Maksimov and Kolovsky, Equation (32)

Percentage Accurate: 75.9% → 96.6%

Time: 14.8s

Alternatives: 9

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: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{m + n}{2} - M\\ \frac{\cos M}{e^{t\_0 \cdot t\_0 + \left(\ell - \left|m - n\right|\right)}} \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (- (/ (+ m n) 2.0) M)))
   (/ (cos M) (exp (+ (* t_0 t_0) (- l (fabs (- m n))))))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) / 2.0) - M;
	return cos(M) / exp(((t_0 * t_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
    real(8) :: t_0
    t_0 = ((m + n) / 2.0d0) - m_1
    code = cos(m_1) / exp(((t_0 * t_0) + (l - abs((m - n)))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = ((m + n) / 2.0) - M;
	return Math.cos(M) / Math.exp(((t_0 * t_0) + (l - Math.abs((m - n)))));
}

Python

def code(K, m, n, M, l):
	t_0 = ((m + n) / 2.0) - M
	return math.cos(M) / math.exp(((t_0 * t_0) + (l - math.fabs((m - n)))))

Julia

function code(K, m, n, M, l)
	t_0 = Float64(Float64(Float64(m + n) / 2.0) - M)
	return Float64(cos(M) / exp(Float64(Float64(t_0 * t_0) + Float64(l - abs(Float64(m - n))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	t_0 = ((m + n) / 2.0) - M;
	tmp = cos(M) / exp(((t_0 * t_0) + (l - abs((m - n)))));
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[(N[(N[(m + n), $MachinePrecision] / 2.0), $MachinePrecision] - M), $MachinePrecision]}, N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(l - N[Abs[N[(m - n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

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. Step-by-step derivation

   1. neg-sub0 N/A

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

   2. associate--l- N/A

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

   3. exp-diff N/A

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

   4. associate-*r/ N/A

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

   5. exp-0 N/A

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

   6. *-rgt-identity N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

3. Simplified 75.7%

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

5. Taylor expanded in K around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
6. Step-by-step derivation

   1. cos-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   2. cos-lowering-cos.f64 97.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

7. Simplified 97.1%

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

Alternative 2: 85.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\cos M}{e^{M \cdot M}}\\ \mathbf{if}\;M \leq -27:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;M \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{-0.001388888888888889 \cdot {M}^{6}}{e^{0.25 \cdot \left(m \cdot m\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (/ (cos M) (exp (* M M)))))
   (if (<= M -27.0)
     t_0
     (if (<= M 1.2e-12)
       (/ (* -0.001388888888888889 (pow M 6.0)) (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((M * M));
	double tmp;
	if (M <= -27.0) {
		tmp = t_0;
	} else if (M <= 1.2e-12) {
		tmp = (-0.001388888888888889 * pow(M, 6.0)) / 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 * m_1))
    if (m_1 <= (-27.0d0)) then
        tmp = t_0
    else if (m_1 <= 1.2d-12) then
        tmp = ((-0.001388888888888889d0) * (m_1 ** 6.0d0)) / 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((M * M));
	double tmp;
	if (M <= -27.0) {
		tmp = t_0;
	} else if (M <= 1.2e-12) {
		tmp = (-0.001388888888888889 * Math.pow(M, 6.0)) / 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((M * M))
	tmp = 0
	if M <= -27.0:
		tmp = t_0
	elif M <= 1.2e-12:
		tmp = (-0.001388888888888889 * math.pow(M, 6.0)) / 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 * M)))
	tmp = 0.0
	if (M <= -27.0)
		tmp = t_0;
	elseif (M <= 1.2e-12)
		tmp = Float64(Float64(-0.001388888888888889 * (M ^ 6.0)) / 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 * M));
	tmp = 0.0;
	if (M <= -27.0)
		tmp = t_0;
	elseif (M <= 1.2e-12)
		tmp = (-0.001388888888888889 * (M ^ 6.0)) / 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[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[M, -27.0], t$95$0, If[LessEqual[M, 1.2e-12], N[(N[(-0.001388888888888889 * N[Power[M, 6.0], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(0.25 * N[(m * m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if M < -27 or 1.19999999999999994e-12 < M

   1. Initial program 80.2%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 80.2%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in M around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left({M}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

      2. *-lowering-*.f64 93.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(M, M\right)\right)\right) \]

   10. Simplified 93.8%

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

   if -27 < M < 1.19999999999999994e-12

   1. Initial program 71.3%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 71.3%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 94.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 94.3%

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

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 62.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 62.2%

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

   11. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({M}^{2}\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\mathsf{*.f64}\left(m, m\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(M \cdot M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {M}^{2}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({M}^{2} \cdot \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({M}^{2}\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(M \cdot M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       15. *-lowering-*.f64 62.2%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   13. Simplified 62.2%

       \[\leadsto \frac{\color{blue}{1 + \left(M \cdot M\right) \cdot \left(\left(M \cdot M\right) \cdot \left(0.041666666666666664 + \left(M \cdot M\right) \cdot -0.001388888888888889\right) + -0.5\right)}}{e^{0.25 \cdot \left(m \cdot m\right)}} \]

   14. Taylor expanded in M around inf

       \[\leadsto \color{blue}{\frac{-1}{720} \cdot \frac{{M}^{6}}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   15. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{720} \cdot {M}^{6}}{\color{blue}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{720} \cdot {M}^{6}\right), \color{blue}{\left(e^{\frac{1}{4} \cdot {m}^{2}}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{720}, \left({M}^{6}\right)\right), \left(e^{\color{blue}{\frac{1}{4} \cdot {m}^{2}}}\right)\right) \]

       4. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{pow.f64}\left(M, 6\right)\right), \left(e^{\frac{1}{4} \cdot \color{blue}{{m}^{2}}}\right)\right) \]

       5. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{pow.f64}\left(M, 6\right)\right), \mathsf{exp.f64}\left(\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{pow.f64}\left(M, 6\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{pow.f64}\left(M, 6\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

       8. *-lowering-*.f64 81.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{720}, \mathsf{pow.f64}\left(M, 6\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   16. Simplified 81.3%

       \[\leadsto \color{blue}{\frac{-0.001388888888888889 \cdot {M}^{6}}{e^{0.25 \cdot \left(m \cdot m\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.88:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -6.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.88)
   (exp (* (* m m) -0.25))
   (if (<= m -6.6e-266)
     (/ (cos M) (exp (* M M)))
     (/ (cos M) (exp (* 0.25 (* n n)))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.88) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -6.6e-266) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = cos(M) / exp((0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-0.88d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-6.6d-266)) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = cos(m_1) / exp((0.25d0 * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.88) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -6.6e-266) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = Math.cos(M) / Math.exp((0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.88:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -6.6e-266:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = math.cos(M) / math.exp((0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.88)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -6.6e-266)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = Float64(cos(M) / exp(Float64(0.25 * Float64(n * n))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -0.88)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -6.6e-266)
		tmp = cos(M) / exp((M * M));
	else
		tmp = cos(M) / exp((0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.88], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -6.6e-266], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.880000000000000004

   1. Initial program 68.6%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 68.6%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 95.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 95.8%

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

   11. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({M}^{2}\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\mathsf{*.f64}\left(m, m\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(M \cdot M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {M}^{2}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({M}^{2} \cdot \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({M}^{2}\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(M \cdot M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       15. *-lowering-*.f64 55.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   13. Simplified 55.8%

       \[\leadsto \frac{\color{blue}{1 + \left(M \cdot M\right) \cdot \left(\left(M \cdot M\right) \cdot \left(0.041666666666666664 + \left(M \cdot M\right) \cdot -0.001388888888888889\right) + -0.5\right)}}{e^{0.25 \cdot \left(m \cdot m\right)}} \]

   14. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   15. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot {m}^{2}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{-1}{4} \cdot {m}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({m}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(m \cdot m\right)\right)\right) \]

       7. *-lowering-*.f64 95.8%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right) \]

   16. Simplified 95.8%

       \[\leadsto \color{blue}{e^{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -0.880000000000000004 < m < -6.6000000000000006e-266

   1. Initial program 72.1%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 72.1%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 91.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 91.0%

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

   8. Taylor expanded in M around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left({M}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

      2. *-lowering-*.f64 55.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(M, M\right)\right)\right) \]

   10. Simplified 55.5%

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

   if -6.6000000000000006e-266 < m

   1. Initial program 80.1%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 80.1%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 97.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 97.4%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {n}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({n}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(n \cdot n\right)\right)\right)\right) \]

      3. *-lowering-*.f64 50.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right)\right)\right) \]

   10. Simplified 50.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.88:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -6.6 \cdot 10^{-266}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos M}{e^{0.25 \cdot \left(n \cdot n\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.88:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -4 \cdot 10^{-266}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.88)
   (exp (* (* m m) -0.25))
   (if (<= m -4e-266) (/ (cos M) (exp (* M M))) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.88) {
		tmp = exp(((m * m) * -0.25));
	} else if (m <= -4e-266) {
		tmp = cos(M) / exp((M * M));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-0.88d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (m <= (-4d-266)) then
        tmp = cos(m_1) / exp((m_1 * m_1))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.88) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (m <= -4e-266) {
		tmp = Math.cos(M) / Math.exp((M * M));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.88:
		tmp = math.exp(((m * m) * -0.25))
	elif m <= -4e-266:
		tmp = math.cos(M) / math.exp((M * M))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.88)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (m <= -4e-266)
		tmp = Float64(cos(M) / exp(Float64(M * M)));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -0.88)
		tmp = exp(((m * m) * -0.25));
	elseif (m <= -4e-266)
		tmp = cos(M) / exp((M * M));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.88], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[m, -4e-266], N[(N[Cos[M], $MachinePrecision] / N[Exp[N[(M * M), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if m < -0.880000000000000004

   1. Initial program 68.6%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 68.6%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 95.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 95.8%

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

   11. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({M}^{2}\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\mathsf{*.f64}\left(m, m\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(M \cdot M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {M}^{2}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({M}^{2} \cdot \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({M}^{2}\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(M \cdot M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       15. *-lowering-*.f64 55.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   13. Simplified 55.8%

       \[\leadsto \frac{\color{blue}{1 + \left(M \cdot M\right) \cdot \left(\left(M \cdot M\right) \cdot \left(0.041666666666666664 + \left(M \cdot M\right) \cdot -0.001388888888888889\right) + -0.5\right)}}{e^{0.25 \cdot \left(m \cdot m\right)}} \]

   14. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   15. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot {m}^{2}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{-1}{4} \cdot {m}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({m}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(m \cdot m\right)\right)\right) \]

       7. *-lowering-*.f64 95.8%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right) \]

   16. Simplified 95.8%

       \[\leadsto \color{blue}{e^{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -0.880000000000000004 < m < -3.9999999999999999e-266

   1. Initial program 72.1%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 72.1%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 91.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 91.0%

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

   8. Taylor expanded in M around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left({M}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\left(M \cdot M\right)\right)\right) \]

      2. *-lowering-*.f64 55.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(M, M\right)\right)\right) \]

   10. Simplified 55.5%

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

   if -3.9999999999999999e-266 < m

   1. Initial program 80.1%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 80.1%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 97.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 97.4%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {n}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({n}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(n \cdot n\right)\right)\right)\right) \]

      3. *-lowering-*.f64 50.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right)\right)\right) \]

   10. Simplified 50.0%

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

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {n}^{2}}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {n}^{2}\right)} \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {n}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({n}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({n}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({n}^{2} \cdot \frac{-1}{4}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({n}^{2}\right), \frac{-1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(n \cdot n\right), \frac{-1}{4}\right)\right) \]

       8. *-lowering-*.f64 50.0%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, n\right), \frac{-1}{4}\right)\right) \]

   13. Simplified 50.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.88:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq -4 \cdot 10^{-266}:\\ \;\;\;\;\frac{\cos M}{e^{M \cdot M}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq 10^{-281}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= n 1e-281)
   (exp (* (* m m) -0.25))
   (if (<= n 2.5e-5) (/ (cos M) (exp l)) (exp (* -0.25 (* n n))))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-281) {
		tmp = exp(((m * m) * -0.25));
	} else if (n <= 2.5e-5) {
		tmp = cos(M) / exp(l);
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (n <= 1d-281) then
        tmp = exp(((m * m) * (-0.25d0)))
    else if (n <= 2.5d-5) then
        tmp = cos(m_1) / exp(l)
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (n <= 1e-281) {
		tmp = Math.exp(((m * m) * -0.25));
	} else if (n <= 2.5e-5) {
		tmp = Math.cos(M) / Math.exp(l);
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if n <= 1e-281:
		tmp = math.exp(((m * m) * -0.25))
	elif n <= 2.5e-5:
		tmp = math.cos(M) / math.exp(l)
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (n <= 1e-281)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	elseif (n <= 2.5e-5)
		tmp = Float64(cos(M) / exp(l));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (n <= 1e-281)
		tmp = exp(((m * m) * -0.25));
	elseif (n <= 2.5e-5)
		tmp = cos(M) / exp(l);
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[n, 1e-281], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[n, 2.5e-5], N[(N[Cos[M], $MachinePrecision] / N[Exp[l], $MachinePrecision]), $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < 1e-281

   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. Step-by-step derivation

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 73.9%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 98.5%

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

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 60.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 60.5%

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

   11. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({M}^{2}\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\mathsf{*.f64}\left(m, m\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(M \cdot M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {M}^{2}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({M}^{2} \cdot \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({M}^{2}\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(M \cdot M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       15. *-lowering-*.f64 39.4%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   13. Simplified 39.4%

       \[\leadsto \frac{\color{blue}{1 + \left(M \cdot M\right) \cdot \left(\left(M \cdot M\right) \cdot \left(0.041666666666666664 + \left(M \cdot M\right) \cdot -0.001388888888888889\right) + -0.5\right)}}{e^{0.25 \cdot \left(m \cdot m\right)}} \]

   14. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   15. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot {m}^{2}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{-1}{4} \cdot {m}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({m}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(m \cdot m\right)\right)\right) \]

       7. *-lowering-*.f64 60.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right) \]

   16. Simplified 60.5%

       \[\leadsto \color{blue}{e^{-0.25 \cdot \left(m \cdot m\right)}} \]

   if 1e-281 < n < 2.50000000000000012e-5

   1. Initial program 84.7%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 84.7%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 92.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 92.7%

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

   8. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 47.2%

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

   if 2.50000000000000012e-5 < n

   1. Initial program 70.6%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 70.6%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 98.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 98.5%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {n}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({n}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(n \cdot n\right)\right)\right)\right) \]

      3. *-lowering-*.f64 92.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right)\right)\right) \]

   10. Simplified 92.4%

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

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {n}^{2}}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {n}^{2}\right)} \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {n}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({n}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({n}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({n}^{2} \cdot \frac{-1}{4}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({n}^{2}\right), \frac{-1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(n \cdot n\right), \frac{-1}{4}\right)\right) \]

       8. *-lowering-*.f64 92.4%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, n\right), \frac{-1}{4}\right)\right) \]

   13. Simplified 92.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq 10^{-281}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\cos M}{e^{\ell}}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{if}\;m \leq -0.88:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;m \leq 54:\\ \;\;\;\;\frac{1}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (let* ((t_0 (exp (* (* m m) -0.25))))
   (if (<= m -0.88)
     t_0
     (if (<= m 54.0)
       (/ 1.0 (+ 1.0 (* l (+ 1.0 (* l (+ 0.5 (* l 0.16666666666666666)))))))
       t_0))))

C

double code(double K, double m, double n, double M, double l) {
	double t_0 = exp(((m * m) * -0.25));
	double tmp;
	if (m <= -0.88) {
		tmp = t_0;
	} else if (m <= 54.0) {
		tmp = 1.0 / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	} 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 = exp(((m * m) * (-0.25d0)))
    if (m <= (-0.88d0)) then
        tmp = t_0
    else if (m <= 54.0d0) then
        tmp = 1.0d0 / (1.0d0 + (l * (1.0d0 + (l * (0.5d0 + (l * 0.16666666666666666d0))))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double t_0 = Math.exp(((m * m) * -0.25));
	double tmp;
	if (m <= -0.88) {
		tmp = t_0;
	} else if (m <= 54.0) {
		tmp = 1.0 / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	t_0 = math.exp(((m * m) * -0.25))
	tmp = 0
	if m <= -0.88:
		tmp = t_0
	elif m <= 54.0:
		tmp = 1.0 / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))))
	else:
		tmp = t_0
	return tmp

Julia

function code(K, m, n, M, l)
	t_0 = exp(Float64(Float64(m * m) * -0.25))
	tmp = 0.0
	if (m <= -0.88)
		tmp = t_0;
	elseif (m <= 54.0)
		tmp = Float64(1.0 / Float64(1.0 + Float64(l * Float64(1.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666)))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	t_0 = exp(((m * m) * -0.25));
	tmp = 0.0;
	if (m <= -0.88)
		tmp = t_0;
	elseif (m <= 54.0)
		tmp = 1.0 / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := Block[{t$95$0 = N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[m, -0.88], t$95$0, If[LessEqual[m, 54.0], N[(1.0 / N[(1.0 + N[(l * N[(1.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if m < -0.880000000000000004 or 54 < m

   1. Initial program 72.3%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 72.3%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 97.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 97.9%

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

   11. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({M}^{2}\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\mathsf{*.f64}\left(m, m\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(M \cdot M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {M}^{2}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({M}^{2} \cdot \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({M}^{2}\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(M \cdot M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       15. *-lowering-*.f64 58.9%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   13. Simplified 58.9%

       \[\leadsto \frac{\color{blue}{1 + \left(M \cdot M\right) \cdot \left(\left(M \cdot M\right) \cdot \left(0.041666666666666664 + \left(M \cdot M\right) \cdot -0.001388888888888889\right) + -0.5\right)}}{e^{0.25 \cdot \left(m \cdot m\right)}} \]

   14. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   15. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot {m}^{2}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{-1}{4} \cdot {m}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({m}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(m \cdot m\right)\right)\right) \]

       7. *-lowering-*.f64 97.9%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right) \]

   16. Simplified 97.9%

       \[\leadsto \color{blue}{e^{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -0.880000000000000004 < m < 54

   1. Initial program 79.8%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 79.8%

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

   5. Taylor expanded in l around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 43.3%

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

   8. Taylor expanded in l around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \color{blue}{\left(1 + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 33.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 33.5%

       \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\color{blue}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}} \]

   11. Taylor expanded in K around 0

       \[\leadsto \color{blue}{\frac{\cos \left(\mathsf{neg}\left(M\right)\right)}{1 + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}} \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\cos \left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(1 + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right) \]

       2. cos-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\cos M, \left(\color{blue}{1} + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)\right) \]

       3. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \left(\color{blue}{1} + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)\right) \]

       4. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right) \]

       8. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

       10. *-lowering-*.f64 35.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   13. Simplified 35.6%

       \[\leadsto \color{blue}{\frac{\cos M}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}} \]

   14. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]
   15. Step-by-step derivation

   16. Simplified 35.6%

       \[\leadsto \frac{\color{blue}{1}}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.88:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{elif}\;m \leq 54:\\ \;\;\;\;\frac{1}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;m \leq -0.095:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (if (<= m -0.095) (exp (* (* m m) -0.25)) (exp (* -0.25 (* n n)))))

C

double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.095) {
		tmp = exp(((m * m) * -0.25));
	} else {
		tmp = exp((-0.25 * (n * n)));
	}
	return tmp;
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    real(8) :: tmp
    if (m <= (-0.095d0)) then
        tmp = exp(((m * m) * (-0.25d0)))
    else
        tmp = exp(((-0.25d0) * (n * n)))
    end if
    code = tmp
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	double tmp;
	if (m <= -0.095) {
		tmp = Math.exp(((m * m) * -0.25));
	} else {
		tmp = Math.exp((-0.25 * (n * n)));
	}
	return tmp;
}

Python

def code(K, m, n, M, l):
	tmp = 0
	if m <= -0.095:
		tmp = math.exp(((m * m) * -0.25))
	else:
		tmp = math.exp((-0.25 * (n * n)))
	return tmp

Julia

function code(K, m, n, M, l)
	tmp = 0.0
	if (m <= -0.095)
		tmp = exp(Float64(Float64(m * m) * -0.25));
	else
		tmp = exp(Float64(-0.25 * Float64(n * n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(K, m, n, M, l)
	tmp = 0.0;
	if (m <= -0.095)
		tmp = exp(((m * m) * -0.25));
	else
		tmp = exp((-0.25 * (n * n)));
	end
	tmp_2 = tmp;
end

Wolfram

code[K_, m_, n_, M_, l_] := If[LessEqual[m, -0.095], N[Exp[N[(N[(m * m), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision], N[Exp[N[(-0.25 * N[(n * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if m < -0.095000000000000001

   1. Initial program 69.0%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 69.0%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 100.0%

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

   8. Taylor expanded in m around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

      3. *-lowering-*.f64 94.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   10. Simplified 94.5%

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

   11. Taylor expanded in M around 0

       \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(1 + {M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \left({M}^{2} \cdot \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({M}^{2}\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\mathsf{*.f64}\left(m, m\right)}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(M \cdot M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) - \frac{1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(\color{blue}{m}, m\right)\right)\right)\right) \]

       5. sub-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right) + \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\left({M}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, \color{blue}{m}\right)\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left({M}^{2}\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       9. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(M \cdot M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \left(\frac{1}{24} + \frac{-1}{720} \cdot {M}^{2}\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       11. +-lowering-+.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{-1}{720} \cdot {M}^{2}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       12. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \left({M}^{2} \cdot \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({M}^{2}\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       14. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(M \cdot M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

       15. *-lowering-*.f64 55.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(M, M\right), \frac{-1}{720}\right)\right)\right), \frac{-1}{2}\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

   13. Simplified 55.0%

       \[\leadsto \frac{\color{blue}{1 + \left(M \cdot M\right) \cdot \left(\left(M \cdot M\right) \cdot \left(0.041666666666666664 + \left(M \cdot M\right) \cdot -0.001388888888888889\right) + -0.5\right)}}{e^{0.25 \cdot \left(m \cdot m\right)}} \]

   14. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {m}^{2}}}} \]
   15. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)} \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {m}^{2}\right)\right)\right) \]

       3. distribute-lft-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot {m}^{2}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\frac{-1}{4} \cdot {m}^{2}\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left({m}^{2}\right)\right)\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \left(m \cdot m\right)\right)\right) \]

       7. *-lowering-*.f64 94.5%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{-1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right) \]

   16. Simplified 94.5%

       \[\leadsto \color{blue}{e^{-0.25 \cdot \left(m \cdot m\right)}} \]

   if -0.095000000000000001 < m

   1. Initial program 78.2%

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

      1. neg-sub0 N/A

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

      2. associate--l- N/A

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

      3. exp-diff N/A

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

      4. associate-*r/ N/A

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

      5. exp-0 N/A

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

      6. *-rgt-identity N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

   3. Simplified 78.2%

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

   5. Taylor expanded in K around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(M\right)\right)}, \mathsf{exp.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\cos M, \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

      2. cos-lowering-cos.f64 96.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right), \mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right), M\right)\right), \mathsf{\_.f64}\left(\ell, \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(m, n\right)\right)\right)\right)}\right)\right) \]

   7. Simplified 96.0%

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

   8. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {n}^{2}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({n}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(n \cdot n\right)\right)\right)\right) \]

      3. *-lowering-*.f64 50.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(n, n\right)\right)\right)\right) \]

   10. Simplified 50.9%

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

   11. Taylor expanded in M around 0

       \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{4} \cdot {n}^{2}}}} \]
   12. Step-by-step derivation

       1. rec-exp N/A

          \[\leadsto e^{\mathsf{neg}\left(\frac{1}{4} \cdot {n}^{2}\right)} \]

       2. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left(\frac{1}{4} \cdot {n}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{exp.f64}\left(\left(\mathsf{neg}\left({n}^{2} \cdot \frac{1}{4}\right)\right)\right) \]

       4. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({n}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{exp.f64}\left(\left({n}^{2} \cdot \frac{-1}{4}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left({n}^{2}\right), \frac{-1}{4}\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\left(n \cdot n\right), \frac{-1}{4}\right)\right) \]

       8. *-lowering-*.f64 50.9%

          \[\leadsto \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(n, n\right), \frac{-1}{4}\right)\right) \]

   13. Simplified 50.9%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;m \leq -0.095:\\ \;\;\;\;e^{\left(m \cdot m\right) \cdot -0.25}\\ \mathbf{else}:\\ \;\;\;\;e^{-0.25 \cdot \left(n \cdot n\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.0% accurate, 28.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)} \end{array} \]

FPCore

(FPCore (K m n M l)
 :precision binary64
 (/ 1.0 (+ 1.0 (* l (+ 1.0 (* l (+ 0.5 (* l 0.16666666666666666))))))))

C

double code(double K, double m, double n, double M, double l) {
	return 1.0 / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
}

Fortran

real(8) function code(k, m, n, m_1, l)
    real(8), intent (in) :: k
    real(8), intent (in) :: m
    real(8), intent (in) :: n
    real(8), intent (in) :: m_1
    real(8), intent (in) :: l
    code = 1.0d0 / (1.0d0 + (l * (1.0d0 + (l * (0.5d0 + (l * 0.16666666666666666d0))))))
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0 / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
}

Python

def code(K, m, n, M, l):
	return 1.0 / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))))

Julia

function code(K, m, n, M, l)
	return Float64(1.0 / Float64(1.0 + Float64(l * Float64(1.0 + Float64(l * Float64(0.5 + Float64(l * 0.16666666666666666)))))))
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0 / (1.0 + (l * (1.0 + (l * (0.5 + (l * 0.16666666666666666))))));
end

Wolfram

code[K_, m_, n_, M_, l_] := N[(1.0 / N[(1.0 + N[(l * N[(1.0 + N[(l * N[(0.5 + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. neg-sub0 N/A

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

   2. associate--l- N/A

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

   3. exp-diff N/A

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

   4. associate-*r/ N/A

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

   5. exp-0 N/A

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

   6. *-rgt-identity N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

3. Simplified 75.7%

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

5. Taylor expanded in l around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{exp.f64}\left(\color{blue}{\ell}\right)\right) \]
6. Step-by-step derivation

7. Simplified 29.3%

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

8. Taylor expanded in l around 0

   \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \color{blue}{\left(1 + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right) \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 27.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 27.8%

    \[\leadsto \frac{\cos \left(K \cdot \frac{m + n}{2} - M\right)}{\color{blue}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}} \]

11. Taylor expanded in K around 0

    \[\leadsto \color{blue}{\frac{\cos \left(\mathsf{neg}\left(M\right)\right)}{1 + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}} \]
12. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\cos \left(\mathsf{neg}\left(M\right)\right), \color{blue}{\left(1 + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right) \]

    2. cos-neg N/A

       \[\leadsto \mathsf{/.f64}\left(\cos M, \left(\color{blue}{1} + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)\right) \]

    3. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \left(\color{blue}{1} + \ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)\right) \]

    4. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)\right)}\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(1 + \ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right) \]

    6. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \color{blue}{\left(\ell \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)\right)}\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \color{blue}{\left(\frac{1}{2} + \frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right) \]

    8. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{1}{6} \cdot \ell\right)}\right)\right)\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \left(\ell \cdot \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

    10. *-lowering-*.f64 33.1%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(M\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \color{blue}{\frac{1}{6}}\right)\right)\right)\right)\right)\right)\right) \]

13. Simplified 33.1%

    \[\leadsto \color{blue}{\frac{\cos M}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)}} \]

14. Taylor expanded in M around 0

    \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\ell, \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\ell, \frac{1}{6}\right)\right)\right)\right)\right)\right)\right) \]
15. Step-by-step derivation

16. Simplified 33.1%

    \[\leadsto \frac{\color{blue}{1}}{1 + \ell \cdot \left(1 + \ell \cdot \left(0.5 + \ell \cdot 0.16666666666666666\right)\right)} \]
17. Add Preprocessing

Alternative 9: 6.8% accurate, 425.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (K m n M l) :precision binary64 1.0)

C

double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

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 = 1.0d0
end function

Java

public static double code(double K, double m, double n, double M, double l) {
	return 1.0;
}

Python

def code(K, m, n, M, l):
	return 1.0

Julia

function code(K, m, n, M, l)
	return 1.0
end

MATLAB

function tmp = code(K, m, n, M, l)
	tmp = 1.0;
end

Wolfram

code[K_, m_, n_, M_, l_] := 1.0

Derivation

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. Step-by-step derivation

   1. neg-sub0 N/A

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

   2. associate--l- N/A

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

   3. exp-diff N/A

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

   4. associate-*r/ N/A

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

   5. exp-0 N/A

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

   6. *-rgt-identity N/A

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

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos \left(\frac{K \cdot \left(m + n\right)}{2} - M\right), \color{blue}{\left(e^{{\left(\frac{m + n}{2} - M\right)}^{2} + \left(\ell - \left|m - n\right|\right)}\right)}\right) \]

3. Simplified 75.7%

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

5. Taylor expanded in m around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{exp.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot {m}^{2}\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left({m}^{2}\right)\right)\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(m \cdot m\right)\right)\right)\right) \]

   3. *-lowering-*.f64 42.7%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(K, \mathsf{/.f64}\left(\mathsf{+.f64}\left(m, n\right), 2\right)\right), M\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{*.f64}\left(m, m\right)\right)\right)\right) \]

7. Simplified 42.7%

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

8. Taylor expanded in m around 0

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

   1. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \left(K \cdot n\right) - M\right)\right) \]

   2. --lowering--.f64 N/A

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\left(\frac{1}{2} \cdot \left(K \cdot n\right)\right), M\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(K \cdot n\right)\right), M\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(n \cdot K\right)\right), M\right)\right) \]

   5. *-lowering-*.f64 5.0%

      \[\leadsto \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(n, K\right)\right), M\right)\right) \]

10. Simplified 5.0%

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

11. Taylor expanded in n around 0

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

    1. cos-neg N/A

       \[\leadsto \cos M \]

    2. cos-lowering-cos.f64 5.5%

       \[\leadsto \mathsf{cos.f64}\left(M\right) \]

13. Simplified 5.5%

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

14. Taylor expanded in M around 0

    \[\leadsto \color{blue}{1} \]
15. Step-by-step derivation

16. Simplified 5.5%

    \[\leadsto \color{blue}{1} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024288 
(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)))))))