Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 76.1% → 96.5%

Time: 28.1s

Alternatives: 11

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 1.0× speedup

Math

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

FPCore

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

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = cos(m_1) * exp(((abs((m - n)) - l) - ((((m + n) * 0.5d0) - m_1) ** 2.0d0)))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return Math.cos(M) * Math.exp(((Math.abs((m - n)) - l) - Math.pow((((m + n) * 0.5) - 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) * 0.5) - 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) * 0.5) - M) ^ 2.0))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = cos(M) * exp(((abs((m - n)) - l) - ((((m + n) * 0.5) - 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] * 0.5), $MachinePrecision] - M), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in K around 0 99.0%

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

   1. cos-neg 99.0%

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

   2. associate--r+ 99.0%

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

   3. *-commutative 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 2: 86.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - n \cdot 0.5\\ \mathbf{if}\;m \leq -2 \cdot 10^{+17}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) + t_0 \cdot \left(m - t_0\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* n 0.5))))
   (if (<= m -2e+17)
     (exp (* (pow m 2.0) -0.25))
     (* (cos M) (exp (+ (- (fabs (- m n)) l) (* t_0 (- m t_0))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -2e+17) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = cos(M) * exp(((fabs((m - n)) - l) + (t_0 * (m - 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 = m_1 - (n * 0.5d0)
    if (m <= (-2d+17)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = cos(m_1) * exp(((abs((m - n)) - l) + (t_0 * (m - 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 = M - (n * 0.5);
	double tmp;
	if (m <= -2e+17) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.cos(M) * Math.exp(((Math.abs((m - n)) - l) + (t_0 * (m - t_0))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (n * 0.5)
	tmp = 0
	if m <= -2e+17:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.cos(M) * math.exp(((math.fabs((m - n)) - l) + (t_0 * (m - t_0))))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(n * 0.5))
	tmp = 0.0
	if (m <= -2e+17)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(abs(Float64(m - n)) - l) + Float64(t_0 * Float64(m - t_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 (m <= -2e+17)
		tmp = exp(((m ^ 2.0) * -0.25));
	else
		tmp = cos(M) * exp(((abs((m - n)) - l) + (t_0 * (m - t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -2e+17], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - l), $MachinePrecision] + N[(t$95$0 * N[(m - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -2e17

   1. Initial program 60.9%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 98.6%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.6%

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

      2. +-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in m around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -2e17 < m

   1. Initial program 79.9%

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

   3. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

      2. associate--r+ 98.6%

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

      3. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in m around 0 81.7%

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

      1. +-commutative 81.7%

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

      2. unpow2 81.7%

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

      3. distribute-rgt-out 86.5%

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

   8. Simplified 86.5%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -2 \cdot 10^{+17}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(\left|m - n\right| - \ell\right) + \left(M - n \cdot 0.5\right) \cdot \left(m - \left(M - n \cdot 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{{m}^{2} \cdot -0.25}\\ \mathbf{if}\;n \leq 3.6 \cdot 10^{-300}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-263}:\\ \;\;\;\;\frac{n}{\frac{e^{\ell}}{\sin \left(m \cdot \left(0.5 \cdot K\right)\right)}} \cdot \left(K \cdot -0.5\right)\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-184}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;-0.5 \cdot \frac{K}{\frac{e^{\ell}}{\sin M \cdot \left(-n\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (pow m 2.0) -0.25))))
   (if (<= n 3.6e-300)
     t_0
     (if (<= n 1.5e-263)
       (* (/ n (/ (exp l) (sin (* m (* 0.5 K))))) (* K -0.5))
       (if (<= n 1.25e-184)
         t_0
         (if (<= n 1.8e-11)
           (* -0.5 (/ K (/ (exp l) (* (sin M) (- n)))))
           (exp (* -0.25 (pow n 2.0)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp((pow(m, 2.0) * -0.25));
	double tmp;
	if (n <= 3.6e-300) {
		tmp = t_0;
	} else if (n <= 1.5e-263) {
		tmp = (n / (exp(l) / sin((m * (0.5 * K))))) * (K * -0.5);
	} else if (n <= 1.25e-184) {
		tmp = t_0;
	} else if (n <= 1.8e-11) {
		tmp = -0.5 * (K / (exp(l) / (sin(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) :: t_0
    real(8) :: tmp
    t_0 = exp(((m ** 2.0d0) * (-0.25d0)))
    if (n <= 3.6d-300) then
        tmp = t_0
    else if (n <= 1.5d-263) then
        tmp = (n / (exp(l) / sin((m * (0.5d0 * k))))) * (k * (-0.5d0))
    else if (n <= 1.25d-184) then
        tmp = t_0
    else if (n <= 1.8d-11) then
        tmp = (-0.5d0) * (k / (exp(l) / (sin(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 t_0 = Math.exp((Math.pow(m, 2.0) * -0.25));
	double tmp;
	if (n <= 3.6e-300) {
		tmp = t_0;
	} else if (n <= 1.5e-263) {
		tmp = (n / (Math.exp(l) / Math.sin((m * (0.5 * K))))) * (K * -0.5);
	} else if (n <= 1.25e-184) {
		tmp = t_0;
	} else if (n <= 1.8e-11) {
		tmp = -0.5 * (K / (Math.exp(l) / (Math.sin(M) * -n)));
	} else {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp((math.pow(m, 2.0) * -0.25))
	tmp = 0
	if n <= 3.6e-300:
		tmp = t_0
	elif n <= 1.5e-263:
		tmp = (n / (math.exp(l) / math.sin((m * (0.5 * K))))) * (K * -0.5)
	elif n <= 1.25e-184:
		tmp = t_0
	elif n <= 1.8e-11:
		tmp = -0.5 * (K / (math.exp(l) / (math.sin(M) * -n)))
	else:
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64((m ^ 2.0) * -0.25))
	tmp = 0.0
	if (n <= 3.6e-300)
		tmp = t_0;
	elseif (n <= 1.5e-263)
		tmp = Float64(Float64(n / Float64(exp(l) / sin(Float64(m * Float64(0.5 * K))))) * Float64(K * -0.5));
	elseif (n <= 1.25e-184)
		tmp = t_0;
	elseif (n <= 1.8e-11)
		tmp = Float64(-0.5 * Float64(K / Float64(exp(l) / Float64(sin(M) * Float64(-n)))));
	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 = exp(((m ^ 2.0) * -0.25));
	tmp = 0.0;
	if (n <= 3.6e-300)
		tmp = t_0;
	elseif (n <= 1.5e-263)
		tmp = (n / (exp(l) / sin((m * (0.5 * K))))) * (K * -0.5);
	elseif (n <= 1.25e-184)
		tmp = t_0;
	elseif (n <= 1.8e-11)
		tmp = -0.5 * (K / (exp(l) / (sin(M) * -n)));
	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[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[n, 3.6e-300], t$95$0, If[LessEqual[n, 1.5e-263], N[(N[(n / N[(N[Exp[l], $MachinePrecision] / N[Sin[N[(m * N[(0.5 * K), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(K * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.25e-184], t$95$0, If[LessEqual[n, 1.8e-11], N[(-0.5 * N[(K / N[(N[Exp[l], $MachinePrecision] / N[(N[Sin[M], $MachinePrecision] * (-n)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if n < 3.60000000000000016e-300 or 1.5e-263 < n < 1.25000000000000001e-184

   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. Add Preprocessing

   3. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

      2. associate--r+ 98.3%

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

      3. *-commutative 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in M around 0 82.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 82.0%

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

      2. +-commutative 82.0%

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

   8. Simplified 82.0%

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

   9. Taylor expanded in m around inf 53.6%

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

       1. *-commutative 53.6%

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

   11. Simplified 53.6%

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

   if 3.60000000000000016e-300 < n < 1.5e-263

   1. Initial program 85.7%

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

   3. Taylor expanded in l around inf 53.0%

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

      1. mul-1-neg 53.0%

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

   5. Simplified 53.0%

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

   6. Taylor expanded in n around 0 53.0%

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

   7. Taylor expanded in K around inf 44.5%

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

      1. associate-*r* 44.5%

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

      2. *-commutative 44.5%

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

      3. associate-*r* 44.5%

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

      4. fma-neg 44.5%

         \[\leadsto \left(\left(n \cdot e^{-\ell}\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(0.5, K \cdot m, -M\right)\right)}\right) \cdot \left(-0.5 \cdot K\right) \]

      5. *-commutative 44.5%

         \[\leadsto \left(\left(n \cdot e^{-\ell}\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \color{blue}{m \cdot K}, -M\right)\right)\right) \cdot \left(-0.5 \cdot K\right) \]

      6. *-commutative 44.5%

         \[\leadsto \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.5, m \cdot K, -M\right)\right) \cdot \left(n \cdot e^{-\ell}\right)\right)} \cdot \left(-0.5 \cdot K\right) \]

      7. *-commutative 44.5%

         \[\leadsto \left(\sin \left(\mathsf{fma}\left(0.5, \color{blue}{K \cdot m}, -M\right)\right) \cdot \left(n \cdot e^{-\ell}\right)\right) \cdot \left(-0.5 \cdot K\right) \]

      8. fma-neg 44.5%

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

      9. associate-*r* 44.5%

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

      10. *-commutative 44.5%

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

      11. exp-neg 44.5%

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

      12. associate-*r/ 44.5%

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

      13. *-rgt-identity 44.5%

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

   9. Simplified 44.5%

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

   10. Taylor expanded in M around 0 58.0%

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

       1. associate-/l* 58.0%

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

       2. associate-*r* 58.0%

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

       3. *-commutative 58.0%

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

   12. Simplified 58.0%

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

   if 1.25000000000000001e-184 < n < 1.79999999999999992e-11

   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. Add Preprocessing

   3. Taylor expanded in l around inf 41.2%

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

      1. mul-1-neg 41.2%

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

   5. Simplified 41.2%

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

   6. Taylor expanded in n around 0 41.1%

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

   7. Taylor expanded in K around inf 49.7%

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

      1. associate-*r* 49.7%

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

      2. *-commutative 49.7%

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

      3. associate-*r* 49.7%

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

      4. fma-neg 49.7%

         \[\leadsto \left(\left(n \cdot e^{-\ell}\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(0.5, K \cdot m, -M\right)\right)}\right) \cdot \left(-0.5 \cdot K\right) \]

      5. *-commutative 49.7%

         \[\leadsto \left(\left(n \cdot e^{-\ell}\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \color{blue}{m \cdot K}, -M\right)\right)\right) \cdot \left(-0.5 \cdot K\right) \]

      6. *-commutative 49.7%

         \[\leadsto \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.5, m \cdot K, -M\right)\right) \cdot \left(n \cdot e^{-\ell}\right)\right)} \cdot \left(-0.5 \cdot K\right) \]

      7. *-commutative 49.7%

         \[\leadsto \left(\sin \left(\mathsf{fma}\left(0.5, \color{blue}{K \cdot m}, -M\right)\right) \cdot \left(n \cdot e^{-\ell}\right)\right) \cdot \left(-0.5 \cdot K\right) \]

      8. fma-neg 49.7%

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

      9. associate-*r* 49.7%

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

      10. *-commutative 49.7%

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

      11. exp-neg 49.7%

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

      12. associate-*r/ 49.7%

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

      13. *-rgt-identity 49.7%

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

   9. Simplified 49.7%

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

   10. Taylor expanded in m around 0 66.8%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{K \cdot \left(n \cdot \sin \left(-M\right)\right)}{e^{\ell}}} \]
   11. Step-by-step derivation

       1. associate-/l* 66.9%

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

       2. sin-neg 66.9%

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

   12. Simplified 66.9%

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

   if 1.79999999999999992e-11 < n

   1. Initial program 65.6%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 95.4%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 95.4%

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

      2. +-commutative 95.4%

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

   8. Simplified 95.4%

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

   9. Taylor expanded in n around inf 91.7%

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

       1. *-commutative 91.7%

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

   11. Simplified 91.7%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 3.6 \cdot 10^{-300}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-263}:\\ \;\;\;\;\frac{n}{\frac{e^{\ell}}{\sin \left(m \cdot \left(0.5 \cdot K\right)\right)}} \cdot \left(K \cdot -0.5\right)\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-184}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;-0.5 \cdot \frac{K}{\frac{e^{\ell}}{\sin M \cdot \left(-n\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := M - n \cdot 0.5\\ \mathbf{if}\;m \leq -1.5 \cdot 10^{+44}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{t_0 \cdot \left(m - t_0\right) - \left(\left(n - m\right) + \ell\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- M (* n 0.5))))
   (if (<= m -1.5e+44)
     (exp (* (pow m 2.0) -0.25))
     (* (cos M) (exp (- (* t_0 (- m t_0)) (+ (- n m) l)))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -1.5e+44) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = cos(M) * exp(((t_0 * (m - t_0)) - ((n - m) + l)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = m_1 - (n * 0.5d0)
    if (m <= (-1.5d+44)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = cos(m_1) * exp(((t_0 * (m - t_0)) - ((n - m) + l)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = M - (n * 0.5);
	double tmp;
	if (m <= -1.5e+44) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.cos(M) * Math.exp(((t_0 * (m - t_0)) - ((n - m) + l)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = M - (n * 0.5)
	tmp = 0
	if m <= -1.5e+44:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.cos(M) * math.exp(((t_0 * (m - t_0)) - ((n - m) + l)))
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = Float64(M - Float64(n * 0.5))
	tmp = 0.0
	if (m <= -1.5e+44)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	else
		tmp = Float64(cos(M) * exp(Float64(Float64(t_0 * Float64(m - t_0)) - Float64(Float64(n - m) + l))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = M - (n * 0.5);
	tmp = 0.0;
	if (m <= -1.5e+44)
		tmp = exp(((m ^ 2.0) * -0.25));
	else
		tmp = cos(M) * exp(((t_0 * (m - t_0)) - ((n - m) + l)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(M - N[(n * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[m, -1.5e+44], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[(N[Cos[M], $MachinePrecision] * N[Exp[N[(N[(t$95$0 * N[(m - t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(n - m), $MachinePrecision] + l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if m < -1.49999999999999993e44

   1. Initial program 60.3%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 100.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in m around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -1.49999999999999993e44 < m

   1. Initial program 79.5%

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

   3. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

      2. associate--r+ 98.6%

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

      3. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in m around 0 80.7%

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

      1. +-commutative 80.7%

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

      2. unpow2 80.7%

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

      3. distribute-rgt-out 86.4%

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

   8. Simplified 86.4%

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

      1. cancel-sign-sub-inv 86.4%

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

      2. add-sqr-sqrt 44.9%

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

      3. fabs-sqr 44.9%

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

      4. add-sqr-sqrt 86.7%

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

      5. *-commutative 86.7%

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

      6. +-commutative 86.7%

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

      7. *-commutative 86.7%

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

   10. Applied egg-rr 86.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -1.5 \cdot 10^{+44}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{\left(M - n \cdot 0.5\right) \cdot \left(m - \left(M - n \cdot 0.5\right)\right) - \left(\left(n - m\right) + \ell\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.15 \cdot 10^{-106} \lor \neg \left(m \leq 2.1 \cdot 10^{-229}\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell} \cdot \cos \left(m \cdot \left(0.5 \cdot K\right) - M\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -3.6e-6)
   (exp (* (pow m 2.0) -0.25))
   (if (or (<= m -2.15e-106) (not (<= m 2.1e-229)))
     (exp (* -0.25 (pow n 2.0)))
     (* (exp (- l)) (cos (- (* m (* 0.5 K)) M))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3.6e-6) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else if ((m <= -2.15e-106) || !(m <= 2.1e-229)) {
		tmp = exp((-0.25 * pow(n, 2.0)));
	} else {
		tmp = exp(-l) * cos(((m * (0.5 * K)) - M));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-3.6d-6)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else if ((m <= (-2.15d-106)) .or. (.not. (m <= 2.1d-229))) then
        tmp = exp(((-0.25d0) * (n ** 2.0d0)))
    else
        tmp = exp(-l) * cos(((m * (0.5d0 * k)) - m_1))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -3.6e-6) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if ((m <= -2.15e-106) || !(m <= 2.1e-229)) {
		tmp = Math.exp((-0.25 * Math.pow(n, 2.0)));
	} else {
		tmp = Math.exp(-l) * Math.cos(((m * (0.5 * K)) - M));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -3.6e-6:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	elif (m <= -2.15e-106) or not (m <= 2.1e-229):
		tmp = math.exp((-0.25 * math.pow(n, 2.0)))
	else:
		tmp = math.exp(-l) * math.cos(((m * (0.5 * K)) - M))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -3.6e-6)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	elseif ((m <= -2.15e-106) || !(m <= 2.1e-229))
		tmp = exp(Float64(-0.25 * (n ^ 2.0)));
	else
		tmp = Float64(exp(Float64(-l)) * cos(Float64(Float64(m * Float64(0.5 * K)) - M)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -3.6e-6)
		tmp = exp(((m ^ 2.0) * -0.25));
	elseif ((m <= -2.15e-106) || ~((m <= 2.1e-229)))
		tmp = exp((-0.25 * (n ^ 2.0)));
	else
		tmp = exp(-l) * cos(((m * (0.5 * K)) - M));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -3.6e-6], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[m, -2.15e-106], N[Not[LessEqual[m, 2.1e-229]], $MachinePrecision]], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[(-l)], $MachinePrecision] * N[Cos[N[(N[(m * N[(0.5 * K), $MachinePrecision]), $MachinePrecision] - M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -3.59999999999999984e-6

   1. Initial program 63.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 97.3%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 97.3%

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

      2. +-commutative 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in m around inf 97.3%

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

       1. *-commutative 97.3%

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

   11. Simplified 97.3%

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

   if -3.59999999999999984e-6 < m < -2.1500000000000001e-106 or 2.09999999999999983e-229 < m

   1. Initial program 76.1%

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

   3. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

      2. associate--r+ 97.9%

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

      3. *-commutative 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in M around 0 85.3%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 85.3%

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

      2. +-commutative 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in n around inf 55.9%

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

       1. *-commutative 55.9%

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

   11. Simplified 55.9%

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

   if -2.1500000000000001e-106 < m < 2.09999999999999983e-229

   1. Initial program 86.4%

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

   3. Taylor expanded in l around inf 41.8%

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

      1. mul-1-neg 41.8%

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

   5. Simplified 41.8%

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

   6. Taylor expanded in n around 0 43.8%

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

      1. *-commutative 43.8%

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

      2. associate-*r* 43.8%

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

   8. Simplified 43.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -2.15 \cdot 10^{-106} \lor \neg \left(m \leq 2.1 \cdot 10^{-229}\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell} \cdot \cos \left(m \cdot \left(0.5 \cdot K\right) - M\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6500000000000:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.5 \cdot \left(n \cdot \left(m + n \cdot 0.5\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -6500000000000.0)
   (exp (* (pow m 2.0) -0.25))
   (exp (- (fabs (- m n)) (+ l (* 0.5 (* n (+ m (* n 0.5)))))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6500000000000.0) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = exp((fabs((m - n)) - (l + (0.5 * (n * (m + (n * 0.5)))))));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-6500000000000.0d0)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = exp((abs((m - n)) - (l + (0.5d0 * (n * (m + (n * 0.5d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -6500000000000.0) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.exp((Math.abs((m - n)) - (l + (0.5 * (n * (m + (n * 0.5)))))));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -6500000000000.0:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.exp((math.fabs((m - n)) - (l + (0.5 * (n * (m + (n * 0.5)))))))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -6500000000000.0)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	else
		tmp = exp(Float64(abs(Float64(m - n)) - Float64(l + Float64(0.5 * Float64(n * Float64(m + Float64(n * 0.5)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -6500000000000.0)
		tmp = exp(((m ^ 2.0) * -0.25));
	else
		tmp = exp((abs((m - n)) - (l + (0.5 * (n * (m + (n * 0.5)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -6500000000000.0], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision] - N[(l + N[(0.5 * N[(n * N[(m + N[(n * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -6.5e12

   1. Initial program 61.4%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 98.6%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.6%

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

      2. +-commutative 98.6%

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

   8. Simplified 98.6%

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

   9. Taylor expanded in m around inf 100.0%

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

       1. *-commutative 100.0%

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

   11. Simplified 100.0%

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

   if -6.5e12 < m

   1. Initial program 79.8%

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

   3. Taylor expanded in K around 0 98.6%

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

      1. cos-neg 98.6%

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

      2. associate--r+ 98.6%

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

      3. *-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in m around 0 81.6%

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

      1. +-commutative 81.6%

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

      2. unpow2 81.6%

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

      3. distribute-rgt-out 86.4%

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

   8. Simplified 86.4%

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

   9. Taylor expanded in M around 0 65.4%

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

       1. *-commutative 65.4%

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

   11. Simplified 65.4%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6500000000000:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{\left|m - n\right| - \left(\ell + 0.5 \cdot \left(n \cdot \left(m + n \cdot 0.5\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -1.4 \cdot 10^{-106} \lor \neg \left(m \leq 2.1 \cdot 10^{-229}\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -3.6e-6)
   (exp (* (pow m 2.0) -0.25))
   (if (or (<= m -1.4e-106) (not (<= m 2.1e-229)))
     (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 (m <= -3.6e-6) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else if ((m <= -1.4e-106) || !(m <= 2.1e-229)) {
		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 (m <= (-3.6d-6)) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else if ((m <= (-1.4d-106)) .or. (.not. (m <= 2.1d-229))) 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 (m <= -3.6e-6) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if ((m <= -1.4e-106) || !(m <= 2.1e-229)) {
		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 m <= -3.6e-6:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	elif (m <= -1.4e-106) or not (m <= 2.1e-229):
		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 (m <= -3.6e-6)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	elseif ((m <= -1.4e-106) || !(m <= 2.1e-229))
		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 (m <= -3.6e-6)
		tmp = exp(((m ^ 2.0) * -0.25));
	elseif ((m <= -1.4e-106) || ~((m <= 2.1e-229)))
		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[m, -3.6e-6], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[m, -1.4e-106], N[Not[LessEqual[m, 2.1e-229]], $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 3 regimes

2. if m < -3.59999999999999984e-6

   1. Initial program 63.0%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 97.3%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 97.3%

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

      2. +-commutative 97.3%

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

   8. Simplified 97.3%

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

   9. Taylor expanded in m around inf 97.3%

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

       1. *-commutative 97.3%

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

   11. Simplified 97.3%

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

   if -3.59999999999999984e-6 < m < -1.39999999999999994e-106 or 2.09999999999999983e-229 < m

   1. Initial program 76.1%

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

   3. Taylor expanded in K around 0 97.9%

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

      1. cos-neg 97.9%

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

      2. associate--r+ 97.9%

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

      3. *-commutative 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in M around 0 85.3%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 85.3%

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

      2. +-commutative 85.3%

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

   8. Simplified 85.3%

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

   9. Taylor expanded in n around inf 55.9%

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

       1. *-commutative 55.9%

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

   11. Simplified 55.9%

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

   if -1.39999999999999994e-106 < m < 2.09999999999999983e-229

   1. Initial program 86.4%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in l around inf 43.8%

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

      1. neg-mul-1 43.8%

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

   8. Simplified 43.8%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;m \leq -1.4 \cdot 10^{-106} \lor \neg \left(m \leq 2.1 \cdot 10^{-229}\right):\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\cos M \cdot e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -6500000000000 \lor \neg \left(m \leq 53\right):\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (or (<= m -6500000000000.0) (not (<= m 53.0)))
   (exp (* (pow m 2.0) -0.25))
   (exp (- l))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -6500000000000.0) || !(m <= 53.0)) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else {
		tmp = exp(-l);
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((m <= (-6500000000000.0d0)) .or. (.not. (m <= 53.0d0))) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else
        tmp = exp(-l)
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if ((m <= -6500000000000.0) || !(m <= 53.0)) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else {
		tmp = Math.exp(-l);
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if (m <= -6500000000000.0) or not (m <= 53.0):
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	else:
		tmp = math.exp(-l)
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if ((m <= -6500000000000.0) || !(m <= 53.0))
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	else
		tmp = exp(Float64(-l));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if ((m <= -6500000000000.0) || ~((m <= 53.0)))
		tmp = exp(((m ^ 2.0) * -0.25));
	else
		tmp = exp(-l);
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[Or[LessEqual[m, -6500000000000.0], N[Not[LessEqual[m, 53.0]], $MachinePrecision]], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[(-l)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -6.5e12 or 53 < m

   1. Initial program 65.6%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 98.5%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 98.5%

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

      2. +-commutative 98.5%

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

   8. Simplified 98.5%

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

   9. Taylor expanded in m around inf 99.2%

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

       1. *-commutative 99.2%

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

   11. Simplified 99.2%

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

   if -6.5e12 < m < 53

   1. Initial program 83.9%

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

   3. Taylor expanded in K around 0 98.0%

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

      1. cos-neg 98.0%

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

      2. associate--r+ 98.0%

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

      3. *-commutative 98.0%

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

   5. Simplified 98.0%

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

   6. Taylor expanded in M around 0 72.8%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 72.8%

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

      2. +-commutative 72.8%

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

   8. Simplified 72.8%

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

   9. Taylor expanded in l around inf 42.6%

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

       1. neg-mul-1 42.6%

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

   11. Simplified 42.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -6500000000000 \lor \neg \left(m \leq 53\right):\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-\ell}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 9.2 \cdot 10^{-195}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 0.0138:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 9.2e-195)
   (exp (* (pow m 2.0) -0.25))
   (if (<= n 0.0138) (exp (- 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-195) {
		tmp = exp((pow(m, 2.0) * -0.25));
	} else if (n <= 0.0138) {
		tmp = exp(-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-195) then
        tmp = exp(((m ** 2.0d0) * (-0.25d0)))
    else if (n <= 0.0138d0) then
        tmp = exp(-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-195) {
		tmp = Math.exp((Math.pow(m, 2.0) * -0.25));
	} else if (n <= 0.0138) {
		tmp = Math.exp(-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-195:
		tmp = math.exp((math.pow(m, 2.0) * -0.25))
	elif n <= 0.0138:
		tmp = math.exp(-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-195)
		tmp = exp(Float64((m ^ 2.0) * -0.25));
	elseif (n <= 0.0138)
		tmp = exp(Float64(-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-195)
		tmp = exp(((m ^ 2.0) * -0.25));
	elseif (n <= 0.0138)
		tmp = exp(-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-195], N[Exp[N[(N[Power[m, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 0.0138], N[Exp[(-l)], $MachinePrecision], N[Exp[N[(-0.25 * N[Power[n, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 9.2000000000000007e-195

   1. Initial program 77.0%

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

   3. Taylor expanded in K around 0 98.3%

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

      1. cos-neg 98.3%

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

      2. associate--r+ 98.3%

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

      3. *-commutative 98.3%

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

   5. Simplified 98.3%

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

   6. Taylor expanded in M around 0 82.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 82.0%

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

      2. +-commutative 82.0%

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

   8. Simplified 82.0%

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

   9. Taylor expanded in m around inf 53.1%

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

       1. *-commutative 53.1%

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

   11. Simplified 53.1%

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

   if 9.2000000000000007e-195 < n < 0.0138

   1. Initial program 82.5%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 83.0%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 83.0%

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

      2. +-commutative 83.0%

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

   8. Simplified 83.0%

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

   9. Taylor expanded in l around inf 50.4%

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

       1. neg-mul-1 50.4%

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

   11. Simplified 50.4%

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

   if 0.0138 < n

   1. Initial program 63.9%

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

   3. Taylor expanded in K around 0 100.0%

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

      1. cos-neg 100.0%

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

      2. associate--r+ 100.0%

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

      3. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in M around 0 96.7%

      \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
   7. Step-by-step derivation

      1. +-commutative 96.7%

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

      2. +-commutative 96.7%

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

   8. Simplified 96.7%

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

   9. Taylor expanded in n around inf 95.2%

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

       1. *-commutative 95.2%

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

   11. Simplified 95.2%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 9.2 \cdot 10^{-195}:\\ \;\;\;\;e^{{m}^{2} \cdot -0.25}\\ \mathbf{elif}\;n \leq 0.0138:\\ \;\;\;\;e^{-\ell}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot {n}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 35.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ e^{-\ell} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[K_, m_, n_, M_, l_] := N[Exp[(-l)], $MachinePrecision]

Derivation

1. Initial program 74.8%

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

3. Taylor expanded in K around 0 99.0%

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

   1. cos-neg 99.0%

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

   2. associate--r+ 99.0%

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

   3. *-commutative 99.0%

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

5. Simplified 99.0%

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

6. Taylor expanded in M around 0 85.6%

   \[\leadsto \color{blue}{e^{\left|m - n\right| - \left(\ell + 0.25 \cdot {\left(m + n\right)}^{2}\right)}} \]
7. Step-by-step derivation

   1. +-commutative 85.6%

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

   2. +-commutative 85.6%

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

8. Simplified 85.6%

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

9. Taylor expanded in l around inf 35.2%

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

    1. neg-mul-1 35.2%

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

11. Simplified 35.2%

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

12. Final simplification 35.2%

    \[\leadsto e^{-\ell} \]
13. Add Preprocessing

Alternative 11: 6.5% 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 74.8%

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

3. Taylor expanded in m around inf 38.9%

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

   1. *-commutative 38.9%

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

5. Simplified 38.9%

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

6. Taylor expanded in m around 0 7.0%

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

7. Taylor expanded in K around 0 7.5%

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

   1. cos-neg 7.5%

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

9. Simplified 7.5%

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

10. Final simplification 7.5%

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

Reproduce

herbie shell --seed 2024023 
(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)))))))