Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.8% → 96.6%

Time: 19.1s

Alternatives: 10

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (fabs (- m n)) (+ 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((m - n)) - (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((m - n)) - (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((m - n)) - (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((m - n)) - (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(m - n)) - 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((m - n)) - (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[(m - n), $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 76.9%

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

   1. associate-/l* 76.9%

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

   2. exp-diff 26.2%

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

   3. sub-neg 26.2%

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

   4. exp-sum 18.7%

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

   5. associate-/r* 18.7%

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

   6. exp-diff 23.0%

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

   7. exp-diff 76.9%

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

   8. sub-neg 76.9%

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

   9. remove-double-neg 76.9%

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

   10. fabs-sub 76.9%

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

3. Simplified 76.9%

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

4. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

6. Simplified 97.5%

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

7. Final simplification 97.5%

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

Alternative 2: 83.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - m \cdot 0.5\\ \mathbf{if}\;n \leq 3.05 \cdot 10^{+35}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{\left|m - n\right| + \left(\left(n - t_0\right) \cdot t_0 - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* m 0.5))))
   (if (<= n 3.05e+35)
     (*
      (cos (- (/ K (/ 2.0 n)) M))
      (exp (+ (fabs (- m n)) (- (* (- n t_0) t_0) l))))
     (* (cos M) (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (m * 0.5);
	double tmp;
	if (n <= 3.05e+35) {
		tmp = cos(((K / (2.0 / n)) - M)) * exp((fabs((m - n)) + (((n - t_0) * t_0) - 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) :: t_0
    real(8) :: tmp
    t_0 = m_1 - (m * 0.5d0)
    if (n <= 3.05d+35) then
        tmp = cos(((k / (2.0d0 / n)) - m_1)) * exp((abs((m - n)) + (((n - t_0) * t_0) - 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 t_0 = M - (m * 0.5);
	double tmp;
	if (n <= 3.05e+35) {
		tmp = Math.cos(((K / (2.0 / n)) - M)) * Math.exp((Math.abs((m - n)) + (((n - t_0) * t_0) - 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):
	t_0 = M - (m * 0.5)
	tmp = 0
	if n <= 3.05e+35:
		tmp = math.cos(((K / (2.0 / n)) - M)) * math.exp((math.fabs((m - n)) + (((n - t_0) * t_0) - 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)
	t_0 = Float64(M - Float64(m * 0.5))
	tmp = 0.0
	if (n <= 3.05e+35)
		tmp = Float64(cos(Float64(Float64(K / Float64(2.0 / n)) - M)) * exp(Float64(abs(Float64(m - n)) + Float64(Float64(Float64(n - t_0) * t_0) - 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)
	t_0 = M - (m * 0.5);
	tmp = 0.0;
	if (n <= 3.05e+35)
		tmp = cos(((K / (2.0 / n)) - M)) * exp((abs((m - n)) + (((n - t_0) * t_0) - l)));
	else
		tmp = cos(M) * exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, 3.05e+35], N[(N[Cos[N[(N[(K / N[(2.0 / n), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(n - t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if n < 3.04999999999999989e35

   1. Initial program 79.4%

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

      1. associate-/l* 79.5%

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

      2. exp-diff 30.5%

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

      3. sub-neg 30.5%

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

      4. exp-sum 22.8%

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

      5. associate-/r* 22.8%

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

      6. exp-diff 28.1%

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

      7. exp-diff 79.5%

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

      8. sub-neg 79.5%

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

      9. remove-double-neg 79.5%

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

      10. fabs-sub 79.5%

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

   3. Simplified 79.5%

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

   4. Taylor expanded in n around 0 65.9%

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

      1. +-commutative 65.9%

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

      2. unpow2 65.9%

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

      3. distribute-rgt-out 68.7%

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

      4. *-commutative 68.7%

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

      5. *-commutative 68.7%

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

   6. Simplified 68.7%

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

   7. Taylor expanded in m around 0 79.4%

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

   if 3.04999999999999989e35 < n

   1. Initial program 65.2%

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

      1. associate-/l* 65.2%

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

      2. exp-diff 6.5%

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

      3. sub-neg 6.5%

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

      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 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 65.2%

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

      8. sub-neg 65.2%

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

      9. remove-double-neg 65.2%

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

      10. fabs-sub 65.2%

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

   3. Simplified 65.2%

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

   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 inf 100.0%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.05 \cdot 10^{+35}:\\ \;\;\;\;\cos \left(\frac{K}{\frac{2}{n}} - M\right) \cdot e^{\left|m - n\right| + \left(\left(n - \left(M - m \cdot 0.5\right)\right) \cdot \left(M - m \cdot 0.5\right) - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]

Alternative 3: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -5.5 \cdot 10^{+65}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell + \left(m - n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -5.5e+65)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (*
    (cos M)
    (exp (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (+ l (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -5.5e+65) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l + (m - n))));
	}
	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 <= (-5.5d+65)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - (l + (m - n))))
    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 <= -5.5e+65) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l + (m - n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -5.5e+65:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l + (m - n))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -5.5e+65)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - Float64(l + Float64(m - n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -5.5e+65)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l + (m - n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -5.5e+65], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - N[(l + N[(m - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -5.4999999999999996e65

   1. Initial program 57.1%

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

      1. associate-/l* 57.1%

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

      2. exp-diff 6.1%

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

      3. sub-neg 6.1%

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

      4. exp-sum 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 2.0%

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

      7. exp-diff 57.1%

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

      8. sub-neg 57.1%

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

      9. remove-double-neg 57.1%

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

      10. fabs-sub 57.1%

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

   3. Simplified 57.1%

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

   4. Taylor expanded in K around 0 100.0%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]
   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 m around inf 98.0%

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

   if -5.4999999999999996e65 < m

   1. Initial program 81.5%

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

      1. associate-/l* 81.6%

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

      2. exp-diff 30.9%

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

      3. sub-neg 30.9%

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

      4. exp-sum 23.2%

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

      5. associate-/r* 23.2%

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

      6. exp-diff 28.0%

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

      7. exp-diff 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 97.0%

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

      1. cos-neg 97.0%

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

   6. Simplified 97.0%

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

   7. Taylor expanded in m around 0 81.2%

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

      1. +-commutative 81.2%

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

      2. unpow2 81.2%

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

      3. distribute-rgt-out 84.6%

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

      4. *-commutative 84.6%

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

      5. *-commutative 84.6%

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

   9. Simplified 84.6%

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

       1. associate-+l- 84.6%

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

       2. neg-sub0 84.6%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - \left(n \cdot 0.5 - M\right) \cdot \left(\left(n \cdot 0.5 - M\right) + m\right)\right)} - \left(\ell - \left|n - m\right|\right)} \]

       3. associate--l- 84.6%

          \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(n \cdot 0.5 - M\right) + m\right) + \left(\ell - \left|n - m\right|\right)\right)}} \]

       4. +-commutative 84.6%

          \[\leadsto \cos M \cdot e^{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \color{blue}{\left(m + \left(n \cdot 0.5 - M\right)\right)} + \left(\ell - \left|n - m\right|\right)\right)} \]

       5. add-sqr-sqrt 35.3%

          \[\leadsto \cos M \cdot e^{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) + \left(\ell - \left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right|\right)\right)} \]

       6. fabs-sqr 35.3%

          \[\leadsto \cos M \cdot e^{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) + \left(\ell - \color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right)\right)} \]

       7. add-sqr-sqrt 89.3%

          \[\leadsto \cos M \cdot e^{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) + \left(\ell - \color{blue}{\left(n - m\right)}\right)\right)} \]

   11. Applied egg-rr 89.3%

       \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) + \left(\ell - \left(n - m\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -5.5 \cdot 10^{+65}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell + \left(m - n\right)\right)}\\ \end{array} \]

Alternative 4: 58.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-282}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(n \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell + \left(m - n\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -2.7e-282)
   (pow (exp m) (* n -0.5))
   (*
    (cos M)
    (exp (- (* (- (* n 0.5) M) (- (- M (* n 0.5)) m)) (+ l (- m n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -2.7e-282) {
		tmp = pow(exp(m), (n * -0.5));
	} else {
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l + (m - n))));
	}
	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 <= (-2.7d-282)) then
        tmp = exp(m) ** (n * (-0.5d0))
    else
        tmp = cos(m_1) * exp(((((n * 0.5d0) - m_1) * ((m_1 - (n * 0.5d0)) - m)) - (l + (m - n))))
    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 <= -2.7e-282) {
		tmp = Math.pow(Math.exp(m), (n * -0.5));
	} else {
		tmp = Math.cos(M) * Math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l + (m - n))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -2.7e-282:
		tmp = math.pow(math.exp(m), (n * -0.5))
	else:
		tmp = math.cos(M) * math.exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l + (m - n))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -2.7e-282)
		tmp = exp(m) ^ Float64(n * -0.5);
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(Float64(Float64(n * 0.5) - M) * Float64(Float64(M - Float64(n * 0.5)) - m)) - Float64(l + Float64(m - n)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -2.7e-282)
		tmp = exp(m) ^ (n * -0.5);
	else
		tmp = cos(M) * exp(((((n * 0.5) - M) * ((M - (n * 0.5)) - m)) - (l + (m - n))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -2.7e-282], N[Power[N[Exp[m], $MachinePrecision], N[(n * -0.5), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[(N[(n * 0.5), $MachinePrecision] - M), $MachinePrecision] * N[(N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision] - m), $MachinePrecision]), $MachinePrecision] - N[(l + N[(m - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.69999999999999982e-282

   1. Initial program 77.3%

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

      1. associate-/l* 77.4%

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

      2. exp-diff 25.0%

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

      3. sub-neg 25.0%

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

      4. exp-sum 15.9%

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

      5. associate-/r* 15.9%

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

      6. exp-diff 18.7%

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

      7. exp-diff 77.4%

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

      8. sub-neg 77.4%

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

      9. remove-double-neg 77.4%

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

      10. fabs-sub 77.4%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in K around 0 97.6%

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

      1. cos-neg 97.6%

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

   6. Simplified 97.6%

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

   7. Taylor expanded in m around 0 73.4%

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

      1. +-commutative 73.4%

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

      2. unpow2 73.4%

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

      3. distribute-rgt-out 78.4%

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

      4. *-commutative 78.4%

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

      5. *-commutative 78.4%

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

   9. Simplified 78.4%

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

   10. Taylor expanded in m around inf 38.9%

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

   11. Taylor expanded in M around 0 32.8%

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

       1. *-commutative 32.8%

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

       2. associate-*r* 32.8%

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

       3. exp-prod 28.1%

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

       4. *-commutative 28.1%

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

   13. Simplified 28.1%

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

   if -2.69999999999999982e-282 < n

   1. Initial program 76.3%

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

      1. associate-/l* 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 27.7%

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

      3. sub-neg 27.7%

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

      4. exp-sum 22.4%

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

      5. associate-/r* 22.4%

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

      6. exp-diff 28.6%

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

      7. exp-diff 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 97.4%

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

      1. cos-neg 97.4%

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

   6. Simplified 97.4%

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

   7. Taylor expanded in m around 0 76.4%

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

      1. +-commutative 76.4%

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

      2. unpow2 76.4%

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

      3. distribute-rgt-out 82.6%

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

      4. *-commutative 82.6%

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

      5. *-commutative 82.6%

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

   9. Simplified 82.6%

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

       1. associate-+l- 82.6%

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

       2. neg-sub0 82.6%

          \[\leadsto \cos M \cdot e^{\color{blue}{\left(0 - \left(n \cdot 0.5 - M\right) \cdot \left(\left(n \cdot 0.5 - M\right) + m\right)\right)} - \left(\ell - \left|n - m\right|\right)} \]

       3. associate--l- 82.6%

          \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(\left(n \cdot 0.5 - M\right) + m\right) + \left(\ell - \left|n - m\right|\right)\right)}} \]

       4. +-commutative 82.6%

          \[\leadsto \cos M \cdot e^{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \color{blue}{\left(m + \left(n \cdot 0.5 - M\right)\right)} + \left(\ell - \left|n - m\right|\right)\right)} \]

       5. add-sqr-sqrt 70.1%

          \[\leadsto \cos M \cdot e^{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) + \left(\ell - \left|\color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right|\right)\right)} \]

       6. fabs-sqr 70.1%

          \[\leadsto \cos M \cdot e^{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) + \left(\ell - \color{blue}{\sqrt{n - m} \cdot \sqrt{n - m}}\right)\right)} \]

       7. add-sqr-sqrt 86.1%

          \[\leadsto \cos M \cdot e^{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) + \left(\ell - \color{blue}{\left(n - m\right)}\right)\right)} \]

   11. Applied egg-rr 86.1%

       \[\leadsto \cos M \cdot e^{\color{blue}{0 - \left(\left(n \cdot 0.5 - M\right) \cdot \left(m + \left(n \cdot 0.5 - M\right)\right) + \left(\ell - \left(n - m\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{-282}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(n \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(n \cdot 0.5 - M\right) \cdot \left(\left(M - n \cdot 0.5\right) - m\right) - \left(\ell + \left(m - n\right)\right)}\\ \end{array} \]

Alternative 5: 48.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-\ell}\\ t_1 := \cos M \cdot e^{M \cdot m}\\ \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{+42}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(n \cdot -0.5\right)}\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-305}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-42}:\\ \;\;\;\;e^{-0.5 \cdot \left(m \cdot n\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- l)))) (t_1 (* (cos M) (exp (* M m)))))
   (if (<= l -2.4e+267)
     t_0
     (if (<= l -1.25e+42)
       (pow (exp m) (* n -0.5))
       (if (<= l -1.3e-305)
         t_1
         (if (<= l 1.1e-42)
           (exp (* -0.5 (* m n)))
           (if (<= l 1.5e-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 = cos(M) * exp((M * m));
	double tmp;
	if (l <= -2.4e+267) {
		tmp = t_0;
	} else if (l <= -1.25e+42) {
		tmp = pow(exp(m), (n * -0.5));
	} else if (l <= -1.3e-305) {
		tmp = t_1;
	} else if (l <= 1.1e-42) {
		tmp = exp((-0.5 * (m * n)));
	} else if (l <= 1.5e-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 = cos(m_1) * exp((m_1 * m))
    if (l <= (-2.4d+267)) then
        tmp = t_0
    else if (l <= (-1.25d+42)) then
        tmp = exp(m) ** (n * (-0.5d0))
    else if (l <= (-1.3d-305)) then
        tmp = t_1
    else if (l <= 1.1d-42) then
        tmp = exp(((-0.5d0) * (m * n)))
    else if (l <= 1.5d-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.cos(M) * Math.exp((M * m));
	double tmp;
	if (l <= -2.4e+267) {
		tmp = t_0;
	} else if (l <= -1.25e+42) {
		tmp = Math.pow(Math.exp(m), (n * -0.5));
	} else if (l <= -1.3e-305) {
		tmp = t_1;
	} else if (l <= 1.1e-42) {
		tmp = Math.exp((-0.5 * (m * n)));
	} else if (l <= 1.5e-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.cos(M) * math.exp((M * m))
	tmp = 0
	if l <= -2.4e+267:
		tmp = t_0
	elif l <= -1.25e+42:
		tmp = math.pow(math.exp(m), (n * -0.5))
	elif l <= -1.3e-305:
		tmp = t_1
	elif l <= 1.1e-42:
		tmp = math.exp((-0.5 * (m * n)))
	elif l <= 1.5e-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 = Float64(cos(M) * exp(Float64(M * m)))
	tmp = 0.0
	if (l <= -2.4e+267)
		tmp = t_0;
	elseif (l <= -1.25e+42)
		tmp = exp(m) ^ Float64(n * -0.5);
	elseif (l <= -1.3e-305)
		tmp = t_1;
	elseif (l <= 1.1e-42)
		tmp = exp(Float64(-0.5 * Float64(m * n)));
	elseif (l <= 1.5e-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 = cos(M) * exp((M * m));
	tmp = 0.0;
	if (l <= -2.4e+267)
		tmp = t_0;
	elseif (l <= -1.25e+42)
		tmp = exp(m) ^ (n * -0.5);
	elseif (l <= -1.3e-305)
		tmp = t_1;
	elseif (l <= 1.1e-42)
		tmp = exp((-0.5 * (m * n)));
	elseif (l <= 1.5e-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[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -2.4e+267], t$95$0, If[LessEqual[l, -1.25e+42], N[Power[N[Exp[m], $MachinePrecision], N[(n * -0.5), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -1.3e-305], t$95$1, If[LessEqual[l, 1.1e-42], N[Exp[N[(-0.5 * N[(m * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, 1.5e-12], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.39999999999999984e267 or 1.5000000000000001e-12 < l

   1. Initial program 80.3%

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

      1. associate-/l* 80.3%

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

      2. exp-diff 55.3%

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

      3. sub-neg 55.3%

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

      4. exp-sum 30.3%

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

      5. associate-/r* 30.3%

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

      6. exp-diff 31.6%

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

      7. exp-diff 80.3%

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

      8. sub-neg 80.3%

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

      9. remove-double-neg 80.3%

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

      10. fabs-sub 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in K around 0 99.6%

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

      1. cos-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in l around inf 91.9%

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

      1. neg-mul-1 91.9%

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

   9. Simplified 91.9%

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

   if -2.39999999999999984e267 < l < -1.25000000000000002e42

   1. Initial program 69.6%

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

      1. associate-/l* 69.6%

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

      2. exp-diff 6.5%

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

      3. sub-neg 6.5%

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

      4. exp-sum 6.5%

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

      5. associate-/r* 6.5%

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

      6. exp-diff 23.9%

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

      7. exp-diff 69.6%

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

      8. sub-neg 69.6%

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

      9. remove-double-neg 69.6%

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

      10. fabs-sub 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in m around 0 67.7%

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

      1. +-commutative 67.7%

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

      2. unpow2 67.7%

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

      3. distribute-rgt-out 74.3%

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

      4. *-commutative 74.3%

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

      5. *-commutative 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in m around inf 40.6%

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

   11. Taylor expanded in M around 0 32.2%

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

       1. *-commutative 32.2%

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

       2. associate-*r* 32.2%

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

       3. exp-prod 23.7%

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

       4. *-commutative 23.7%

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

   13. Simplified 23.7%

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

   if -1.25000000000000002e42 < l < -1.3000000000000001e-305 or 1.10000000000000003e-42 < l < 1.5000000000000001e-12

   1. Initial program 71.3%

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

      1. associate-/l* 71.6%

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

      2. exp-diff 12.3%

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

      3. sub-neg 12.3%

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

      4. exp-sum 12.3%

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

      5. associate-/r* 12.3%

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

      6. exp-diff 14.8%

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

      7. exp-diff 71.6%

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

      8. sub-neg 71.6%

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

      9. remove-double-neg 71.6%

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

      10. fabs-sub 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in K around 0 96.1%

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

      1. cos-neg 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in m around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. unpow2 69.2%

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

      3. distribute-rgt-out 73.0%

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

      4. *-commutative 73.0%

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

      5. *-commutative 73.0%

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

   9. Simplified 73.0%

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

   10. Taylor expanded in m around inf 40.1%

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

   11. Taylor expanded in M around inf 41.5%

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

   if -1.3000000000000001e-305 < l < 1.10000000000000003e-42

   1. Initial program 86.8%

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

      1. associate-/l* 86.8%

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

      2. exp-diff 22.7%

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

      3. sub-neg 22.7%

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

      4. exp-sum 22.7%

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

      5. associate-/r* 22.7%

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

      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 86.8%

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

      8. sub-neg 86.8%

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

      9. remove-double-neg 86.8%

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

      10. fabs-sub 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in K around 0 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 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. distribute-rgt-out 81.7%

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

      4. *-commutative 81.7%

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

      5. *-commutative 81.7%

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

   9. Simplified 81.7%

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

   10. Taylor expanded in m around inf 50.4%

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

   11. Taylor expanded in M around 0 41.5%

       \[\leadsto \color{blue}{e^{-0.5 \cdot \left(m \cdot n\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.4 \cdot 10^{+267}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;\ell \leq -1.25 \cdot 10^{+42}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(n \cdot -0.5\right)}\\ \mathbf{elif}\;\ell \leq -1.3 \cdot 10^{-305}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{elif}\;\ell \leq 1.1 \cdot 10^{-42}:\\ \;\;\;\;e^{-0.5 \cdot \left(m \cdot n\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 6: 48.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-\ell}\\ t_1 := \cos M \cdot e^{M \cdot m}\\ \mathbf{if}\;\ell \leq -1.52 \cdot 10^{+267}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\ell \leq -2.65 \cdot 10^{+41}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(n \cdot -0.5\right)}\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-42}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(n \cdot -0.5\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- l)))) (t_1 (* (cos M) (exp (* M m)))))
   (if (<= l -1.52e+267)
     t_0
     (if (<= l -2.65e+41)
       (pow (exp m) (* n -0.5))
       (if (<= l -5.2e-300)
         t_1
         (if (<= l 1.65e-42)
           (* (cos M) (exp (* m (* n -0.5))))
           (if (<= l 1.5e-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 = cos(M) * exp((M * m));
	double tmp;
	if (l <= -1.52e+267) {
		tmp = t_0;
	} else if (l <= -2.65e+41) {
		tmp = pow(exp(m), (n * -0.5));
	} else if (l <= -5.2e-300) {
		tmp = t_1;
	} else if (l <= 1.65e-42) {
		tmp = cos(M) * exp((m * (n * -0.5)));
	} else if (l <= 1.5e-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 = cos(m_1) * exp((m_1 * m))
    if (l <= (-1.52d+267)) then
        tmp = t_0
    else if (l <= (-2.65d+41)) then
        tmp = exp(m) ** (n * (-0.5d0))
    else if (l <= (-5.2d-300)) then
        tmp = t_1
    else if (l <= 1.65d-42) then
        tmp = cos(m_1) * exp((m * (n * (-0.5d0))))
    else if (l <= 1.5d-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.cos(M) * Math.exp((M * m));
	double tmp;
	if (l <= -1.52e+267) {
		tmp = t_0;
	} else if (l <= -2.65e+41) {
		tmp = Math.pow(Math.exp(m), (n * -0.5));
	} else if (l <= -5.2e-300) {
		tmp = t_1;
	} else if (l <= 1.65e-42) {
		tmp = Math.cos(M) * Math.exp((m * (n * -0.5)));
	} else if (l <= 1.5e-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.cos(M) * math.exp((M * m))
	tmp = 0
	if l <= -1.52e+267:
		tmp = t_0
	elif l <= -2.65e+41:
		tmp = math.pow(math.exp(m), (n * -0.5))
	elif l <= -5.2e-300:
		tmp = t_1
	elif l <= 1.65e-42:
		tmp = math.cos(M) * math.exp((m * (n * -0.5)))
	elif l <= 1.5e-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 = Float64(cos(M) * exp(Float64(M * m)))
	tmp = 0.0
	if (l <= -1.52e+267)
		tmp = t_0;
	elseif (l <= -2.65e+41)
		tmp = exp(m) ^ Float64(n * -0.5);
	elseif (l <= -5.2e-300)
		tmp = t_1;
	elseif (l <= 1.65e-42)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(n * -0.5))));
	elseif (l <= 1.5e-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 = cos(M) * exp((M * m));
	tmp = 0.0;
	if (l <= -1.52e+267)
		tmp = t_0;
	elseif (l <= -2.65e+41)
		tmp = exp(m) ^ (n * -0.5);
	elseif (l <= -5.2e-300)
		tmp = t_1;
	elseif (l <= 1.65e-42)
		tmp = cos(M) * exp((m * (n * -0.5)));
	elseif (l <= 1.5e-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[(N[Cos[M], $MachinePrecision] * N[Exp[N[(M * m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[l, -1.52e+267], t$95$0, If[LessEqual[l, -2.65e+41], N[Power[N[Exp[m], $MachinePrecision], N[(n * -0.5), $MachinePrecision]], $MachinePrecision], If[LessEqual[l, -5.2e-300], t$95$1, If[LessEqual[l, 1.65e-42], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(n * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 1.5e-12], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.5199999999999999e267 or 1.5000000000000001e-12 < l

   1. Initial program 80.3%

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

      1. associate-/l* 80.3%

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

      2. exp-diff 55.3%

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

      3. sub-neg 55.3%

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

      4. exp-sum 30.3%

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

      5. associate-/r* 30.3%

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

      6. exp-diff 31.6%

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

      7. exp-diff 80.3%

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

      8. sub-neg 80.3%

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

      9. remove-double-neg 80.3%

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

      10. fabs-sub 80.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in K around 0 99.6%

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

      1. cos-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in l around inf 91.9%

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

      1. neg-mul-1 91.9%

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

   9. Simplified 91.9%

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

   if -1.5199999999999999e267 < l < -2.6499999999999998e41

   1. Initial program 69.6%

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

      1. associate-/l* 69.6%

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

      2. exp-diff 6.5%

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

      3. sub-neg 6.5%

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

      4. exp-sum 6.5%

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

      5. associate-/r* 6.5%

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

      6. exp-diff 23.9%

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

      7. exp-diff 69.6%

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

      8. sub-neg 69.6%

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

      9. remove-double-neg 69.6%

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

      10. fabs-sub 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in K around 0 95.7%

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

      1. cos-neg 95.7%

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

   6. Simplified 95.7%

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

   7. Taylor expanded in m around 0 67.7%

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

      1. +-commutative 67.7%

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

      2. unpow2 67.7%

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

      3. distribute-rgt-out 74.3%

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

      4. *-commutative 74.3%

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

      5. *-commutative 74.3%

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

   9. Simplified 74.3%

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

   10. Taylor expanded in m around inf 40.6%

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

   11. Taylor expanded in M around 0 32.2%

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

       1. *-commutative 32.2%

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

       2. associate-*r* 32.2%

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

       3. exp-prod 23.7%

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

       4. *-commutative 23.7%

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

   13. Simplified 23.7%

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

   if -2.6499999999999998e41 < l < -5.19999999999999993e-300 or 1.6500000000000001e-42 < l < 1.5000000000000001e-12

   1. Initial program 71.3%

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

      1. associate-/l* 71.6%

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

      2. exp-diff 12.3%

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

      3. sub-neg 12.3%

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

      4. exp-sum 12.3%

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

      5. associate-/r* 12.3%

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

      6. exp-diff 14.8%

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

      7. exp-diff 71.6%

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

      8. sub-neg 71.6%

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

      9. remove-double-neg 71.6%

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

      10. fabs-sub 71.6%

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

   3. Simplified 71.6%

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

   4. Taylor expanded in K around 0 96.1%

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

      1. cos-neg 96.1%

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

   6. Simplified 96.1%

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

   7. Taylor expanded in m around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. unpow2 69.2%

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

      3. distribute-rgt-out 73.0%

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

      4. *-commutative 73.0%

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

      5. *-commutative 73.0%

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

   9. Simplified 73.0%

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

   10. Taylor expanded in m around inf 40.1%

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

   11. Taylor expanded in M around inf 41.5%

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

   if -5.19999999999999993e-300 < l < 1.6500000000000001e-42

   1. Initial program 86.8%

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

      1. associate-/l* 86.8%

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

      2. exp-diff 22.7%

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

      3. sub-neg 22.7%

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

      4. exp-sum 22.7%

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

      5. associate-/r* 22.7%

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

      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 86.8%

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

      8. sub-neg 86.8%

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

      9. remove-double-neg 86.8%

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

      10. fabs-sub 86.8%

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

   3. Simplified 86.8%

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

   4. Taylor expanded in K around 0 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 76.0%

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

      1. +-commutative 76.0%

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

      2. unpow2 76.0%

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

      3. distribute-rgt-out 81.7%

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

      4. *-commutative 81.7%

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

      5. *-commutative 81.7%

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

   9. Simplified 81.7%

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

   10. Taylor expanded in m around inf 50.4%

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

   11. Taylor expanded in M around 0 41.5%

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

       1. *-commutative 41.5%

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

       2. associate-*r* 41.5%

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

       3. *-commutative 41.5%

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

   13. Simplified 41.5%

       \[\leadsto \cos M \cdot e^{\color{blue}{m \cdot \left(-0.5 \cdot n\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.52 \cdot 10^{+267}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;\ell \leq -2.65 \cdot 10^{+41}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(n \cdot -0.5\right)}\\ \mathbf{elif}\;\ell \leq -5.2 \cdot 10^{-300}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{-42}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(n \cdot -0.5\right)}\\ \mathbf{elif}\;\ell \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\cos M \cdot e^{M \cdot m}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 7: 54.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l 1.5e-12)
   (* (cos M) (exp (* m (- M (* n 0.5)))))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.5e-12) {
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= 1.5d-12) then
        tmp = cos(m_1) * exp((m * (m_1 - (n * 0.5d0))))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= 1.5e-12) {
		tmp = Math.cos(M) * Math.exp((m * (M - (n * 0.5))));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= 1.5e-12:
		tmp = math.cos(M) * math.exp((m * (M - (n * 0.5))))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= 1.5e-12)
		tmp = Float64(cos(M) * exp(Float64(m * Float64(M - Float64(n * 0.5)))));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= 1.5e-12)
		tmp = cos(M) * exp((m * (M - (n * 0.5))));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, 1.5e-12], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(m * N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 1.5000000000000001e-12

   1. Initial program 75.2%

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

      1. associate-/l* 75.2%

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

      2. exp-diff 14.7%

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

      3. sub-neg 14.7%

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

      4. exp-sum 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 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 75.2%

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

      8. sub-neg 75.2%

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

      9. remove-double-neg 75.2%

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

      10. fabs-sub 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in K around 0 96.8%

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

      1. cos-neg 96.8%

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

   6. Simplified 96.8%

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

   7. Taylor expanded in m around 0 71.3%

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

      1. +-commutative 71.3%

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

      2. unpow2 71.3%

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

      3. distribute-rgt-out 76.1%

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

      4. *-commutative 76.1%

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

      5. *-commutative 76.1%

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

   9. Simplified 76.1%

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

   10. Taylor expanded in m around inf 42.7%

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

   if 1.5000000000000001e-12 < l

   1. Initial program 81.8%

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

      1. associate-/l* 81.8%

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

      2. exp-diff 59.1%

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

      3. sub-neg 59.1%

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

      4. exp-sum 30.3%

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

      5. associate-/r* 30.3%

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

      6. exp-diff 30.3%

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

      7. exp-diff 81.8%

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

      8. sub-neg 81.8%

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

      9. remove-double-neg 81.8%

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

      10. fabs-sub 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in K around 0 99.6%

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

      1. cos-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in l around inf 98.1%

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

      1. neg-mul-1 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\cos M \cdot e^{m \cdot \left(M - n \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]

Alternative 8: 48.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{+262} \lor \neg \left(\ell \leq 3.4 \cdot 10^{-14}\right):\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.5 \cdot \left(m \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= l -1.6e+262) (not (<= l 3.4e-14)))
   (* (cos M) (exp (- l)))
   (exp (* -0.5 (* m n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((l <= -1.6e+262) || !(l <= 3.4e-14)) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = exp((-0.5 * (m * n)));
	}
	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 <= (-1.6d+262)) .or. (.not. (l <= 3.4d-14))) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = exp(((-0.5d0) * (m * n)))
    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 <= -1.6e+262) || !(l <= 3.4e-14)) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.exp((-0.5 * (m * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (l <= -1.6e+262) or not (l <= 3.4e-14):
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.exp((-0.5 * (m * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((l <= -1.6e+262) || !(l <= 3.4e-14))
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = exp(Float64(-0.5 * Float64(m * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((l <= -1.6e+262) || ~((l <= 3.4e-14)))
		tmp = cos(M) * exp(-l);
	else
		tmp = exp((-0.5 * (m * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[l, -1.6e+262], N[Not[LessEqual[l, 3.4e-14]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.5 * N[(m * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.5999999999999999e262 or 3.40000000000000003e-14 < l

   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* 78.8%

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

      2. exp-diff 52.5%

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

      3. sub-neg 52.5%

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

      4. exp-sum 28.8%

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

      5. associate-/r* 28.8%

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

      6. exp-diff 30.0%

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

      7. exp-diff 78.8%

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

      8. sub-neg 78.8%

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

      9. remove-double-neg 78.8%

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

      10. fabs-sub 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in K around 0 99.6%

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

      1. cos-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in l around inf 88.6%

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

      1. neg-mul-1 88.6%

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

   9. Simplified 88.6%

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

   if -1.5999999999999999e262 < l < 3.40000000000000003e-14

   1. Initial program 76.0%

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

      1. associate-/l* 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 14.2%

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

      3. sub-neg 14.2%

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

      4. exp-sum 14.2%

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

      5. associate-/r* 14.2%

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

      6. exp-diff 19.9%

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

      7. exp-diff 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 96.6%

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

      1. cos-neg 96.6%

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

   6. Simplified 96.6%

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

   7. Taylor expanded in m around 0 70.2%

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

      1. +-commutative 70.2%

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

      2. unpow2 70.2%

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

      3. distribute-rgt-out 75.3%

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

      4. *-commutative 75.3%

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

      5. *-commutative 75.3%

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

   9. Simplified 75.3%

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

   10. Taylor expanded in m around inf 43.6%

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

   11. Taylor expanded in M around 0 32.7%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1.6 \cdot 10^{+262} \lor \neg \left(\ell \leq 3.4 \cdot 10^{-14}\right):\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.5 \cdot \left(m \cdot n\right)}\\ \end{array} \]

Alternative 9: 48.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+269} \lor \neg \left(\ell \leq 1.5 \cdot 10^{-12}\right):\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(n \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= l -4.4e+269) (not (<= l 1.5e-12)))
   (* (cos M) (exp (- l)))
   (pow (exp m) (* n -0.5))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((l <= -4.4e+269) || !(l <= 1.5e-12)) {
		tmp = cos(M) * exp(-l);
	} else {
		tmp = pow(exp(m), (n * -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((l <= (-4.4d+269)) .or. (.not. (l <= 1.5d-12))) then
        tmp = cos(m_1) * exp(-l)
    else
        tmp = exp(m) ** (n * (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((l <= -4.4e+269) || !(l <= 1.5e-12)) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else {
		tmp = Math.pow(Math.exp(m), (n * -0.5));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (l <= -4.4e+269) or not (l <= 1.5e-12):
		tmp = math.cos(M) * math.exp(-l)
	else:
		tmp = math.pow(math.exp(m), (n * -0.5))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((l <= -4.4e+269) || !(l <= 1.5e-12))
		tmp = Float64(cos(M) * exp(Float64(-l)));
	else
		tmp = exp(m) ^ Float64(n * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((l <= -4.4e+269) || ~((l <= 1.5e-12)))
		tmp = cos(M) * exp(-l);
	else
		tmp = exp(m) ^ (n * -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[l, -4.4e+269], N[Not[LessEqual[l, 1.5e-12]], $MachinePrecision]], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], N[Power[N[Exp[m], $MachinePrecision], N[(n * -0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.3999999999999997e269 or 1.5000000000000001e-12 < l

   1. Initial program 80.0%

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

      1. associate-/l* 80.0%

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

      2. exp-diff 56.0%

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

      3. sub-neg 56.0%

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

      4. exp-sum 30.7%

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

      5. associate-/r* 30.7%

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

      6. exp-diff 32.0%

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

      7. exp-diff 80.0%

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

      8. sub-neg 80.0%

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

      9. remove-double-neg 80.0%

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

      10. fabs-sub 80.0%

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

   3. Simplified 80.0%

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

   4. Taylor expanded in K around 0 99.6%

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

      1. cos-neg 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in l around inf 93.1%

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

      1. neg-mul-1 93.1%

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

   9. Simplified 93.1%

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

   if -4.3999999999999997e269 < l < 1.5000000000000001e-12

   1. Initial program 75.6%

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

      1. associate-/l* 75.7%

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

      2. exp-diff 13.8%

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

      3. sub-neg 13.8%

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

      4. exp-sum 13.8%

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

      5. associate-/r* 13.8%

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

      6. exp-diff 19.3%

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

      7. exp-diff 75.7%

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

      8. sub-neg 75.7%

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

      9. remove-double-neg 75.7%

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

      10. fabs-sub 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in K around 0 96.7%

      \[\leadsto \color{blue}{\cos \left(-M\right)} \cdot e^{\left(\left(-{\left(\frac{m + n}{2} - M\right)}^{2}\right) - \ell\right) + \left|n - m\right|} \]
   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 m around 0 71.0%

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

      1. +-commutative 71.0%

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

      2. unpow2 71.0%

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

      3. distribute-rgt-out 76.0%

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

      4. *-commutative 76.0%

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

      5. *-commutative 76.0%

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

   9. Simplified 76.0%

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

   10. Taylor expanded in m around inf 43.6%

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

   11. Taylor expanded in M around 0 31.9%

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

       1. *-commutative 31.9%

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

       2. associate-*r* 31.9%

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

       3. exp-prod 28.6%

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

       4. *-commutative 28.6%

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

   13. Simplified 28.6%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.4 \cdot 10^{+269} \lor \neg \left(\ell \leq 1.5 \cdot 10^{-12}\right):\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;{\left(e^{m}\right)}^{\left(n \cdot -0.5\right)}\\ \end{array} \]

Alternative 10: 31.2% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (K m n M l) :precision binary64 (exp (* -0.5 (* m n))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((-0.5 * (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 = exp(((-0.5d0) * (m * n)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(-0.5 * N[(m * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.9%

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

   1. associate-/l* 76.9%

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

   2. exp-diff 26.2%

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

   3. sub-neg 26.2%

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

   4. exp-sum 18.7%

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

   5. associate-/r* 18.7%

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

   6. exp-diff 23.0%

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

   7. exp-diff 76.9%

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

   8. sub-neg 76.9%

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

   9. remove-double-neg 76.9%

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

   10. fabs-sub 76.9%

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

3. Simplified 76.9%

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

4. Taylor expanded in K around 0 97.5%

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

   1. cos-neg 97.5%

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

6. Simplified 97.5%

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

7. Taylor expanded in m around 0 74.7%

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

   1. +-commutative 74.7%

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

   2. unpow2 74.7%

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

   3. distribute-rgt-out 80.2%

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

   4. *-commutative 80.2%

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

   5. *-commutative 80.2%

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

9. Simplified 80.2%

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

10. Taylor expanded in m around inf 39.0%

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

11. Taylor expanded in M around 0 28.9%

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

12. Final simplification 28.9%

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

Reproduce

herbie shell --seed 2023318 
(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)))))))