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

   10. Taylor expanded in m around inf 93.3%

       \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {m}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 93.3%

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

   12. Simplified 93.3%

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

   if -1e14 < m

   1. Initial program 78.8%

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

      1. associate-/l* 78.3%

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

      2. exp-diff 27.5%

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

      3. sub-neg 27.5%

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

      4. exp-sum 21.9%

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

      5. associate-/r* 21.9%

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

      6. exp-diff 29.6%

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

      7. exp-diff 78.3%

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

      8. sub-neg 78.3%

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

      9. remove-double-neg 78.3%

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

      10. fabs-sub 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in m around 0 75.1%

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

      1. +-commutative 75.1%

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

      2. unpow2 75.1%

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

      3. distribute-rgt-out 82.2%

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

      4. *-commutative 82.2%

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

      5. *-commutative 82.2%

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

   9. Simplified 82.2%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\cos M \cdot e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left|n - m\right| + \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \ell\right)}\\ \end{array} \]

Alternative 3: 87.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 98:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(m \cdot 0.5 - M\right) \cdot \left(M - \left(n + m \cdot 0.5\right)\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 98.0)
   (exp (+ (fabs (- n m)) (- (* (- (* m 0.5) M) (- M (+ n (* m 0.5)))) l)))
   (* (cos M) (exp (* -0.25 (pow n 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 98.0) {
		tmp = exp((fabs((n - m)) + ((((m * 0.5) - M) * (M - (n + (m * 0.5)))) - l)));
	} else {
		tmp = cos(M) * exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 98.0d0) then
        tmp = exp((abs((n - m)) + ((((m * 0.5d0) - m_1) * (m_1 - (n + (m * 0.5d0)))) - l)))
    else
        tmp = cos(m_1) * exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 98.0) {
		tmp = Math.exp((Math.abs((n - m)) + ((((m * 0.5) - M) * (M - (n + (m * 0.5)))) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 98.0:
		tmp = math.exp((math.fabs((n - m)) + ((((m * 0.5) - M) * (M - (n + (m * 0.5)))) - l)))
	else:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 98.0)
		tmp = exp(Float64(abs(Float64(n - m)) + Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(M - Float64(n + Float64(m * 0.5)))) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (n ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 98.0)
		tmp = exp((abs((n - m)) + ((((m * 0.5) - M) * (M - (n + (m * 0.5)))) - l)));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 98.0], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(M - N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < 98

   1. Initial program 78.1%

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

      1. associate-/l* 77.6%

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

      2. exp-diff 28.8%

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

      3. sub-neg 28.8%

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

      4. exp-sum 22.2%

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

      5. associate-/r* 22.2%

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

      6. exp-diff 30.4%

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

      7. exp-diff 77.6%

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

      8. sub-neg 77.6%

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

      9. remove-double-neg 77.6%

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

      10. fabs-sub 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in n around 0 69.1%

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

      1. +-commutative 82.6%

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

      2. unpow2 82.6%

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

      3. distribute-rgt-out 84.7%

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

      4. *-commutative 84.7%

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

      5. *-commutative 84.7%

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

   6. Simplified 70.1%

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

   7. Taylor expanded in M around 0 70.1%

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

      1. *-commutative 70.1%

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

      2. *-commutative 70.1%

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

      3. associate-*r* 70.1%

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

   9. Simplified 70.1%

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

   10. Taylor expanded in K around 0 84.7%

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

   if 98 < n

   1. Initial program 72.1%

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

      1. associate-/l* 72.1%

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

      2. exp-diff 4.9%

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

      3. sub-neg 4.9%

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

      4. exp-sum 0.0%

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

      5. associate-/r* 0.0%

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

      6. exp-diff 4.9%

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

      7. exp-diff 72.1%

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

      8. sub-neg 72.1%

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

      9. remove-double-neg 72.1%

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

      10. fabs-sub 72.1%

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

   3. Simplified 72.1%

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

   4. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   6. Simplified 98.4%

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

   7. Taylor expanded in m around 0 85.4%

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

      1. +-commutative 85.4%

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

      2. unpow2 85.4%

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

      3. distribute-rgt-out 90.3%

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

      4. *-commutative 90.3%

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

      5. *-commutative 90.3%

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

   9. Simplified 90.3%

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

   10. Taylor expanded in n around inf 93.5%

       \[\leadsto \cos M \cdot e^{\color{blue}{-0.25 \cdot {n}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 93.5%

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

   12. Simplified 93.5%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 98:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(m \cdot 0.5 - M\right) \cdot \left(M - \left(n + m \cdot 0.5\right)\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]

Alternative 4: 77.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq 0.00072:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(m \cdot 0.5 - M\right) \cdot \left(M - \left(n + m \cdot 0.5\right)\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M 0.00072)
   (exp (+ (fabs (- n m)) (- (* (- (* m 0.5) M) (- M (+ n (* m 0.5)))) l)))
   (* (cos M) (exp (* M (- m M))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= 0.00072) {
		tmp = exp((fabs((n - m)) + ((((m * 0.5) - M) * (M - (n + (m * 0.5)))) - l)));
	} else {
		tmp = cos(M) * exp((M * (m - M)));
	}
	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 <= 0.00072d0) then
        tmp = exp((abs((n - m)) + ((((m * 0.5d0) - m_1) * (m_1 - (n + (m * 0.5d0)))) - l)))
    else
        tmp = cos(m_1) * exp((m_1 * (m - m_1)))
    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 <= 0.00072) {
		tmp = Math.exp((Math.abs((n - m)) + ((((m * 0.5) - M) * (M - (n + (m * 0.5)))) - l)));
	} else {
		tmp = Math.cos(M) * Math.exp((M * (m - M)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= 0.00072:
		tmp = math.exp((math.fabs((n - m)) + ((((m * 0.5) - M) * (M - (n + (m * 0.5)))) - l)))
	else:
		tmp = math.cos(M) * math.exp((M * (m - M)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= 0.00072)
		tmp = exp(Float64(abs(Float64(n - m)) + Float64(Float64(Float64(Float64(m * 0.5) - M) * Float64(M - Float64(n + Float64(m * 0.5)))) - l)));
	else
		tmp = Float64(cos(M) * exp(Float64(M * Float64(m - M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= 0.00072)
		tmp = exp((abs((n - m)) + ((((m * 0.5) - M) * (M - (n + (m * 0.5)))) - l)));
	else
		tmp = cos(M) * exp((M * (m - M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, 0.00072], N[Exp[N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(M - N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if M < 7.20000000000000045e-4

   1. Initial program 74.3%

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

      1. associate-/l* 73.7%

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

      2. exp-diff 20.4%

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

      3. sub-neg 20.4%

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

      4. exp-sum 14.7%

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

      5. associate-/r* 14.7%

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

      6. exp-diff 22.6%

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

      7. exp-diff 73.7%

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

      8. sub-neg 73.7%

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

      9. remove-double-neg 73.7%

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

      10. fabs-sub 73.7%

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

   3. Simplified 73.7%

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

   4. Taylor expanded in n around 0 58.9%

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

      1. +-commutative 73.8%

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

      2. unpow2 73.8%

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

      3. distribute-rgt-out 76.6%

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

      4. *-commutative 76.6%

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

      5. *-commutative 76.6%

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

   6. Simplified 60.6%

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

   7. Taylor expanded in M around 0 60.6%

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

      1. *-commutative 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*r* 60.6%

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

   9. Simplified 60.6%

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

   10. Taylor expanded in K around 0 75.5%

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

   if 7.20000000000000045e-4 < M

   1. Initial program 82.1%

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

      1. associate-/l* 82.1%

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

      2. exp-diff 29.5%

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

      3. sub-neg 29.5%

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

      4. exp-sum 21.8%

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

      5. associate-/r* 21.8%

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

      6. exp-diff 28.2%

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

      7. exp-diff 82.1%

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

      8. sub-neg 82.1%

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

      9. remove-double-neg 82.1%

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

      10. fabs-sub 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in n around 0 79.7%

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

      1. +-commutative 79.7%

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

      2. unpow2 79.7%

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

      3. distribute-rgt-out 86.1%

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

      4. *-commutative 86.1%

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

      5. *-commutative 86.1%

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

   9. Simplified 86.1%

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

   10. Taylor expanded in M around -inf 61.8%

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

       1. +-commutative 61.8%

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

       2. mul-1-neg 61.8%

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

       3. unsub-neg 61.8%

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

       4. +-commutative 61.8%

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

   12. Simplified 61.8%

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

   13. Taylor expanded in n around 0 72.1%

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

       1. unpow2 72.1%

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

       2. distribute-lft-out-- 79.8%

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

   15. Simplified 79.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq 0.00072:\\ \;\;\;\;e^{\left|n - m\right| + \left(\left(m \cdot 0.5 - M\right) \cdot \left(M - \left(n + m \cdot 0.5\right)\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \]

Alternative 5: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{M \cdot \left(m - M\right)}\\ t_1 := e^{-\ell}\\ \mathbf{if}\;M \leq -2.7 \cdot 10^{+161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq -4 \cdot 10^{+124}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;M \leq -1.65 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;M \leq -215000:\\ \;\;\;\;\cos M \cdot t_1\\ \mathbf{elif}\;M \leq -1.4 \cdot 10^{-177}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(M + m \cdot -0.5\right)}\\ \mathbf{elif}\;M \leq 1.45 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (* M (- m M))))) (t_1 (exp (- l))))
   (if (<= M -2.7e+161)
     t_0
     (if (<= M -4e+124)
       (exp (* n (- M (* m 0.5))))
       (if (<= M -1.65e+64)
         t_0
         (if (<= M -215000.0)
           (* (cos M) t_1)
           (if (<= M -1.4e-177)
             (pow (exp n) (+ M (* m -0.5)))
             (if (<= M 1.45e-103) t_1 t_0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp((M * (m - M)));
	double t_1 = exp(-l);
	double tmp;
	if (M <= -2.7e+161) {
		tmp = t_0;
	} else if (M <= -4e+124) {
		tmp = exp((n * (M - (m * 0.5))));
	} else if (M <= -1.65e+64) {
		tmp = t_0;
	} else if (M <= -215000.0) {
		tmp = cos(M) * t_1;
	} else if (M <= -1.4e-177) {
		tmp = pow(exp(n), (M + (m * -0.5)));
	} else if (M <= 1.45e-103) {
		tmp = t_1;
	} 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 = cos(m_1) * exp((m_1 * (m - m_1)))
    t_1 = exp(-l)
    if (m_1 <= (-2.7d+161)) then
        tmp = t_0
    else if (m_1 <= (-4d+124)) then
        tmp = exp((n * (m_1 - (m * 0.5d0))))
    else if (m_1 <= (-1.65d+64)) then
        tmp = t_0
    else if (m_1 <= (-215000.0d0)) then
        tmp = cos(m_1) * t_1
    else if (m_1 <= (-1.4d-177)) then
        tmp = exp(n) ** (m_1 + (m * (-0.5d0)))
    else if (m_1 <= 1.45d-103) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp((M * (m - M)));
	double t_1 = Math.exp(-l);
	double tmp;
	if (M <= -2.7e+161) {
		tmp = t_0;
	} else if (M <= -4e+124) {
		tmp = Math.exp((n * (M - (m * 0.5))));
	} else if (M <= -1.65e+64) {
		tmp = t_0;
	} else if (M <= -215000.0) {
		tmp = Math.cos(M) * t_1;
	} else if (M <= -1.4e-177) {
		tmp = Math.pow(Math.exp(n), (M + (m * -0.5)));
	} else if (M <= 1.45e-103) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp((M * (m - M)))
	t_1 = math.exp(-l)
	tmp = 0
	if M <= -2.7e+161:
		tmp = t_0
	elif M <= -4e+124:
		tmp = math.exp((n * (M - (m * 0.5))))
	elif M <= -1.65e+64:
		tmp = t_0
	elif M <= -215000.0:
		tmp = math.cos(M) * t_1
	elif M <= -1.4e-177:
		tmp = math.pow(math.exp(n), (M + (m * -0.5)))
	elif M <= 1.45e-103:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(M * Float64(m - M))))
	t_1 = exp(Float64(-l))
	tmp = 0.0
	if (M <= -2.7e+161)
		tmp = t_0;
	elseif (M <= -4e+124)
		tmp = exp(Float64(n * Float64(M - Float64(m * 0.5))));
	elseif (M <= -1.65e+64)
		tmp = t_0;
	elseif (M <= -215000.0)
		tmp = Float64(cos(M) * t_1);
	elseif (M <= -1.4e-177)
		tmp = exp(n) ^ Float64(M + Float64(m * -0.5));
	elseif (M <= 1.45e-103)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp((M * (m - M)));
	t_1 = exp(-l);
	tmp = 0.0;
	if (M <= -2.7e+161)
		tmp = t_0;
	elseif (M <= -4e+124)
		tmp = exp((n * (M - (m * 0.5))));
	elseif (M <= -1.65e+64)
		tmp = t_0;
	elseif (M <= -215000.0)
		tmp = cos(M) * t_1;
	elseif (M <= -1.4e-177)
		tmp = exp(n) ^ (M + (m * -0.5));
	elseif (M <= 1.45e-103)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-l)], $MachinePrecision]}, If[LessEqual[M, -2.7e+161], t$95$0, If[LessEqual[M, -4e+124], N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[M, -1.65e+64], t$95$0, If[LessEqual[M, -215000.0], N[(N[Cos[M], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[M, -1.4e-177], N[Power[N[Exp[n], $MachinePrecision], N[(M + N[(m * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[M, 1.45e-103], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if M < -2.6999999999999998e161 or -3.99999999999999979e124 < M < -1.64999999999999994e64 or 1.4499999999999999e-103 < M

   1. Initial program 76.6%

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

      1. associate-/l* 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. exp-diff 22.6%

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

      3. sub-neg 22.6%

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

      4. exp-sum 16.1%

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

      5. associate-/r* 16.1%

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

      6. exp-diff 23.4%

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

      7. exp-diff 76.6%

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

      8. sub-neg 76.6%

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

      9. remove-double-neg 76.6%

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

      10. fabs-sub 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in K around 0 99.3%

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

      1. cos-neg 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in n around 0 81.9%

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

      1. +-commutative 81.9%

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

      2. unpow2 81.9%

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

      3. distribute-rgt-out 87.1%

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

      4. *-commutative 87.1%

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

      5. *-commutative 87.1%

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

   9. Simplified 87.1%

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

   10. Taylor expanded in M around -inf 63.1%

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

       1. +-commutative 63.1%

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

       2. mul-1-neg 63.1%

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

       3. unsub-neg 63.1%

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

       4. +-commutative 63.1%

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

   12. Simplified 63.1%

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

   13. Taylor expanded in n around 0 72.6%

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

       1. unpow2 72.6%

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

       2. distribute-lft-out-- 78.5%

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

   15. Simplified 78.5%

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

   if -2.6999999999999998e161 < M < -3.99999999999999979e124

   1. Initial program 28.6%

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

      1. associate-/l* 28.6%

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

      2. exp-diff 14.3%

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

      3. sub-neg 14.3%

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

      4. exp-sum 0.0%

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

      5. associate-/r* 0.0%

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

      6. exp-diff 0.0%

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

      7. exp-diff 28.6%

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

      8. sub-neg 28.6%

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

      9. remove-double-neg 28.6%

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

      10. fabs-sub 28.6%

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

   3. Simplified 28.6%

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

   4. Taylor expanded in n around 0 28.6%

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

      1. +-commutative 85.7%

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

      2. unpow2 85.7%

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

      3. distribute-rgt-out 85.9%

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

      4. *-commutative 85.9%

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

      5. *-commutative 85.9%

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

   6. Simplified 28.6%

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

   7. Taylor expanded in M around 0 28.6%

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

      1. *-commutative 28.6%

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

      2. *-commutative 28.6%

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

      3. associate-*r* 28.6%

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

   9. Simplified 28.6%

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

   10. Taylor expanded in n around inf 28.6%

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

   11. Taylor expanded in K around 0 85.9%

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

   if -1.64999999999999994e64 < M < -215000

   1. Initial program 73.3%

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

      1. associate-/l* 73.3%

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

      2. exp-diff 26.7%

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

      3. sub-neg 26.7%

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

      4. exp-sum 6.7%

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

      5. associate-/r* 6.7%

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

      6. exp-diff 26.7%

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

      7. exp-diff 73.3%

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

      8. sub-neg 73.3%

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

      9. remove-double-neg 73.3%

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

      10. fabs-sub 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in K around 0 93.3%

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

      1. cos-neg 93.3%

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

   6. Simplified 93.3%

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

   7. Taylor expanded in n around 0 80.2%

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

      1. +-commutative 80.2%

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

      2. unpow2 80.2%

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

      3. distribute-rgt-out 80.2%

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

      4. *-commutative 80.2%

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

      5. *-commutative 80.2%

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

   9. Simplified 80.2%

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

   10. Taylor expanded in l around inf 54.4%

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

       1. neg-mul-1 54.4%

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

   12. Simplified 54.4%

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

   if -215000 < M < -1.39999999999999993e-177

   1. Initial program 78.8%

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

      1. associate-/l* 75.6%

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

      2. exp-diff 25.6%

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

      3. sub-neg 25.6%

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

      4. exp-sum 22.5%

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

      5. associate-/r* 22.5%

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

      6. exp-diff 25.6%

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

      7. exp-diff 75.6%

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

      8. sub-neg 75.6%

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

      9. remove-double-neg 75.6%

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

      10. fabs-sub 75.6%

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

   3. Simplified 75.6%

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

   4. Taylor expanded in n around 0 48.9%

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

      1. +-commutative 51.6%

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

      2. unpow2 51.6%

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

      3. distribute-rgt-out 54.7%

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

      4. *-commutative 54.7%

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

      5. *-commutative 54.7%

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

   6. Simplified 48.9%

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

   7. Taylor expanded in M around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*r* 52.1%

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

   9. Simplified 52.1%

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

   10. Taylor expanded in n around inf 39.9%

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

   11. Taylor expanded in K around 0 42.6%

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

       1. exp-prod 45.6%

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

       2. cancel-sign-sub-inv 45.6%

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

       3. metadata-eval 45.6%

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

       4. *-commutative 45.6%

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

   13. Simplified 45.6%

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

   if -1.39999999999999993e-177 < M < 1.4499999999999999e-103

   1. Initial program 81.6%

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

      1. associate-/l* 81.6%

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

      2. exp-diff 23.1%

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

      3. sub-neg 23.1%

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

      4. exp-sum 20.0%

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

      5. associate-/r* 20.0%

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

      6. exp-diff 27.7%

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

      7. exp-diff 81.6%

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

      8. sub-neg 81.6%

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

      9. remove-double-neg 81.6%

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

      10. fabs-sub 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   6. Simplified 96.2%

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

   7. Taylor expanded in n around 0 71.9%

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

      1. +-commutative 71.9%

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

      2. unpow2 71.9%

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

      3. distribute-rgt-out 75.0%

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

      4. *-commutative 75.0%

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

      5. *-commutative 75.0%

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

   9. Simplified 75.0%

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

   10. Taylor expanded in l around inf 54.4%

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

       1. neg-mul-1 54.4%

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

   12. Simplified 54.4%

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

   13. Taylor expanded in M around 0 54.4%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -2.7 \cdot 10^{+161}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{elif}\;M \leq -4 \cdot 10^{+124}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;M \leq -1.65 \cdot 10^{+64}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \mathbf{elif}\;M \leq -215000:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;M \leq -1.4 \cdot 10^{-177}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(M + m \cdot -0.5\right)}\\ \mathbf{elif}\;M \leq 1.45 \cdot 10^{-103}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \]

Alternative 6: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -3.1 \cdot 10^{-83}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 0.00075:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -3.1e-83)
   (* (cos M) (exp (* M (- n M))))
   (if (<= M 0.00075)
     (exp (- (- (fabs (- n m)) l) (* (* m 0.5) (+ n (* m 0.5)))))
     (* (cos M) (exp (* M (- m M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -3.1e-83) {
		tmp = cos(M) * exp((M * (n - M)));
	} else if (M <= 0.00075) {
		tmp = exp(((fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	} else {
		tmp = cos(M) * exp((M * (m - M)));
	}
	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 <= (-3.1d-83)) then
        tmp = cos(m_1) * exp((m_1 * (n - m_1)))
    else if (m_1 <= 0.00075d0) then
        tmp = exp(((abs((n - m)) - l) - ((m * 0.5d0) * (n + (m * 0.5d0)))))
    else
        tmp = cos(m_1) * exp((m_1 * (m - m_1)))
    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 <= -3.1e-83) {
		tmp = Math.cos(M) * Math.exp((M * (n - M)));
	} else if (M <= 0.00075) {
		tmp = Math.exp(((Math.abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	} else {
		tmp = Math.cos(M) * Math.exp((M * (m - M)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -3.1e-83:
		tmp = math.cos(M) * math.exp((M * (n - M)))
	elif M <= 0.00075:
		tmp = math.exp(((math.fabs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))))
	else:
		tmp = math.cos(M) * math.exp((M * (m - M)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -3.1e-83)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(n - M))));
	elseif (M <= 0.00075)
		tmp = exp(Float64(Float64(abs(Float64(n - m)) - l) - Float64(Float64(m * 0.5) * Float64(n + Float64(m * 0.5)))));
	else
		tmp = Float64(cos(M) * exp(Float64(M * Float64(m - M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -3.1e-83)
		tmp = cos(M) * exp((M * (n - M)));
	elseif (M <= 0.00075)
		tmp = exp(((abs((n - m)) - l) - ((m * 0.5) * (n + (m * 0.5)))));
	else
		tmp = cos(M) * exp((M * (m - M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -3.1e-83], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 0.00075], N[Exp[N[(N[(N[Abs[N[(n - m), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[(N[(m * 0.5), $MachinePrecision] * N[(n + N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -3.09999999999999992e-83

   1. Initial program 65.7%

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

      1. associate-/l* 64.5%

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

      2. exp-diff 16.9%

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

      3. sub-neg 16.9%

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

      4. exp-sum 9.8%

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

      5. associate-/r* 9.8%

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

      6. exp-diff 20.5%

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

      7. exp-diff 64.5%

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

      8. sub-neg 64.5%

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

      9. remove-double-neg 64.5%

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

      10. fabs-sub 64.5%

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

   3. Simplified 64.5%

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

   4. Taylor expanded in K around 0 94.1%

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

      1. cos-neg 94.1%

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

   6. Simplified 94.1%

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

   7. Taylor expanded in n around 0 75.2%

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

      1. +-commutative 75.2%

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

      2. unpow2 75.2%

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

      3. distribute-rgt-out 77.7%

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

      4. *-commutative 77.7%

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

      5. *-commutative 77.7%

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

   9. Simplified 77.7%

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

   10. Taylor expanded in M around -inf 54.2%

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

       1. +-commutative 54.2%

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

       2. mul-1-neg 54.2%

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

       3. unsub-neg 54.2%

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

       4. +-commutative 54.2%

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

   12. Simplified 54.2%

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

   13. Taylor expanded in m around 0 68.3%

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

       1. unpow2 68.3%

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

       2. distribute-lft-out-- 70.7%

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

   15. Simplified 70.7%

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

   if -3.09999999999999992e-83 < M < 7.5000000000000002e-4

   1. Initial program 81.9%

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

      1. associate-/l* 81.9%

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

      2. exp-diff 23.4%

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

      3. sub-neg 23.4%

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

      4. exp-sum 19.2%

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

      5. associate-/r* 19.2%

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

      6. exp-diff 24.5%

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

      7. exp-diff 81.9%

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

      8. sub-neg 81.9%

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

      9. remove-double-neg 81.9%

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

      10. fabs-sub 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in n around 0 61.3%

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

      1. +-commutative 72.5%

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

      2. unpow2 72.5%

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

      3. distribute-rgt-out 75.7%

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

      4. *-commutative 75.7%

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

      5. *-commutative 75.7%

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

   6. Simplified 63.4%

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

   7. Taylor expanded in K around 0 70.2%

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

      1. cos-neg 70.2%

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

      2. associate-*r* 70.2%

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

      3. sin-neg 70.2%

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

   9. Simplified 70.2%

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

   10. Taylor expanded in M around 0 75.7%

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

       1. associate--r+ 75.7%

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

       2. fabs-sub 75.7%

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

       3. associate-*r* 75.7%

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

   12. Simplified 75.7%

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

   if 7.5000000000000002e-4 < M

   1. Initial program 82.1%

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

      1. associate-/l* 82.1%

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

      2. exp-diff 29.5%

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

      3. sub-neg 29.5%

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

      4. exp-sum 21.8%

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

      5. associate-/r* 21.8%

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

      6. exp-diff 28.2%

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

      7. exp-diff 82.1%

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

      8. sub-neg 82.1%

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

      9. remove-double-neg 82.1%

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

      10. fabs-sub 82.1%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in K around 0 98.7%

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

      1. cos-neg 98.7%

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

   6. Simplified 98.7%

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

   7. Taylor expanded in n around 0 79.7%

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

      1. +-commutative 79.7%

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

      2. unpow2 79.7%

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

      3. distribute-rgt-out 86.1%

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

      4. *-commutative 86.1%

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

      5. *-commutative 86.1%

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

   9. Simplified 86.1%

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

   10. Taylor expanded in M around -inf 61.8%

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

       1. +-commutative 61.8%

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

       2. mul-1-neg 61.8%

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

       3. unsub-neg 61.8%

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

       4. +-commutative 61.8%

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

   12. Simplified 61.8%

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

   13. Taylor expanded in n around 0 72.1%

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

       1. unpow2 72.1%

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

       2. distribute-lft-out-- 79.8%

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

   15. Simplified 79.8%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -3.1 \cdot 10^{-83}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 0.00075:\\ \;\;\;\;e^{\left(\left|n - m\right| - \ell\right) - \left(m \cdot 0.5\right) \cdot \left(n + m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \]

Alternative 7: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-\ell}\\ t_1 := e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+258}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(M + m \cdot -0.5\right)}\\ \mathbf{elif}\;\ell \leq -9.8 \cdot 10^{+174}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-81}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- l)))) (t_1 (exp (* n (- M (* m 0.5))))))
   (if (<= l -4.5e+258)
     (pow (exp n) (+ M (* m -0.5)))
     (if (<= l -9.8e+174)
       t_0
       (if (<= l -4.8e+41)
         t_1
         (if (<= l -2.6e-81)
           (* (cos M) (exp (* M m)))
           (if (<= l 5.9e-12) t_1 t_0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(-l);
	double t_1 = exp((n * (M - (m * 0.5))));
	double tmp;
	if (l <= -4.5e+258) {
		tmp = pow(exp(n), (M + (m * -0.5)));
	} else if (l <= -9.8e+174) {
		tmp = t_0;
	} else if (l <= -4.8e+41) {
		tmp = t_1;
	} else if (l <= -2.6e-81) {
		tmp = cos(M) * exp((M * m));
	} else if (l <= 5.9e-12) {
		tmp = t_1;
	} 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 = cos(m_1) * exp(-l)
    t_1 = exp((n * (m_1 - (m * 0.5d0))))
    if (l <= (-4.5d+258)) then
        tmp = exp(n) ** (m_1 + (m * (-0.5d0)))
    else if (l <= (-9.8d+174)) then
        tmp = t_0
    else if (l <= (-4.8d+41)) then
        tmp = t_1
    else if (l <= (-2.6d-81)) then
        tmp = cos(m_1) * exp((m_1 * m))
    else if (l <= 5.9d-12) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.cos(M) * Math.exp(-l);
	double t_1 = Math.exp((n * (M - (m * 0.5))));
	double tmp;
	if (l <= -4.5e+258) {
		tmp = Math.pow(Math.exp(n), (M + (m * -0.5)));
	} else if (l <= -9.8e+174) {
		tmp = t_0;
	} else if (l <= -4.8e+41) {
		tmp = t_1;
	} else if (l <= -2.6e-81) {
		tmp = Math.cos(M) * Math.exp((M * m));
	} else if (l <= 5.9e-12) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(-l)
	t_1 = math.exp((n * (M - (m * 0.5))))
	tmp = 0
	if l <= -4.5e+258:
		tmp = math.pow(math.exp(n), (M + (m * -0.5)))
	elif l <= -9.8e+174:
		tmp = t_0
	elif l <= -4.8e+41:
		tmp = t_1
	elif l <= -2.6e-81:
		tmp = math.cos(M) * math.exp((M * m))
	elif l <= 5.9e-12:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-l)))
	t_1 = exp(Float64(n * Float64(M - Float64(m * 0.5))))
	tmp = 0.0
	if (l <= -4.5e+258)
		tmp = exp(n) ^ Float64(M + Float64(m * -0.5));
	elseif (l <= -9.8e+174)
		tmp = t_0;
	elseif (l <= -4.8e+41)
		tmp = t_1;
	elseif (l <= -2.6e-81)
		tmp = Float64(cos(M) * exp(Float64(M * m)));
	elseif (l <= 5.9e-12)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(-l);
	t_1 = exp((n * (M - (m * 0.5))));
	tmp = 0.0;
	if (l <= -4.5e+258)
		tmp = exp(n) ^ (M + (m * -0.5));
	elseif (l <= -9.8e+174)
		tmp = t_0;
	elseif (l <= -4.8e+41)
		tmp = t_1;
	elseif (l <= -2.6e-81)
		tmp = cos(M) * exp((M * m));
	elseif (l <= 5.9e-12)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -4.5e+258], N[Power[N[Exp[n], $MachinePrecision], N[(M + N[(m * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -9.8e+174], t$95$0, If[LessEqual[l, -4.8e+41], t$95$1, If[LessEqual[l, -2.6e-81], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.9e-12], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -4.5000000000000004e258

   1. Initial program 75.0%

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

      1. associate-/l* 75.0%

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

      2. exp-diff 6.3%

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

      3. sub-neg 6.3%

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

      4. exp-sum 6.3%

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

      5. associate-/r* 6.3%

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

      6. exp-diff 43.8%

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

      7. exp-diff 75.0%

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

      8. sub-neg 75.0%

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

      9. remove-double-neg 75.0%

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

      10. fabs-sub 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in n around 0 62.6%

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

      1. +-commutative 62.7%

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

      2. unpow2 62.7%

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

      3. distribute-rgt-out 68.9%

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

      4. *-commutative 68.9%

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

      5. *-commutative 68.9%

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

   6. Simplified 68.8%

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

   7. Taylor expanded in M around 0 68.8%

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

      1. *-commutative 68.8%

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

      2. *-commutative 68.8%

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

      3. associate-*r* 68.8%

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

   9. Simplified 68.8%

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

   10. Taylor expanded in n around inf 32.3%

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

   11. Taylor expanded in K around 0 44.8%

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

       1. exp-prod 44.9%

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

       2. cancel-sign-sub-inv 44.9%

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

       3. metadata-eval 44.9%

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

       4. *-commutative 44.9%

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

   13. Simplified 44.9%

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

   if -4.5000000000000004e258 < l < -9.7999999999999993e174 or 5.9e-12 < l

   1. Initial program 74.4%

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

      1. associate-/l* 74.4%

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

      2. exp-diff 36.0%

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

      3. sub-neg 36.0%

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

      4. exp-sum 17.4%

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

      5. associate-/r* 17.4%

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

      6. exp-diff 24.4%

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

      7. exp-diff 74.4%

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

      8. sub-neg 74.4%

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

      9. remove-double-neg 74.4%

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

      10. fabs-sub 74.4%

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

   3. Simplified 74.4%

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

   4. Taylor expanded in K around 0 98.8%

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

      1. cos-neg 98.8%

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

   6. Simplified 98.8%

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

   7. Taylor expanded in n around 0 85.0%

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

      1. +-commutative 85.0%

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

      2. unpow2 85.0%

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

      3. distribute-rgt-out 89.7%

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

      4. *-commutative 89.7%

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

      5. *-commutative 89.7%

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

   9. Simplified 89.7%

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

   10. Taylor expanded in l around inf 89.7%

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

       1. neg-mul-1 89.7%

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

   12. Simplified 89.7%

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

   if -9.7999999999999993e174 < l < -4.8000000000000003e41 or -2.5999999999999999e-81 < l < 5.9e-12

   1. Initial program 78.3%

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

      1. associate-/l* 77.7%

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

      2. exp-diff 18.0%

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

      3. sub-neg 18.0%

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

      4. exp-sum 18.0%

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

      5. associate-/r* 18.0%

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

      6. exp-diff 20.3%

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

      7. exp-diff 77.7%

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

      8. sub-neg 77.7%

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

      9. remove-double-neg 77.7%

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

      10. fabs-sub 77.7%

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

   3. Simplified 77.7%

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

   4. Taylor expanded in n around 0 60.3%

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

      1. +-commutative 72.1%

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

      2. unpow2 72.1%

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

      3. distribute-rgt-out 76.0%

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

      4. *-commutative 76.0%

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

      5. *-commutative 76.0%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in M around 0 62.5%

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

      1. *-commutative 62.5%

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

      2. *-commutative 62.5%

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

      3. associate-*r* 62.5%

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

   9. Simplified 62.5%

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

   10. Taylor expanded in n around inf 36.9%

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

   11. Taylor expanded in K around 0 46.6%

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

   if -4.8000000000000003e41 < l < -2.5999999999999999e-81

   1. Initial program 76.8%

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

      1. associate-/l* 76.1%

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

      2. exp-diff 16.1%

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

      3. sub-neg 16.1%

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

      4. exp-sum 16.1%

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

      5. associate-/r* 16.1%

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

      6. exp-diff 32.1%

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

      7. exp-diff 76.1%

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

      8. sub-neg 76.1%

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

      9. remove-double-neg 76.1%

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

      10. fabs-sub 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in K around 0 93.1%

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

      1. cos-neg 93.1%

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

   6. Simplified 93.1%

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

   7. Taylor expanded in n around 0 69.4%

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

      1. +-commutative 69.4%

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

      2. unpow2 69.4%

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

      3. distribute-rgt-out 69.5%

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

      4. *-commutative 69.5%

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

      5. *-commutative 69.5%

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

   9. Simplified 69.5%

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

   10. Taylor expanded in M around -inf 46.0%

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

       1. +-commutative 46.0%

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

       2. mul-1-neg 46.0%

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

       3. unsub-neg 46.0%

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

       4. +-commutative 46.0%

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

   12. Simplified 46.0%

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

   13. Taylor expanded in m around inf 19.1%

       \[\leadsto \cos M \cdot e^{\color{blue}{M \cdot m}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+258}:\\ \;\;\;\;{\left(e^{n}\right)}^{\left(M + m \cdot -0.5\right)}\\ \mathbf{elif}\;\ell \leq -9.8 \cdot 10^{+174}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;\ell \leq -4.8 \cdot 10^{+41}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq -2.6 \cdot 10^{-81}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 8: 60.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;M \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 5 \cdot 10^{-100}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= M -4e-187)
   (* (cos M) (exp (* M (- n M))))
   (if (<= M 5e-100) (exp (- l)) (* (cos M) (exp (* M (- m M)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (M <= -4e-187) {
		tmp = cos(M) * exp((M * (n - M)));
	} else if (M <= 5e-100) {
		tmp = exp(-l);
	} else {
		tmp = cos(M) * exp((M * (m - M)));
	}
	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 <= (-4d-187)) then
        tmp = cos(m_1) * exp((m_1 * (n - m_1)))
    else if (m_1 <= 5d-100) then
        tmp = exp(-l)
    else
        tmp = cos(m_1) * exp((m_1 * (m - m_1)))
    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 <= -4e-187) {
		tmp = Math.cos(M) * Math.exp((M * (n - M)));
	} else if (M <= 5e-100) {
		tmp = Math.exp(-l);
	} else {
		tmp = Math.cos(M) * Math.exp((M * (m - M)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if M <= -4e-187:
		tmp = math.cos(M) * math.exp((M * (n - M)))
	elif M <= 5e-100:
		tmp = math.exp(-l)
	else:
		tmp = math.cos(M) * math.exp((M * (m - M)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (M <= -4e-187)
		tmp = Float64(cos(M) * exp(Float64(M * Float64(n - M))));
	elseif (M <= 5e-100)
		tmp = exp(Float64(-l));
	else
		tmp = Float64(cos(M) * exp(Float64(M * Float64(m - M))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (M <= -4e-187)
		tmp = cos(M) * exp((M * (n - M)));
	elseif (M <= 5e-100)
		tmp = exp(-l);
	else
		tmp = cos(M) * exp((M * (m - M)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[M, -4e-187], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(n - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[M, 5e-100], N[Exp[(-l)], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * N[(m - M), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if M < -4.0000000000000001e-187

   1. Initial program 70.5%

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

      1. associate-/l* 69.5%

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

      2. exp-diff 20.0%

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

      3. sub-neg 20.0%

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

      4. exp-sum 13.1%

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

      5. associate-/r* 13.1%

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

      6. exp-diff 22.0%

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

      7. exp-diff 69.5%

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

      8. sub-neg 69.5%

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

      9. remove-double-neg 69.5%

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

      10. fabs-sub 69.5%

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

   3. Simplified 69.5%

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

   4. Taylor expanded in K around 0 95.1%

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

      1. cos-neg 95.1%

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

   6. Simplified 95.1%

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

   7. Taylor expanded in n around 0 73.8%

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

      1. +-commutative 73.8%

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

      2. unpow2 73.8%

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

      3. distribute-rgt-out 76.9%

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

      4. *-commutative 76.9%

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

      5. *-commutative 76.9%

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

   9. Simplified 76.9%

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

   10. Taylor expanded in M around -inf 52.4%

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

       1. +-commutative 52.4%

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

       2. mul-1-neg 52.4%

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

       3. unsub-neg 52.4%

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

       4. +-commutative 52.4%

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

   12. Simplified 52.4%

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

   13. Taylor expanded in m around 0 59.4%

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

       1. unpow2 59.4%

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

       2. distribute-lft-out-- 61.4%

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

   15. Simplified 61.4%

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

   if -4.0000000000000001e-187 < M < 5.0000000000000001e-100

   1. Initial program 80.7%

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

      1. associate-/l* 80.7%

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

      2. exp-diff 22.6%

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

      3. sub-neg 22.6%

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

      4. exp-sum 19.4%

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

      5. associate-/r* 19.4%

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

      6. exp-diff 27.5%

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

      7. exp-diff 80.7%

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

      8. sub-neg 80.7%

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

      9. remove-double-neg 80.7%

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

      10. fabs-sub 80.7%

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

   3. Simplified 80.7%

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

   4. Taylor expanded in K around 0 96.0%

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

      1. cos-neg 96.0%

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

   6. Simplified 96.0%

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

   7. Taylor expanded in n around 0 72.1%

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

      1. +-commutative 72.1%

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

      2. unpow2 72.1%

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

      3. distribute-rgt-out 75.4%

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

      4. *-commutative 75.4%

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

      5. *-commutative 75.4%

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

   9. Simplified 75.4%

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

   10. Taylor expanded in l around inf 53.8%

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

       1. neg-mul-1 53.8%

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

   12. Simplified 53.8%

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

   13. Taylor expanded in M around 0 53.8%

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

   if 5.0000000000000001e-100 < M

   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* 80.6%

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

      2. exp-diff 26.9%

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

      3. sub-neg 26.9%

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

      4. exp-sum 19.4%

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

      5. associate-/r* 19.4%

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

      6. exp-diff 24.7%

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

      7. exp-diff 80.6%

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

      8. sub-neg 80.6%

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

      9. remove-double-neg 80.6%

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

      10. fabs-sub 80.6%

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

   3. Simplified 80.6%

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

   4. Taylor expanded in K around 0 98.9%

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

      1. cos-neg 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in n around 0 79.8%

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

      1. +-commutative 79.8%

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

      2. unpow2 79.8%

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

      3. distribute-rgt-out 85.2%

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

      4. *-commutative 85.2%

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

      5. *-commutative 85.2%

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

   9. Simplified 85.2%

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

   10. Taylor expanded in M around -inf 58.5%

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

       1. +-commutative 58.5%

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

       2. mul-1-neg 58.5%

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

       3. unsub-neg 58.5%

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

       4. +-commutative 58.5%

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

   12. Simplified 58.5%

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

   13. Taylor expanded in n around 0 66.1%

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

       1. unpow2 66.1%

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

       2. distribute-lft-out-- 72.6%

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

   15. Simplified 72.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -4 \cdot 10^{-187}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(n - M\right)}\\ \mathbf{elif}\;M \leq 5 \cdot 10^{-100}:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{M \cdot \left(m - M\right)}\\ \end{array} \]

Alternative 9: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+265}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-82}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* n (- M (* m 0.5))))))
   (if (<= l -3.1e+265)
     t_0
     (if (<= l -1.65e-82)
       (* (cos M) (exp (* M m)))
       (if (<= l 5.9e-12) t_0 (* (cos M) (exp (- l))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((n * (M - (m * 0.5))));
	double tmp;
	if (l <= -3.1e+265) {
		tmp = t_0;
	} else if (l <= -1.65e-82) {
		tmp = cos(M) * exp((M * m));
	} else if (l <= 5.9e-12) {
		tmp = t_0;
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp((n * (m_1 - (m * 0.5d0))))
    if (l <= (-3.1d+265)) then
        tmp = t_0
    else if (l <= (-1.65d-82)) then
        tmp = cos(m_1) * exp((m_1 * m))
    else if (l <= 5.9d-12) then
        tmp = t_0
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp((n * (M - (m * 0.5))));
	double tmp;
	if (l <= -3.1e+265) {
		tmp = t_0;
	} else if (l <= -1.65e-82) {
		tmp = Math.cos(M) * Math.exp((M * m));
	} else if (l <= 5.9e-12) {
		tmp = t_0;
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((n * (M - (m * 0.5))))
	tmp = 0
	if l <= -3.1e+265:
		tmp = t_0
	elif l <= -1.65e-82:
		tmp = math.cos(M) * math.exp((M * m))
	elif l <= 5.9e-12:
		tmp = t_0
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(n * Float64(M - Float64(m * 0.5))))
	tmp = 0.0
	if (l <= -3.1e+265)
		tmp = t_0;
	elseif (l <= -1.65e-82)
		tmp = Float64(cos(M) * exp(Float64(M * m)));
	elseif (l <= 5.9e-12)
		tmp = t_0;
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp((n * (M - (m * 0.5))));
	tmp = 0.0;
	if (l <= -3.1e+265)
		tmp = t_0;
	elseif (l <= -1.65e-82)
		tmp = cos(M) * exp((M * m));
	elseif (l <= 5.9e-12)
		tmp = t_0;
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[l, -3.1e+265], t$95$0, If[LessEqual[l, -1.65e-82], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 5.9e-12], t$95$0, N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if l < -3.10000000000000008e265 or -1.65000000000000011e-82 < l < 5.9e-12

   1. Initial program 79.2%

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

      1. associate-/l* 79.3%

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

      2. exp-diff 17.7%

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

      3. sub-neg 17.7%

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

      4. exp-sum 17.7%

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

      5. associate-/r* 17.7%

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

      6. exp-diff 22.7%

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

      7. exp-diff 79.3%

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

      8. sub-neg 79.3%

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

      9. remove-double-neg 79.3%

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

      10. fabs-sub 79.3%

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

   3. Simplified 79.3%

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

   4. Taylor expanded in n around 0 61.5%

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

      1. +-commutative 70.8%

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

      2. unpow2 70.8%

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

      3. distribute-rgt-out 74.2%

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

      4. *-commutative 74.2%

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

      5. *-commutative 74.2%

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

   6. Simplified 64.0%

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

   7. Taylor expanded in M around 0 63.9%

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

      1. *-commutative 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*r* 63.9%

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

   9. Simplified 63.9%

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

   10. Taylor expanded in n around inf 39.5%

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

   11. Taylor expanded in K around 0 49.0%

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

   if -3.10000000000000008e265 < l < -1.65000000000000011e-82

   1. Initial program 74.6%

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

      1. associate-/l* 72.8%

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

      2. exp-diff 12.2%

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

      3. sub-neg 12.2%

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

      4. exp-sum 12.2%

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

      5. associate-/r* 12.2%

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

      6. exp-diff 31.9%

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

      7. exp-diff 72.8%

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

      8. sub-neg 72.8%

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

      9. remove-double-neg 72.8%

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

      10. fabs-sub 72.8%

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

   3. Simplified 72.8%

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

   4. Taylor expanded in K around 0 94.4%

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

      1. cos-neg 94.4%

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

   6. Simplified 94.4%

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

   7. Taylor expanded in n around 0 74.8%

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

      1. +-commutative 74.8%

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

      2. unpow2 74.8%

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

      3. distribute-rgt-out 79.5%

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

      4. *-commutative 79.5%

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

      5. *-commutative 79.5%

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

   9. Simplified 79.5%

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

   10. Taylor expanded in M around -inf 37.9%

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

       1. +-commutative 37.9%

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

       2. mul-1-neg 37.9%

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

       3. unsub-neg 37.9%

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

       4. +-commutative 37.9%

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

   12. Simplified 37.9%

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

   13. Taylor expanded in m around inf 19.1%

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

   if 5.9e-12 < l

   1. Initial program 74.3%

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

      1. associate-/l* 74.3%

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

      2. exp-diff 42.9%

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

      3. sub-neg 42.9%

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

      4. exp-sum 20.0%

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

      5. associate-/r* 20.0%

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

      6. exp-diff 20.0%

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

      7. exp-diff 74.3%

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

      8. sub-neg 74.3%

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

      9. remove-double-neg 74.3%

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

      10. fabs-sub 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 84.4%

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

      1. +-commutative 84.4%

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

      2. unpow2 84.4%

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

      3. distribute-rgt-out 88.8%

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

      4. *-commutative 88.8%

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

      5. *-commutative 88.8%

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

   9. Simplified 88.8%

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

   10. Taylor expanded in l around inf 98.6%

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

       1. neg-mul-1 98.6%

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

   12. Simplified 98.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3.1 \cdot 10^{+265}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{elif}\;\ell \leq -1.65 \cdot 10^{-82}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{elif}\;\ell \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 10: 54.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 5.9e-12) (exp (* n (- M (* m 0.5)))) (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 5.9e-12) {
		tmp = exp((n * (M - (m * 0.5))));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 5.9d-12) then
        tmp = exp((n * (m_1 - (m * 0.5d0))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 5.9e-12) {
		tmp = Math.exp((n * (M - (m * 0.5))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 5.9e-12:
		tmp = math.exp((n * (M - (m * 0.5))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 5.9e-12)
		tmp = exp(Float64(n * Float64(M - Float64(m * 0.5))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 5.9e-12)
		tmp = exp((n * (M - (m * 0.5))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 5.9e-12], N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.9e-12

   1. Initial program 77.5%

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

      1. associate-/l* 77.0%

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

      2. exp-diff 15.7%

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

      3. sub-neg 15.7%

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

      4. exp-sum 15.7%

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

      5. associate-/r* 15.7%

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

      6. exp-diff 25.9%

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

      7. exp-diff 77.0%

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

      8. sub-neg 77.0%

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

      9. remove-double-neg 77.0%

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

      10. fabs-sub 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in n around 0 61.2%

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

      1. +-commutative 72.2%

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

      2. unpow2 72.2%

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

      3. distribute-rgt-out 76.1%

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

      4. *-commutative 76.1%

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

      5. *-commutative 76.1%

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

   6. Simplified 63.4%

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

   7. Taylor expanded in M around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

   9. Simplified 63.4%

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

   10. Taylor expanded in n around inf 33.1%

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

   11. Taylor expanded in K around 0 41.5%

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

   if 5.9e-12 < l

   1. Initial program 74.3%

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

      1. associate-/l* 74.3%

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

      2. exp-diff 42.9%

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

      3. sub-neg 42.9%

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

      4. exp-sum 20.0%

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

      5. associate-/r* 20.0%

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

      6. exp-diff 20.0%

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

      7. exp-diff 74.3%

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

      8. sub-neg 74.3%

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

      9. remove-double-neg 74.3%

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

      10. fabs-sub 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 84.4%

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

      1. +-commutative 84.4%

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

      2. unpow2 84.4%

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

      3. distribute-rgt-out 88.8%

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

      4. *-commutative 88.8%

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

      5. *-commutative 88.8%

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

   9. Simplified 88.8%

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

   10. Taylor expanded in l around inf 98.6%

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

       1. neg-mul-1 98.6%

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

   12. Simplified 98.6%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 11: 54.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 5.9e-12) (exp (* n (- M (* m 0.5)))) (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 5.9e-12) {
		tmp = exp((n * (M - (m * 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 <= 5.9d-12) then
        tmp = exp((n * (m_1 - (m * 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 <= 5.9e-12) {
		tmp = Math.exp((n * (M - (m * 0.5))));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 5.9e-12:
		tmp = math.exp((n * (M - (m * 0.5))))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 5.9e-12)
		tmp = exp(Float64(n * Float64(M - Float64(m * 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 <= 5.9e-12)
		tmp = exp((n * (M - (m * 0.5))));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 5.9e-12], N[Exp[N[(n * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 5.9e-12

   1. Initial program 77.5%

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

      1. associate-/l* 77.0%

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

      2. exp-diff 15.7%

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

      3. sub-neg 15.7%

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

      4. exp-sum 15.7%

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

      5. associate-/r* 15.7%

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

      6. exp-diff 25.9%

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

      7. exp-diff 77.0%

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

      8. sub-neg 77.0%

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

      9. remove-double-neg 77.0%

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

      10. fabs-sub 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in n around 0 61.2%

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

      1. +-commutative 72.2%

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

      2. unpow2 72.2%

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

      3. distribute-rgt-out 76.1%

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

      4. *-commutative 76.1%

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

      5. *-commutative 76.1%

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

   6. Simplified 63.4%

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

   7. Taylor expanded in M around 0 63.4%

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

      1. *-commutative 63.4%

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

      2. *-commutative 63.4%

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

      3. associate-*r* 63.4%

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

   9. Simplified 63.4%

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

   10. Taylor expanded in n around inf 33.1%

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

   11. Taylor expanded in K around 0 41.5%

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

   if 5.9e-12 < l

   1. Initial program 74.3%

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

      1. associate-/l* 74.3%

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

      2. exp-diff 42.9%

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

      3. sub-neg 42.9%

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

      4. exp-sum 20.0%

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

      5. associate-/r* 20.0%

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

      6. exp-diff 20.0%

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

      7. exp-diff 74.3%

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

      8. sub-neg 74.3%

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

      9. remove-double-neg 74.3%

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

      10. fabs-sub 74.3%

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

   3. Simplified 74.3%

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

   4. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in n around 0 84.4%

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

      1. +-commutative 84.4%

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

      2. unpow2 84.4%

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

      3. distribute-rgt-out 88.8%

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

      4. *-commutative 88.8%

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

      5. *-commutative 88.8%

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

   9. Simplified 88.8%

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

   10. Taylor expanded in l around inf 98.6%

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

       1. neg-mul-1 98.6%

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

   12. Simplified 98.6%

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

   13. Taylor expanded in M around 0 98.6%

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

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 5.9 \cdot 10^{-12}:\\ \;\;\;\;e^{n \cdot \left(M - m \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]

Alternative 12: 35.0% 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 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. Step-by-step derivation

   1. associate-/l* 76.3%

      \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. exp-diff 23.1%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell - \left|m - n\right|}}} \]

   3. sub-neg 23.1%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\color{blue}{\ell + \left(-\left|m - n\right|\right)}}} \]

   4. exp-sum 16.9%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{\color{blue}{e^{\ell} \cdot e^{-\left|m - n\right|}}} \]

   5. associate-/r* 16.9%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \color{blue}{\frac{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}{e^{-\left|m - n\right|}}} \]

   6. exp-diff 24.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \frac{\color{blue}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}{e^{-\left|m - n\right|}} \]

   7. exp-diff 76.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \color{blue}{e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

   8. sub-neg 76.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left(-\left(-\left|m - n\right|\right)\right)}} \]

   9. remove-double-neg 76.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \color{blue}{\left|m - n\right|}} \]

   10. fabs-sub 76.3%

       \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \color{blue}{\left|n - m\right|}} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|}} \]

4. Taylor expanded in K around 0 96.7%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]
5. Step-by-step derivation

   1. cos-neg 96.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

6. Simplified 96.7%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

7. Taylor expanded in n around 0 75.6%

   \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \ell\right) + \left|n - m\right|} \]
8. Step-by-step derivation

   1. +-commutative 75.6%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \ell\right) + \left|n - m\right|} \]

   2. unpow2 75.6%

      \[\leadsto \cos M \cdot e^{\left(\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \ell\right) + \left|n - m\right|} \]

   3. distribute-rgt-out 79.5%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \ell\right) + \left|n - m\right|} \]

   4. *-commutative 79.5%

      \[\leadsto \cos M \cdot e^{\left(\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \ell\right) + \left|n - m\right|} \]

   5. *-commutative 79.5%

      \[\leadsto \cos M \cdot e^{\left(\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \ell\right) + \left|n - m\right|} \]

9. Simplified 79.5%

   \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \ell\right) + \left|n - m\right|} \]

10. Taylor expanded in l around inf 38.7%

    \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
11. Step-by-step derivation

    1. neg-mul-1 38.7%

       \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

12. Simplified 38.7%

    \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

13. Taylor expanded in M around 0 37.9%

    \[\leadsto \color{blue}{e^{-\ell}} \]

14. Final simplification 37.9%

    \[\leadsto e^{-\ell} \]

Alternative 13: 6.9% 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 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. Step-by-step derivation

   1. associate-/l* 76.3%

      \[\leadsto \cos \left(\color{blue}{\frac{K}{\frac{2}{m + n}}} - M\right) \cdot e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \left(\ell - \left|m - n\right|\right)} \]

   2. exp-diff 23.1%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \color{blue}{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell - \left|m - n\right|}}} \]

   3. sub-neg 23.1%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\color{blue}{\ell + \left(-\left|m - n\right|\right)}}} \]

   4. exp-sum 16.9%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{\color{blue}{e^{\ell} \cdot e^{-\left|m - n\right|}}} \]

   5. associate-/r* 16.9%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \color{blue}{\frac{\frac{e^{-{\left(\frac{m + n}{2} - M\right)}^{2}}}{e^{\ell}}}{e^{-\left|m - n\right|}}} \]

   6. exp-diff 24.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \frac{\color{blue}{e^{\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell}}}{e^{-\left|m - n\right|}} \]

   7. exp-diff 76.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot \color{blue}{e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) - \left(-\left|m - n\right|\right)}} \]

   8. sub-neg 76.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\color{blue}{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left(-\left(-\left|m - n\right|\right)\right)}} \]

   9. remove-double-neg 76.3%

      \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \color{blue}{\left|m - n\right|}} \]

   10. fabs-sub 76.3%

       \[\leadsto \cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \color{blue}{\left|n - m\right|}} \]

3. Simplified 76.3%

   \[\leadsto \color{blue}{\cos \left(\frac{K}{\frac{2}{m + n}} - M\right) \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|}} \]

4. Taylor expanded in K around 0 96.7%

   \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]
5. Step-by-step derivation

   1. cos-neg 96.7%

      \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

6. Simplified 96.7%

   \[\leadsto \color{blue}{\cos M} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]

7. Taylor expanded in n around 0 75.6%

   \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\left(n \cdot \left(0.5 \cdot m - M\right) + {\left(0.5 \cdot m - M\right)}^{2}\right)}\right) - \ell\right) + \left|n - m\right|} \]
8. Step-by-step derivation

   1. +-commutative 75.6%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\left({\left(0.5 \cdot m - M\right)}^{2} + n \cdot \left(0.5 \cdot m - M\right)\right)}\right) - \ell\right) + \left|n - m\right|} \]

   2. unpow2 75.6%

      \[\leadsto \cos M \cdot e^{\left(\left(-\left(\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(0.5 \cdot m - M\right)} + n \cdot \left(0.5 \cdot m - M\right)\right)\right) - \ell\right) + \left|n - m\right|} \]

   3. distribute-rgt-out 79.5%

      \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\left(0.5 \cdot m - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)}\right) - \ell\right) + \left|n - m\right|} \]

   4. *-commutative 79.5%

      \[\leadsto \cos M \cdot e^{\left(\left(-\left(\color{blue}{m \cdot 0.5} - M\right) \cdot \left(\left(0.5 \cdot m - M\right) + n\right)\right) - \ell\right) + \left|n - m\right|} \]

   5. *-commutative 79.5%

      \[\leadsto \cos M \cdot e^{\left(\left(-\left(m \cdot 0.5 - M\right) \cdot \left(\left(\color{blue}{m \cdot 0.5} - M\right) + n\right)\right) - \ell\right) + \left|n - m\right|} \]

9. Simplified 79.5%

   \[\leadsto \cos M \cdot e^{\left(\left(-\color{blue}{\left(m \cdot 0.5 - M\right) \cdot \left(\left(m \cdot 0.5 - M\right) + n\right)}\right) - \ell\right) + \left|n - m\right|} \]

10. Taylor expanded in l around inf 38.7%

    \[\leadsto \cos M \cdot e^{\color{blue}{-1 \cdot \ell}} \]
11. Step-by-step derivation

    1. neg-mul-1 38.7%

       \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

12. Simplified 38.7%

    \[\leadsto \cos M \cdot e^{\color{blue}{-\ell}} \]

13. Taylor expanded in l around 0 6.7%

    \[\leadsto \color{blue}{\cos M} \]

14. Final simplification 6.7%

    \[\leadsto \cos M \]

Reproduce

herbie shell --seed 2023299 
(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)))))))