Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.2% → 96.4%

Time: 26.3s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% 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: 96.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (K m n M l)
 :precision binary64
 (* (cos M) (exp (- (- (fabs (- 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(Float64(abs(Float64(m - n)) - 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[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] - N[Power[N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in K around 0 96.7%

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

   1. cos-neg 96.7%

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

5. Simplified 96.7%

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

6. Final simplification 96.7%

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

Alternative 2: 75.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(m + n\right) \cdot 0.5\\ \mathbf{if}\;n \leq -9.2 \cdot 10^{-129}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 950000:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(t\_0 - M\right) \cdot \left(M - t\_0\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (+ m n) 0.5)))
   (if (<= n -9.2e-129)
     (* (cos M) (exp (* -0.25 (pow m 2.0))))
     (if (<= n 950000.0)
       (*
        (cos (- (/ (* (+ m n) K) 2.0) M))
        (exp (+ (* (- t_0 M) (- M t_0)) (- (fabs (- m n)) l))))
       (exp (* -0.25 (pow n 2.0)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = (m + n) * 0.5;
	double tmp;
	if (n <= -9.2e-129) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (n <= 950000.0) {
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp((((t_0 - M) * (M - t_0)) + (fabs((m - n)) - l)));
	} else {
		tmp = 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 + n) * 0.5d0
    if (n <= (-9.2d-129)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (n <= 950000.0d0) then
        tmp = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp((((t_0 - m_1) * (m_1 - t_0)) + (abs((m - n)) - l)))
    else
        tmp = 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 + n) * 0.5;
	double tmp;
	if (n <= -9.2e-129) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (n <= 950000.0) {
		tmp = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp((((t_0 - M) * (M - t_0)) + (Math.abs((m - n)) - l)));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = (m + n) * 0.5
	tmp = 0
	if n <= -9.2e-129:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif n <= 950000.0:
		tmp = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp((((t_0 - M) * (M - t_0)) + (math.fabs((m - n)) - l)))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(m + n) * 0.5)
	tmp = 0.0
	if (n <= -9.2e-129)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (n <= 950000.0)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(Float64(t_0 - M) * Float64(M - t_0)) + Float64(abs(Float64(m - n)) - l))));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = (m + n) * 0.5;
	tmp = 0.0;
	if (n <= -9.2e-129)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (n <= 950000.0)
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp((((t_0 - M) * (M - t_0)) + (abs((m - n)) - l)));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(m + n), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[n, -9.2e-129], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 950000.0], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(t$95$0 - M), $MachinePrecision] * N[(M - t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.1999999999999998e-129

   1. Initial program 70.4%

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

   3. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   5. Simplified 95.5%

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

   6. Taylor expanded in m around inf 54.1%

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

   if -9.1999999999999998e-129 < n < 9.5e5

   1. Initial program 89.3%

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

      1. unpow2 89.3%

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

      2. div-inv 89.3%

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

      3. metadata-eval 89.3%

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

      4. div-inv 89.3%

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

      5. metadata-eval 89.3%

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

   4. Applied egg-rr 89.3%

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

   if 9.5e5 < n

   1. Initial program 66.7%

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

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in n around inf 98.4%

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

   7. Taylor expanded in M around 0 98.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-129}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 950000:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(\left(m + n\right) \cdot 0.5 - M\right) \cdot \left(M - \left(m + n\right) \cdot 0.5\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-129}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 920000:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -9.2e-129)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= n 920000.0)
     (*
      (cos (- (/ (* (+ m n) K) 2.0) M))
      (exp (+ (* (+ n (- (* m 0.5) M)) (- M (* m 0.5))) (- (fabs (- m n)) l))))
     (exp (* -0.25 (pow n 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -9.2e-129) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (n <= 920000.0) {
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp((((n + ((m * 0.5) - M)) * (M - (m * 0.5))) + (fabs((m - n)) - l)));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= (-9.2d-129)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (n <= 920000.0d0) then
        tmp = cos(((((m + n) * k) / 2.0d0) - m_1)) * exp((((n + ((m * 0.5d0) - m_1)) * (m_1 - (m * 0.5d0))) + (abs((m - n)) - l)))
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -9.2e-129) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (n <= 920000.0) {
		tmp = Math.cos(((((m + n) * K) / 2.0) - M)) * Math.exp((((n + ((m * 0.5) - M)) * (M - (m * 0.5))) + (Math.abs((m - n)) - l)));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -9.2e-129:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif n <= 920000.0:
		tmp = math.cos(((((m + n) * K) / 2.0) - M)) * math.exp((((n + ((m * 0.5) - M)) * (M - (m * 0.5))) + (math.fabs((m - n)) - l)))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -9.2e-129)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (n <= 920000.0)
		tmp = Float64(cos(Float64(Float64(Float64(Float64(m + n) * K) / 2.0) - M)) * exp(Float64(Float64(Float64(n + Float64(Float64(m * 0.5) - M)) * Float64(M - Float64(m * 0.5))) + Float64(abs(Float64(m - n)) - l))));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -9.2e-129)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (n <= 920000.0)
		tmp = cos(((((m + n) * K) / 2.0) - M)) * exp((((n + ((m * 0.5) - M)) * (M - (m * 0.5))) + (abs((m - n)) - l)));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -9.2e-129], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 920000.0], N[(N[Cos[N[(N[(N[(N[(m + n), $MachinePrecision] * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[(n + N[(N[(m * 0.5), $MachinePrecision] - M), $MachinePrecision]), $MachinePrecision] * N[(M - N[(m * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -9.1999999999999998e-129

   1. Initial program 70.4%

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

   3. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   5. Simplified 95.5%

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

   6. Taylor expanded in m around inf 54.1%

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

   if -9.1999999999999998e-129 < n < 9.2e5

   1. Initial program 89.3%

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

   3. Taylor expanded in n around 0 89.3%

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

      1. +-commutative 89.3%

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

      2. unpow2 89.3%

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

      3. distribute-rgt-out 89.3%

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

      4. *-commutative 89.3%

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

      5. *-commutative 89.3%

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

   5. Simplified 89.3%

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

   if 9.2e5 < n

   1. Initial program 66.7%

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

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in n around inf 98.4%

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

   7. Taylor expanded in M around 0 98.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -9.2 \cdot 10^{-129}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 920000:\\ \;\;\;\;\cos \left(\frac{\left(m + n\right) \cdot K}{2} - M\right) \cdot e^{\left(n + \left(m \cdot 0.5 - M\right)\right) \cdot \left(M - m \cdot 0.5\right) + \left(\left|m - n\right| - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(K \cdot \frac{m}{2} - M\right) \cdot e^{\left(m - n\right) + \left({n}^{2} \cdot 0.25 - \ell\right)}\\ \mathbf{if}\;n \leq -1.02 \cdot 10^{-200}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-172}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 750000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0
         (*
          (cos (- (* K (/ m 2.0)) M))
          (exp (+ (- m n) (- (* (pow n 2.0) 0.25) l))))))
   (if (<= n -1.02e-200)
     (* (cos M) (exp (* -0.25 (pow m 2.0))))
     (if (<= n 4.4e-172)
       t_0
       (if (<= n 1.15e-7)
         (* (cos M) (exp (- (pow M 2.0))))
         (if (<= n 750000.0) t_0 (exp (* -0.25 (pow n 2.0)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(((K * (m / 2.0)) - M)) * exp(((m - n) + ((pow(n, 2.0) * 0.25) - l)));
	double tmp;
	if (n <= -1.02e-200) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (n <= 4.4e-172) {
		tmp = t_0;
	} else if (n <= 1.15e-7) {
		tmp = cos(M) * exp(-pow(M, 2.0));
	} else if (n <= 750000.0) {
		tmp = t_0;
	} else {
		tmp = 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 = cos(((k * (m / 2.0d0)) - m_1)) * exp(((m - n) + (((n ** 2.0d0) * 0.25d0) - l)))
    if (n <= (-1.02d-200)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (n <= 4.4d-172) then
        tmp = t_0
    else if (n <= 1.15d-7) then
        tmp = cos(m_1) * exp(-(m_1 ** 2.0d0))
    else if (n <= 750000.0d0) then
        tmp = t_0
    else
        tmp = 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 = Math.cos(((K * (m / 2.0)) - M)) * Math.exp(((m - n) + ((Math.pow(n, 2.0) * 0.25) - l)));
	double tmp;
	if (n <= -1.02e-200) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (n <= 4.4e-172) {
		tmp = t_0;
	} else if (n <= 1.15e-7) {
		tmp = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	} else if (n <= 750000.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(((K * (m / 2.0)) - M)) * math.exp(((m - n) + ((math.pow(n, 2.0) * 0.25) - l)))
	tmp = 0
	if n <= -1.02e-200:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif n <= 4.4e-172:
		tmp = t_0
	elif n <= 1.15e-7:
		tmp = math.cos(M) * math.exp(-math.pow(M, 2.0))
	elif n <= 750000.0:
		tmp = t_0
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(Float64(Float64(K * Float64(m / 2.0)) - M)) * exp(Float64(Float64(m - n) + Float64(Float64((n ^ 2.0) * 0.25) - l))))
	tmp = 0.0
	if (n <= -1.02e-200)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (n <= 4.4e-172)
		tmp = t_0;
	elseif (n <= 1.15e-7)
		tmp = Float64(cos(M) * exp(Float64(-(M ^ 2.0))));
	elseif (n <= 750000.0)
		tmp = t_0;
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(((K * (m / 2.0)) - M)) * exp(((m - n) + (((n ^ 2.0) * 0.25) - l)));
	tmp = 0.0;
	if (n <= -1.02e-200)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (n <= 4.4e-172)
		tmp = t_0;
	elseif (n <= 1.15e-7)
		tmp = cos(M) * exp(-(M ^ 2.0));
	elseif (n <= 750000.0)
		tmp = t_0;
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[N[(N[(K * N[(m / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(m - n), $MachinePrecision] + N[(N[(N[Power[n, 2.0], $MachinePrecision] * 0.25), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.02e-200], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 4.4e-172], t$95$0, If[LessEqual[n, 1.15e-7], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 750000.0], t$95$0, N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.02e-200

   1. Initial program 73.9%

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

   3. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in m around inf 52.8%

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

   if -1.02e-200 < n < 4.40000000000000018e-172 or 1.14999999999999997e-7 < n < 7.5e5

   1. Initial program 88.8%

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

   3. Taylor expanded in m around inf 89.1%

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

   4. Taylor expanded in n around inf 41.1%

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

      1. *-commutative 41.1%

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

   6. Simplified 41.1%

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

      1. pow1 41.1%

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

   8. Applied egg-rr 59.3%

      \[\leadsto \color{blue}{{\left(\cos \left(K \cdot \frac{m}{2} - M\right) \cdot e^{0.25 \cdot {n}^{2} - \left(\ell - \left(m - n\right)\right)}\right)}^{1}} \]
   9. Step-by-step derivation

      1. unpow1 59.3%

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

      2. associate--r- 59.3%

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

      3. *-commutative 59.3%

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

   10. Simplified 59.3%

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

   if 4.40000000000000018e-172 < n < 1.14999999999999997e-7

   1. Initial program 88.5%

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

   3. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in M around inf 62.7%

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

      1. mul-1-neg 62.7%

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

   8. Simplified 62.7%

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

   if 7.5e5 < n

   1. Initial program 66.7%

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

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in n around inf 98.4%

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

   7. Taylor expanded in M around 0 98.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.02 \cdot 10^{-200}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 4.4 \cdot 10^{-172}:\\ \;\;\;\;\cos \left(K \cdot \frac{m}{2} - M\right) \cdot e^{\left(m - n\right) + \left({n}^{2} \cdot 0.25 - \ell\right)}\\ \mathbf{elif}\;n \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 750000:\\ \;\;\;\;\cos \left(K \cdot \frac{m}{2} - M\right) \cdot e^{\left(m - n\right) + \left({n}^{2} \cdot 0.25 - \ell\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.35 \cdot 10^{-201}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 750000:\\ \;\;\;\;\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right) \cdot e^{\left|m - n\right| - \left(\ell + M \cdot \left(M - n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n -2.35e-201)
   (* (cos M) (exp (* -0.25 (pow m 2.0))))
   (if (<= n 750000.0)
     (*
      (cos (- (* 0.5 (* n K)) M))
      (exp (- (fabs (- m n)) (+ l (* M (- M n))))))
     (exp (* -0.25 (pow n 2.0))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -2.35e-201) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (n <= 750000.0) {
		tmp = cos(((0.5 * (n * K)) - M)) * exp((fabs((m - n)) - (l + (M * (M - n)))));
	} else {
		tmp = exp((-0.25 * pow(n, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= (-2.35d-201)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (n <= 750000.0d0) then
        tmp = cos(((0.5d0 * (n * k)) - m_1)) * exp((abs((m - n)) - (l + (m_1 * (m_1 - n)))))
    else
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= -2.35e-201) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (n <= 750000.0) {
		tmp = Math.cos(((0.5 * (n * K)) - M)) * Math.exp((Math.abs((m - n)) - (l + (M * (M - n)))));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= -2.35e-201:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif n <= 750000.0:
		tmp = math.cos(((0.5 * (n * K)) - M)) * math.exp((math.fabs((m - n)) - (l + (M * (M - n)))))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= -2.35e-201)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (n <= 750000.0)
		tmp = Float64(cos(Float64(Float64(0.5 * Float64(n * K)) - M)) * exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(M * Float64(M - n))))));
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= -2.35e-201)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (n <= 750000.0)
		tmp = cos(((0.5 * (n * K)) - M)) * exp((abs((m - n)) - (l + (M * (M - n)))));
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, -2.35e-201], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 750000.0], N[(N[Cos[N[(N[(0.5 * N[(n * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(M * N[(M - n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.34999999999999997e-201

   1. Initial program 73.9%

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

   3. Taylor expanded in K around 0 96.2%

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

      1. cos-neg 96.2%

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

   5. Simplified 96.2%

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

   6. Taylor expanded in m around inf 52.8%

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

   if -2.34999999999999997e-201 < n < 7.5e5

   1. Initial program 88.7%

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

   3. Taylor expanded in n around 0 88.7%

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

      1. +-commutative 88.7%

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

      2. unpow2 88.7%

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

      3. distribute-rgt-out 88.7%

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

      4. *-commutative 88.7%

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

      5. *-commutative 88.7%

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

   5. Simplified 88.7%

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

   6. Taylor expanded in m around 0 71.2%

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

      1. associate-*r* 71.2%

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

      2. neg-mul-1 71.2%

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

   8. Simplified 71.2%

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

   if 7.5e5 < n

   1. Initial program 66.7%

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

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in n around inf 98.4%

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

   7. Taylor expanded in M around 0 98.4%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.35 \cdot 10^{-201}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 750000:\\ \;\;\;\;\cos \left(0.5 \cdot \left(n \cdot K\right) - M\right) \cdot e^{\left|m - n\right| - \left(\ell + M \cdot \left(M - n\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-{M}^{2}}\\ \mathbf{if}\;n \leq -1.5 \cdot 10^{-247}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 7.8 \cdot 10^{-237}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-172}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 750000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- (pow M 2.0))))))
   (if (<= n -1.5e-247)
     (* (cos M) (exp (* -0.25 (pow m 2.0))))
     (if (<= n 7.8e-237)
       t_0
       (if (<= n 3.7e-172)
         (* (cos M) (exp (- l)))
         (if (<= n 750000.0) t_0 (exp (* -0.25 (pow n 2.0)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(-pow(M, 2.0));
	double tmp;
	if (n <= -1.5e-247) {
		tmp = cos(M) * exp((-0.25 * pow(m, 2.0)));
	} else if (n <= 7.8e-237) {
		tmp = t_0;
	} else if (n <= 3.7e-172) {
		tmp = cos(M) * exp(-l);
	} else if (n <= 750000.0) {
		tmp = t_0;
	} else {
		tmp = 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 = cos(m_1) * exp(-(m_1 ** 2.0d0))
    if (n <= (-1.5d-247)) then
        tmp = cos(m_1) * exp(((-0.25d0) * (m ** 2.0d0)))
    else if (n <= 7.8d-237) then
        tmp = t_0
    else if (n <= 3.7d-172) then
        tmp = cos(m_1) * exp(-l)
    else if (n <= 750000.0d0) then
        tmp = t_0
    else
        tmp = 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 = Math.cos(M) * Math.exp(-Math.pow(M, 2.0));
	double tmp;
	if (n <= -1.5e-247) {
		tmp = Math.cos(M) * Math.exp((-0.25 * Math.pow(m, 2.0)));
	} else if (n <= 7.8e-237) {
		tmp = t_0;
	} else if (n <= 3.7e-172) {
		tmp = Math.cos(M) * Math.exp(-l);
	} else if (n <= 750000.0) {
		tmp = t_0;
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(-math.pow(M, 2.0))
	tmp = 0
	if n <= -1.5e-247:
		tmp = math.cos(M) * math.exp((-0.25 * math.pow(m, 2.0)))
	elif n <= 7.8e-237:
		tmp = t_0
	elif n <= 3.7e-172:
		tmp = math.cos(M) * math.exp(-l)
	elif n <= 750000.0:
		tmp = t_0
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-(M ^ 2.0))))
	tmp = 0.0
	if (n <= -1.5e-247)
		tmp = Float64(cos(M) * exp(Float64(-0.25 * (m ^ 2.0))));
	elseif (n <= 7.8e-237)
		tmp = t_0;
	elseif (n <= 3.7e-172)
		tmp = Float64(cos(M) * exp(Float64(-l)));
	elseif (n <= 750000.0)
		tmp = t_0;
	else
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(-(M ^ 2.0));
	tmp = 0.0;
	if (n <= -1.5e-247)
		tmp = cos(M) * exp((-0.25 * (m ^ 2.0)));
	elseif (n <= 7.8e-237)
		tmp = t_0;
	elseif (n <= 3.7e-172)
		tmp = cos(M) * exp(-l);
	elseif (n <= 750000.0)
		tmp = t_0;
	else
		tmp = exp((-0.25 * (n ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.5e-247], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(-0.25 * N[Power[m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7.8e-237], t$95$0, If[LessEqual[n, 3.7e-172], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 750000.0], t$95$0, N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < -1.4999999999999999e-247

   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. Add Preprocessing

   3. Taylor expanded in K around 0 96.6%

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

      1. cos-neg 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in m around inf 53.6%

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

   if -1.4999999999999999e-247 < n < 7.7999999999999997e-237 or 3.70000000000000001e-172 < n < 7.5e5

   1. Initial program 87.0%

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

   3. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in M around inf 61.1%

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

      1. mul-1-neg 61.1%

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

   8. Simplified 61.1%

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

   if 7.7999999999999997e-237 < n < 3.70000000000000001e-172

   1. Initial program 88.9%

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

   3. Taylor expanded in K around 0 94.4%

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

      1. cos-neg 94.4%

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

   5. Simplified 94.4%

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

   6. Taylor expanded in l around inf 67.4%

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

      1. mul-1-neg 67.4%

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

   8. Simplified 67.4%

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

   if 7.5e5 < n

   1. Initial program 66.7%

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

   3. Taylor expanded in K around 0 98.4%

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

      1. cos-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in n around inf 98.4%

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

   7. Taylor expanded in M around 0 98.4%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.5 \cdot 10^{-247}:\\ \;\;\;\;\cos M \cdot e^{-0.25 \cdot {m}^{2}}\\ \mathbf{elif}\;n \leq 7.8 \cdot 10^{-237}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;n \leq 3.7 \cdot 10^{-172}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \mathbf{elif}\;n \leq 750000:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos M \cdot e^{-{M}^{2}}\\ \mathbf{if}\;M \leq -31000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 2.3 \cdot 10^{-184}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;M \leq 0.0205:\\ \;\;\;\;\cos \left(\frac{m \cdot K}{2} - M\right) \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (* (cos M) (exp (- (pow M 2.0))))))
   (if (<= M -31000000.0)
     t_0
     (if (<= M 2.3e-184)
       (exp (* -0.25 (pow n 2.0)))
       (if (<= M 0.0205)
         (* (cos (- (/ (* m K) 2.0) M)) (exp (* -0.25 (* m m))))
         t_0)))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = cos(M) * exp(-pow(M, 2.0));
	double tmp;
	if (M <= -31000000.0) {
		tmp = t_0;
	} else if (M <= 2.3e-184) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else if (M <= 0.0205) {
		tmp = cos((((m * K) / 2.0) - M)) * exp((-0.25 * (m * m)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(m_1) * exp(-(m_1 ** 2.0d0))
    if (m_1 <= (-31000000.0d0)) then
        tmp = t_0
    else if (m_1 <= 2.3d-184) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else if (m_1 <= 0.0205d0) then
        tmp = cos((((m * k) / 2.0d0) - m_1)) * exp(((-0.25d0) * (m * m)))
    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(-Math.pow(M, 2.0));
	double tmp;
	if (M <= -31000000.0) {
		tmp = t_0;
	} else if (M <= 2.3e-184) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else if (M <= 0.0205) {
		tmp = Math.cos((((m * K) / 2.0) - M)) * Math.exp((-0.25 * (m * m)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.cos(M) * math.exp(-math.pow(M, 2.0))
	tmp = 0
	if M <= -31000000.0:
		tmp = t_0
	elif M <= 2.3e-184:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	elif M <= 0.0205:
		tmp = math.cos((((m * K) / 2.0) - M)) * math.exp((-0.25 * (m * m)))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(cos(M) * exp(Float64(-(M ^ 2.0))))
	tmp = 0.0
	if (M <= -31000000.0)
		tmp = t_0;
	elseif (M <= 2.3e-184)
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	elseif (M <= 0.0205)
		tmp = Float64(cos(Float64(Float64(Float64(m * K) / 2.0) - M)) * exp(Float64(-0.25 * Float64(m * m))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = cos(M) * exp(-(M ^ 2.0));
	tmp = 0.0;
	if (M <= -31000000.0)
		tmp = t_0;
	elseif (M <= 2.3e-184)
		tmp = exp((-0.25 * (n ^ 2.0)));
	elseif (M <= 0.0205)
		tmp = cos((((m * K) / 2.0) - M)) * exp((-0.25 * (m * m)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[Cos[M], $MachinePrecision] * N[Exp[(-N[Power[M, 2.0], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -31000000.0], t$95$0, If[LessEqual[M, 2.3e-184], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[M, 0.0205], N[(N[Cos[N[(N[(N[(m * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if M < -3.1e7 or 0.0205000000000000009 < M

   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. Add Preprocessing

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around inf 97.1%

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

      1. mul-1-neg 97.1%

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

   8. Simplified 97.1%

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

   if -3.1e7 < M < 2.2999999999999999e-184

   1. Initial program 68.8%

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

   3. Taylor expanded in K around 0 92.4%

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

      1. cos-neg 92.4%

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

   5. Simplified 92.4%

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

   6. Taylor expanded in n around inf 63.9%

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

   7. Taylor expanded in M around 0 63.9%

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

   if 2.2999999999999999e-184 < M < 0.0205000000000000009

   1. Initial program 92.0%

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

   3. Taylor expanded in m around inf 94.3%

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

   4. Taylor expanded in m around inf 52.8%

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

      1. unpow2 52.8%

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

   6. Applied egg-rr 52.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -31000000:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \mathbf{elif}\;M \leq 2.3 \cdot 10^{-184}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{elif}\;M \leq 0.0205:\\ \;\;\;\;\cos \left(\frac{m \cdot K}{2} - M\right) \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-{M}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+23}:\\ \;\;\;\;\cos \left(K \cdot \frac{m}{2} - M\right) \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-308}:\\ \;\;\;\;\cos \left(\frac{m \cdot K}{2} - M\right) \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -2.7e+23)
   (* (cos (- (* K (/ m 2.0)) M)) (exp l))
   (if (<= l -6e-308)
     (* (cos (- (/ (* m K) 2.0) M)) (exp (* -0.25 (* m m))))
     (if (<= l 8.5e-12) (exp (* -0.25 (pow n 2.0))) (* (cos M) (exp (- l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -2.7e+23) {
		tmp = cos(((K * (m / 2.0)) - M)) * exp(l);
	} else if (l <= -6e-308) {
		tmp = cos((((m * K) / 2.0) - M)) * exp((-0.25 * (m * m)));
	} else if (l <= 8.5e-12) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-2.7d+23)) then
        tmp = cos(((k * (m / 2.0d0)) - m_1)) * exp(l)
    else if (l <= (-6d-308)) then
        tmp = cos((((m * k) / 2.0d0) - m_1)) * exp(((-0.25d0) * (m * m)))
    else if (l <= 8.5d-12) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -2.7e+23) {
		tmp = Math.cos(((K * (m / 2.0)) - M)) * Math.exp(l);
	} else if (l <= -6e-308) {
		tmp = Math.cos((((m * K) / 2.0) - M)) * Math.exp((-0.25 * (m * m)));
	} else if (l <= 8.5e-12) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -2.7e+23:
		tmp = math.cos(((K * (m / 2.0)) - M)) * math.exp(l)
	elif l <= -6e-308:
		tmp = math.cos((((m * K) / 2.0) - M)) * math.exp((-0.25 * (m * m)))
	elif l <= 8.5e-12:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -2.7e+23)
		tmp = Float64(cos(Float64(Float64(K * Float64(m / 2.0)) - M)) * exp(l));
	elseif (l <= -6e-308)
		tmp = Float64(cos(Float64(Float64(Float64(m * K) / 2.0) - M)) * exp(Float64(-0.25 * Float64(m * m))));
	elseif (l <= 8.5e-12)
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -2.7e+23)
		tmp = cos(((K * (m / 2.0)) - M)) * exp(l);
	elseif (l <= -6e-308)
		tmp = cos((((m * K) / 2.0) - M)) * exp((-0.25 * (m * m)));
	elseif (l <= 8.5e-12)
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -2.7e+23], N[(N[Cos[N[(N[(K * N[(m / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -6e-308], N[(N[Cos[N[(N[(N[(m * K), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(-0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.5e-12], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if l < -2.6999999999999999e23

   1. Initial program 73.1%

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

   3. Taylor expanded in m around inf 82.1%

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

   4. Taylor expanded in l around inf 20.4%

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

      1. mul-1-neg 19.1%

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

   6. Simplified 20.4%

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

      1. pow1 20.4%

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

      2. associate-/l* 20.4%

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

      3. add-sqr-sqrt 20.4%

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

      4. sqrt-unprod 20.4%

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

      5. sqr-neg 20.4%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 63.0%

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

   8. Applied egg-rr 63.0%

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

      1. unpow1 63.0%

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

   10. Simplified 63.0%

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

   if -2.6999999999999999e23 < l < -6.00000000000000044e-308

   1. Initial program 86.7%

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

   3. Taylor expanded in m around inf 90.1%

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

   4. Taylor expanded in m around inf 54.5%

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

      1. unpow2 54.5%

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

   6. Applied egg-rr 54.5%

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

   if -6.00000000000000044e-308 < l < 8.4999999999999997e-12

   1. Initial program 66.3%

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

   3. Taylor expanded in K around 0 94.2%

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

      1. cos-neg 94.2%

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

   5. Simplified 94.2%

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

   6. Taylor expanded in n around inf 55.9%

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

   7. Taylor expanded in M around 0 55.9%

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

   if 8.4999999999999997e-12 < l

   1. Initial program 84.7%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.7 \cdot 10^{+23}:\\ \;\;\;\;\cos \left(K \cdot \frac{m}{2} - M\right) \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq -6 \cdot 10^{-308}:\\ \;\;\;\;\cos \left(\frac{m \cdot K}{2} - M\right) \cdot e^{-0.25 \cdot \left(m \cdot m\right)}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-88} \lor \neg \left(n \leq 750000\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= n -1.65e-88) (not (<= n 750000.0)))
   (exp (* -0.25 (pow n 2.0)))
   (* (cos M) (exp (- l)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -1.65e-88) || !(n <= 750000.0)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((n <= (-1.65d-88)) .or. (.not. (n <= 750000.0d0))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((n <= -1.65e-88) || !(n <= 750000.0)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (n <= -1.65e-88) or not (n <= 750000.0):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((n <= -1.65e-88) || !(n <= 750000.0))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((n <= -1.65e-88) || ~((n <= 750000.0)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[n, -1.65e-88], N[Not[LessEqual[n, 750000.0]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.64999999999999997e-88 or 7.5e5 < n

   1. Initial program 68.7%

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

   3. Taylor expanded in K around 0 97.3%

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

      1. cos-neg 97.3%

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

   5. Simplified 97.3%

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

   6. Taylor expanded in n around inf 90.7%

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

   7. Taylor expanded in M around 0 90.7%

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

   if -1.64999999999999997e-88 < n < 7.5e5

   1. Initial program 88.3%

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

   3. Taylor expanded in K around 0 95.9%

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

      1. cos-neg 95.9%

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

   5. Simplified 95.9%

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

   6. Taylor expanded in l around inf 44.1%

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

      1. mul-1-neg 44.1%

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

   8. Simplified 44.1%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.65 \cdot 10^{-88} \lor \neg \left(n \leq 750000\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -0.012:\\ \;\;\;\;\cos \left(K \cdot \frac{m}{2} - M\right) \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= l -0.012)
   (* (cos (- (* K (/ m 2.0)) M)) (exp l))
   (if (<= l 8.5e-12) (exp (* -0.25 (pow n 2.0))) (* (cos M) (exp (- l))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -0.012) {
		tmp = cos(((K * (m / 2.0)) - M)) * exp(l);
	} else if (l <= 8.5e-12) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = cos(M) * exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (l <= (-0.012d0)) then
        tmp = cos(((k * (m / 2.0d0)) - m_1)) * exp(l)
    else if (l <= 8.5d-12) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = cos(m_1) * exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (l <= -0.012) {
		tmp = Math.cos(((K * (m / 2.0)) - M)) * Math.exp(l);
	} else if (l <= 8.5e-12) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.cos(M) * Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if l <= -0.012:
		tmp = math.cos(((K * (m / 2.0)) - M)) * math.exp(l)
	elif l <= 8.5e-12:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.cos(M) * math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (l <= -0.012)
		tmp = Float64(cos(Float64(Float64(K * Float64(m / 2.0)) - M)) * exp(l));
	elseif (l <= 8.5e-12)
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(cos(M) * exp(Float64(-l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (l <= -0.012)
		tmp = cos(((K * (m / 2.0)) - M)) * exp(l);
	elseif (l <= 8.5e-12)
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = cos(M) * exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[l, -0.012], N[(N[Cos[N[(N[(K * N[(m / 2.0), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision] * N[Exp[l], $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 8.5e-12], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if l < -0.012

   1. Initial program 75.7%

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

   3. Taylor expanded in m around inf 83.8%

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

   4. Taylor expanded in l around inf 18.6%

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

      1. mul-1-neg 17.5%

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

   6. Simplified 18.6%

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

      1. pow1 18.6%

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

      2. associate-/l* 18.6%

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

      3. add-sqr-sqrt 18.6%

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

      4. sqrt-unprod 18.6%

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

      5. sqr-neg 18.6%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 65.2%

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

   8. Applied egg-rr 65.2%

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

      1. unpow1 65.2%

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

   10. Simplified 65.2%

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

   if -0.012 < l < 8.4999999999999997e-12

   1. Initial program 75.4%

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

   3. Taylor expanded in K around 0 95.5%

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

      1. cos-neg 95.5%

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

   5. Simplified 95.5%

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

   6. Taylor expanded in n around inf 58.0%

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

   7. Taylor expanded in M around 0 58.0%

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

   if 8.4999999999999997e-12 < l

   1. Initial program 84.7%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -0.012:\\ \;\;\;\;\cos \left(K \cdot \frac{m}{2} - M\right) \cdot e^{\ell}\\ \mathbf{elif}\;\ell \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ e^{-0.25 \cdot {n}^{2}} \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 (exp (* -0.25 (pow n 2.0))))

C

double code(double K, double m, double n, double M, double l) {
	return exp((-0.25 * pow(n, 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 = exp(((-0.25d0) * (n ** 2.0d0)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.exp((-0.25 * Math.pow(n, 2.0)));
}

Python

def code(K, m, n, M, l):
	return math.exp((-0.25 * math.pow(n, 2.0)))

Julia

function code(K, m, n, M, l)
	return exp(Float64(-0.25 * (n ^ 2.0)))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = exp((-0.25 * (n ^ 2.0)));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in K around 0 96.7%

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

   1. cos-neg 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in n around inf 53.6%

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

7. Taylor expanded in M around 0 53.5%

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

8. Final simplification 53.5%

   \[\leadsto e^{-0.25 \cdot {n}^{2}} \]
9. Add Preprocessing

Alternative 12: 7.1% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.6%

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

3. Taylor expanded in K around 0 96.7%

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

   1. cos-neg 96.7%

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

5. Simplified 96.7%

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

6. Taylor expanded in n around inf 53.6%

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

7. Taylor expanded in n around 0 7.4%

   \[\leadsto \cos M \cdot \color{blue}{1} \]

8. Final simplification 7.4%

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

Reproduce

herbie shell --seed 2024053 
(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)))))))